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+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
+
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+17
+

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+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
+[email protected]
+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.
+

2NFAT4oBgHgl3EQfkh06/content/tmp_files/load_file.txt

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+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'}
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+page_content='  Molecular and nanoscale computing and  technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'}
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+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'}
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+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. Commun. 3, 719 (2012).
+[6] Chabaud, M. et al. Cell migration and antigen capture
+are antagonistic processes coupled by myosin II in den-
+dritic cells. Nature Commun. 6, 7526 (2015).
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+pel, Periodic migration in a physical model of cells on
+micropatterns, Phys. Rev. Lett., 111, 158102 (2013).
+
+6
+[8] P. Sens, Stick-slip model for actin-driven cell protrusions,
+cell protrusions, cell polarization, and crawling, Proc.
+Natl. Acad. Sci. 117, 24670 (2020).
+[9] J.E. Ron, P. Monzo, M.C. Gauthier, R. Voituriez, and
+N.S. Gov, One-dimensional cell motility patterns, Phys.
+Rev. Res., 2, 033237 (2020).
+[10] F. J¨ulicher, K. Kruse, J. Prost, and J.-F. Joanny, Active
+behavior of the Cytoskeleton, Phys. Rep. 449, 3 (2007).
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+driven cell motility, Phys. Rev. Lett. 111, 108102 (2013).
+[12] K. Tawada, K. Sekimoto, Protein friction exerted by
+motor enzymes through a weak-binding interaction’, J.
+Theo. Biol., 150, 193 (1991).
+[13] P. Recho and L. Truskinovsky, “Cell locomotion in one
+dimension,” in Physical Models of Cell Motility, p. 135,
+Springer, 2016.
+[14] T. Putelat, P. Recho, and L. Truskinovsky, Mechanical
+stress as a regulator of cell motility,’ Phys. Rev. E, 97,
+012410 (2018).
+[15] R. Meili and R.A. Firtel, Two poles and a compass, Cell,
+114, 153 (2003).
+[16] Y. T. Maeda, J. Inose, M. Y. Matsuo, S. Iwaya, and
+M. Sano, Ordered patterns of cell shape and orientational
+correlation during spontaneous cell migration, PLoS one,
+3, e3734 (2008).
+[17] T. Y.-C. Tsai, S.R. Collins, C.K. Chan, A. Hadjitheoro-
+rou, P.-Y. Lam, S.S. Lou, H.W. Yang, J. Jorgensen, F.
+Ellett, D. Irimia, M.W. Davidson, R.S. Fischer, A. Hut-
+tenlocher, T. Meyer, J.E. Ferrell Jr, and J.A. Theriot,
+Efficient front-rear coupling in neutrophil chemotaxis by
+dynamic myosin II localization, Dev. Cell 49, 189 (2019).
+[18] M. Nickaeen, I. L. Novak, S. Pulford, A. Rumack,
+J. Brandon, B. M. Slepchenko, and A. Mogilner, A free-
+boundary model of a motile cell explains turning behav-
+ior, PLoS Comp. Biol, 13, e1005862 (2017).
+[19] G. Horton and S. Vandewalle, A space-time multigrid
+method for parabolic partial differential equations, SIAM
+Journal on Scientific Computing, 16, 848 (1995).
+[20] See our Supplementary Material.
+[21] T.P. Lele, J. Pendse, S. Kumar, M. Salanga, J. Karavitis,
+and D.E. Ingber, Mechanical forces alter zyxin unbind-
+ing kinetics within focal adhesions of living cells, J. Cell
+Physiol., 207, 187 (2006).
+[22] J.-Y. Lo and H.-Y. Chen, manuscript in preparation.
+[23] H. Tanimoto and M. Sano, A simple force-motion rela-
+tion for migrating cells revealed by multipole analysis of
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+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.
+

6tFJT4oBgHgl3EQflyzE/content/tmp_files/2301.11585v1.pdf.txt

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+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.O. 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 
+ 
+[email protected] 
+[email protected] 
+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 
+ 
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+

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@@ -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 ([email protected])
+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]. We
+determined an optimal β as the empirical best.
+Table B3 showed SQI performance
+variation about β in the dataset where the scale and size normalization using linear
+interpolation were applied.
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+β “ 2
+β “ 3
+β “ 1
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+0.9092
+0.9066
+0.8951
+0.9221
+0.9208
+0.9298
+0.9292
+0.9195
+0.9189
+PPV
+0.9694
+0.9680
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+0.9397
+0.9389
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+0.8095
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+0.9439
+0.9513
+0.9426
+0.9489
+0.9531
+0.9603
+0.9528
+0.9528
+0.9471
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8NFQT4oBgHgl3EQfHzW5/content/tmp_files/2301.13250v1.pdf.txt

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+.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
+[email protected]
+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)
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+Wilson, C. D., Elmegreen, B. G., Bemis, A., & Brunetti, N.
+2019, The Astrophysical Journal, 882, 5,
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+Wilson, C. D., Scoville, N., Madden, S. C., &
+Charmandaris, V. 2000, The Astrophysical Journal, 542,
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+Observatory, 10
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+Astrophysical Journal, 588, 243, doi: 10.1086/368353
+
+20
+He et al.
+APPENDIX
+A. 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. For the simulated control
+galaxies, we pick the inclination angle of 30 degrees in
+Fig.
+1.
+By increasing inclination angle we might see
+significant increase in Σmol. Fig. A1 shows the data for
+the G3 galaxy viewed with inclination angles of 30, 60
+and 80 degrees. We see little increase in Σmol even for
+an inclination of 80 degrees. This is consistent with our
+expectation that individual clouds are resolved in the
+simulated data. 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
+

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+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,
+[email protected]
+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.
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+on Knowledge and Data Engineering, 29(12):2724–
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+Percy Liang, and Jure Leskovec. 2021. QA-GNN:
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+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
+

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+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
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+[27] Tiwari, SK, Identities with generalized derivations in prime rings, Rendiconti del Circolo
+Matematico di Palermo Series 2, 1–17, 2021.
+[28] Jacobson, Nathan, American Mathematical Society, 37, 1956. 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: [email protected]
+B. Prajapati, School of Liberal Studies, Ambedkar University Delhi, Delhi-110006,
+INDIA.
+Email address: [email protected]
+

DdFRT4oBgHgl3EQfxTg4/content/tmp_files/load_file.txt

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+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'}
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+page_content='  [31] 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'}
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+page_content=' Differential identities with automorphisms and antiautomorphisms, ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'}
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+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'}

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@@ -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
+[email protected]
+Sunayana Sitaram
+Microsoft Research India, Bangalore, India
+[email protected]
+Barbara E. Bullock
+UT at Austin, Austin, USA
+[email protected]
+Almeida Jacqueline Toribio
+UT at Austin, Austin, USA
+[email protected]
+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.
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+

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+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
+† [email protected]
+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.
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+
+SUPPLEMENTARY MATERIALS
+I.
+UNPINNING FOR E > Eth
+FIG. 1. Unpinning at E > Eth (sin−1( Eth
+E ) = 48.950): Spiral waves with different φ0 are unpinned
+in a CPEF with both under-drive (p < 1) and over-drive pacing (p > 1). 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
+

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@@ -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.
+

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+Targeted 𝑘-node Collapse Problem: Towards Understanding the
+Robustness of Local 𝑘-core Structure
+Yuqian Lv
+Zhejiang University of Technology
+[email protected]
+Bo Zhou
+Zhejiang University of Technology
+[email protected]
+Jinhuan Wang
+Zhejiang University of Technology
+[email protected]
+Qi Xuan
+Zhejiang University of Technology
+[email protected]
+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 [email protected]. 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 node�� ∈ 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
+
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+13
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+Improved knowledge distillation by utilizing
+backward pass knowledge in neural networks
+Aref Jafari∗
+University of Waterloo
+[email protected]
+Mehdi Rezagholizadeh
+Huawei Noah’s Ark Lab
+[email protected]
+Ali Ghodsi
+University of Waterloo
+[email protected]
+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
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+2018), arXiv:1803.00443.
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+benchmark and analysis platform for natural language understanding. arXiv preprint arXiv:1804.07461
+(2018).
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+arXiv preprint arXiv:1909.12579 (2019).
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+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
+

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+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<j
+e2
+|ri − rj|,
+(1)
+where i and j enumerate valence electrons and Nv is the
+total number of valence electrons, e is electron charge,
+and r is the distance. ˆHHF
+i
+is the single-electron Hartree-
+Fock (HF) Hamiltonian, which has the form
+ˆHHF
+i
+= cαi · ˆpi + (β − 1)mc2 + Vnuc(ri) + V N−1(ri). (2)
+Here c is the speed of light, αi and β are the Dirac
+matrixes, ˆpi is the electron momentum, m is the elec-
+tron mass, Vnuc(i) is the nuclear potential obtained by
+integrating Fermi distribution of nuclear charge density,
+and V N−1(i) is the self-consistent HF potential obtained
+for the configuration with one 7s (or 6s) electron re-
+moved from the ground state configuration of the con-
+sidered atom. This corresponds to the V N−1 approxima-
+tion [19, 20] which is convenient for generating a single-
+electron basis. Single-electron basis states are calculated
+in the frozen V N−1 potential, so that they correspond
+to the atom with one electron excited from the ground
+
+2
+state. External electron wave functions are expressed in
+terms of coefficients of expansion over single-determinant
+basis state functions
+Ψ(r1, . . . , rM) =
+(3)
+N1
+�
+i=1
+XiΦi(r1, . . . , rM) +
+N2
+�
+j=1
+YjΦj(r1, . . . , rM).
+Here M is the number of valence electrons, N1 is the
+number of low-energy basis states, N2 is the number of
+high-energy basis states.
+Then the CI matrix equation can be written in a block
+form
+�
+A B
+C D
+� �
+X
+Y
+�
+= Ea
+�
+X
+Y
+�
+.
+(4)
+Here block A corresponds to low-energy states, block D
+corresponds to high-energy states, and blocks B and C
+correspond to cross terms. Note that since the total CI
+matrix is symmetric, we have C = B′, i.e., cij = bji.
+Vectors X and Y contain the coefficients of expansion
+of the valence wave function over the single-determinant
+many-electron basis functions (see Eq. 3).
+Finding Y from the second equation of (4) leads to
+Y = (EaI − D)−1CX.
+(5)
+Substituting Y to the first equation of (4) leads to
+�
+A + B(EaI − D)−1C
+�
+X = EaX,
+(6)
+where I is the unit matrix. Then, following Ref. [9] we
+neglect off-diagonal matrix elements in block D.
+This
+leads to a very simple structure of the (EaI − D)−1
+matrix, (EaI − D)−1
+ik
+= δik/(Ea − Ek), where Ek =
+⟨k|HCI|k⟩.Matrix elements of the effective CI matrix (6)
+have the form
+⟨i| ˆHeff|j⟩ = ⟨i| ˆHCI|j⟩ +
+�
+k
+⟨i| ˆHCI|k⟩⟨k| ˆHCI|j⟩
+Ea − Ek
+.
+(7)
+We see that the standard CI matrix elements between
+low-energy states are corrected by an expression which
+is very similar to the second-order perturbation theory
+correction to the energy. This justifies the name of the
+method.
+To calculate this second-order correction we
+need to know the energy of the state Ea which must come
+as the result of the solution of the equation, i.e. it is not
+known in advance. Therefore, iterations are needed. We
+start from any reasonable guess for the energy. For exam-
+ple, it may come from the solution of the equation with
+neglected second-order correction. Note that the energy
+independent numerators of the second-order correction
+can be calculated only once, on the first iteration, kept
+on disk and reused on every consequent iteration. This
+means that only the first iteration takes some time while
+all other iterations are very fast. As a rule, less than ten
+iterations are needed for full convergence. As a result, we
+have an energy of the state Ea and expansion coefficients
+X and Y .
+B.
+Basis states
+To solve the CI equations we need many-electron basis
+states which are constructed from single-electron states.
+For single-electron basis states we use the B-spline tech-
+nique [21, 22]. These states are defined as linear combina-
+tions of B-splines that are eigenstates of the HF Hamilto-
+nian (2). Forty B-splines of the order nine are calculated
+within a box of radius Rmax = 40aB (where aB repre-
+sents Bohr’s radius) and an orbital angular momentum
+of 0 ≤ l ≤ 4. 14 states above the core in each partial
+wave are used. It has been found that by selecting the
+values of lmax, Rmax, and the number of B-splines, a basis
+is adequately saturated for low-lying states. The many-
+electron states are found by making all possible single and
+double electron excitations from a few reference config-
+urations. One, two or three configurations, correspond-
+ing to the low-lying states of an atom are considered as
+reference configurations. One configuration of the same
+parity is considered at a time. For each configuration,
+all possible values of the projection of the total angular
+momentum j of the single-electron states are considered
+and many-electron states with fixed values of total many-
+electron angular momentum J and its projection M are
+constructed. Usually, we take M = J.
+C.
+Calculation of hyperfine structure
+In this section, we mostly follow our previous work on
+hafnium and rutherfordium [23]. To calculate HFS, we
+use the time-dependent Hartree-Fock (TDHF) method,
+which is equivalent to the well-known random-phase ap-
+proximation (RPA). The RPA equations are the follow-
+ing:
+�
+ˆHRHF − ǫc
+�
+δψc = −
+�
+ˆf + δV f
+core
+�
+ψc
+(8)
+where ˆf is an operator of an external field (nuclear mag-
+netic dipole or electric quadrupole fields). Index c in (8)
+numerates states in the core, ψc is a single-electron wave
+function of the state c in the core, δψc is the correction
+to this wave function caused by an external field, and
+δV f
+core is the correction to the self-consistent RHF po-
+tential caused by changing of all core states.
+Eq. (8)
+are solved self-consistently for all states in the core. As
+a result, an effective operator of the interaction of va-
+lence electrons with an external field is constructed as
+ˆf + δV f
+core. The energy shift of a many-electron state a is
+given by
+δǫa = ⟨a|
+M
+�
+i=1
+�
+ˆf + δV f
+core
+�
+i |a⟩.
+(9)
+Here M is the number of valence electrons.
+When the wave function for the valence electrons
+
+3
+comes as a solution of Eq. (6), Eq. (9) is reduced to
+δǫa =
+�
+ij
+xixj⟨Φi| ˆHhfs|Φj⟩,
+(10)
+where ˆHhfs = �M
+i=1( ˆf + δV f
+core)i. For better accuracy of
+the results, the full expansion (3) might be used. Then it
+is convenient to introduce a new vector Z, which contains
+both X and Y , Z ≡ {X, Y }.
+Note that the solution
+of (6) is normalized by the condition �
+i x2
+i = 1. The
+normalization condition for the total wave function (3)
+is different, �
+i x2
+i + �
+j y2
+j ≡ �
+i z2
+i = 1.
+Therefore,
+when X is found from (6), and Y is found from (5), both
+vectors should be renormalized. Then the HFS matrix
+element is given by the expression, which is similar to
+(10) but has much more terms
+δǫa =
+�
+ij
+zizj⟨Φi| ˆHhfs|Φj⟩.
+(11)
+Energy shift (9) is used to calculate HFS constants A
+and B using textbook formulas
+Aa =
+gIδǫ(A)
+a
+�
+Ja(Ja + 1)(2Ja + 1)
+,
+(12)
+and
+Ba = −2Qδǫ(B)
+a
+�
+Ja(2Ja − 1)
+(2Ja + 3)(2Ja + 1)(Ja + 1).
+(13)
+Here δǫ(A)
+a
+is the energy shift (9) caused by the interaction
+of atomic electrons with the nuclear magnetic moment
+µ, gI = µ/I, I is nuclear spin; δǫ(B)
+a
+is the energy shift
+(9) caused by the interaction of atomic electrons with
+the nuclear electric quadrupole moment Q (Q in (13) is
+measured in barns).
+III.
+ENERGY LEVELS AND HFS OF
+DYSPROSIUM AND HOLMIUM
+For the purpose of testing the accuracy of the method,
+we start calculating the energy levels for some low-lying
+states of Dy and Ho. The results are shown in Table I.
+As can be seen, our results are consistent with the exper-
+imental results compiled in Ref. [16] of respective atomic
+systems. The difference between theoretical calculations
+and measurements is within a few hundred cm−1. Cal-
+culated and experimental Land´e g-factors are also pre-
+sented. A comparison of Land´e g-factors calculated with
+non-relativistic expressions is helpful for identifying state
+labels.
+gNR = 1 + J(J + 1) − L(L + 1) + S(S + 1)
+2J(J + 1)
+.
+(14)
+Total orbital momentum L, and total spin S in (14)
+cannot come from relativistic calculations. Instead, we
+TABLE I. Excitation energies (E, cm−1), and g-factors for
+some low states of Dy, and Ho atoms.
+This work
+NIST [16]
+Conf.
+Term
+J
+E
+g
+E
+g
+Dy
+4f 106s2
+5I
+8
+0.000
+1.242
+0.000
+1.2416
+4f 106s2
+7
+3933
+1.175
+4134.2
+1.1735
+4f 106s2
+6
+7179
+1.073
+7050.6
+1.0716
+4f 95d6s2
+7Ho
+8
+7818
+1.347
+7565.610
+1.35246
+4f 95d6s2
+7
+9474
+1.353
+8519.210
+1.336
+4f 106s2
+5I
+5
+9589
+0.909
+9211.6
+0.911
+4f 95d6s2
+7Io
+9
+10048
+1.316
+9990.974
+1.32
+4f 95d6s2
+7Ho
+6
+11052
+1.417
+10088.802
+1.36
+4f 106s2
+5I
+4
+11299
+0.613
+10925.3
+0.618
+Ho
+4f 116s2
+4Io
+15/2
+0.00
+1.196
+0.00
+1.1951
+4f 116s2
+13/2
+5205
+1.107
+5419.7
+−
+4f 105d6s2
+(8, 3
+2)
+17/2
+8344
+1.262
+8378.91
+−
+4f 105d6s2
+15/2
+8385
+1.280
+8427.11
+−
+4f 116s2
+4Io
+11/2
+8501
+0.979
+8605.2
+1.012
+4f 105d6s2
+(8, 3
+2)
+13/2
+8989
+1.336
+9147.08
+−
+4f 105d6s2
+19/2
+8952
+1.231
+9741.50
+−
+4f 116s2
+4Io
+9/2
+10550
+0.780
+10695.8
+0.866
+choose their values from the condition that formula (14)
+gives values very close to the calculated g-factors. This
+allows us to link the state to the non-relativistic nota-
+tion 2S+1LJ.
+Here, J is the total angular momentum
+(J = L + S). A good agreement is also observed be-
+tween current calculations and experimental g-factors of
+Dy and Ho whenever experimental data are available. In
+order to identify the states correctly, it is essential to take
+this into consideration. An exception stands out in state
+4f 116s2 4Io
+9/2 of Ho, where the theory differs significantly
+from the experiment. Based on the NIST database [16]
+of Ho spectrum, we can observe that there are multiple
+states with the same parity and total angular momentum
+J, separated only by small energy intervals and domi-
+nated by different electron configurations. Due to this
+vigorous mixing, the calculations of the g-factor become
+unstable.
+The hyperfine structures of the ground states and some
+low-lying states of Dy and Ho have also been calculated.
+The Dy atom has two stable isotopes, 161Dy and 163Dy,
+and the Ho atom has one stable isotope, 165Ho.
+The
+results of calculations and corresponding nuclear param-
+eters are presented in Table II. One can see that we have
+good agreement between theory and experiment for mag-
+netic dipole constant A and electric quadrupole constant
+B for most states of Dy and Ho. The difference between
+theory and experiment is within 3% for the A constant
+of Dy and Ho, within 4% for the B constants of Dy and
+∼ 20% for the B constant of Ho. A similar agreement
+between theory and experiment was found earlier for the
+HFS constants of Er [7]. Two states of Ho present an ex-
+ception. These are the 4f 105d6s2 (8, 3
+2)13/2 state, and the
+
+4
+TABLE II. Hyperfine structure constants A and B (in MHz) for low-lying states of Dy and Ho. Nuclear spin I, nuclear magnetic
+moment µ(µN), and nuclear electric quadrupole moment Q(b) values for the isotopes of the 161,163Dy and 165Ho are taken from
+Ref. [24], gI = µ/I. Last column presents references to experimental data for A and B.
+Isotope
+This work
+Experimental results.
+Nuclear Parameters
+Conf.
+Term
+J
+A
+B
+A
+B
+Ref.
+161Dy
+µ= -0.480, I= 5/2, Q= 2.51
+4f 106s2
+5I
+8
+-113
+1127
+-116.231
+1091.577
+[10]
+4f 106s2
+7
+-125
+1057
+-126.787
+1009.742
+[10]
+4f 106s2
+6
+-140
+991
+-139.635
+960.889
+[10]
+4f 95d6s2
+7Ho
+8
+-88
+2256
+-
+-
+-
+4f 95d6s2
+7
+-104
+2397
+-
+-
+-
+4f 106s2
+5I
+5
+-166
+928
+-161.971
+894.027
+[10]
+4f 95d6s2
+7Io
+9
+-80
+2663
+-
+-
+-
+4f 95d6s2
+7Ho
+6
+-122
+2901
+-
+-
+-
+4f 106s2
+5I
+4
+-216
+997
+-205.340
+961.156
+[10]
+163Dy
+µ= 0.673, I= 5/2, Q= 2.65
+4f 106s2
+5I
+8
+158
+1190
+162.754
+1152.869
+[10]
+4f 106s2
+7
+176
+1116
+177.535
+1066.430
+[10]
+4f 106s2
+6
+196
+1046
+-
+-
+-
+4f 95d6s2
+7Ho
+8
+123
+2381
+-
+-
+-
+4f 95d6s2
+7
+146
+2531
+-
+-
+-
+4f 106s2
+5I
+5
+233
+979
+-
+-
+-
+4f 95d6s2
+7Io
+9
+112
+2812
+-
+-
+-
+4f 95d6s2
+7Ho
+6
+170
+3063
+-
+-
+-
+4f 106s2
+5I
+4
+303
+1053
+-
+-
+-
+165Ho
+µ= 4.17, I= 7/2, Q= 3.58
+4f 116s2
+4Io
+15/2
+787
+-1943
+800.583
+-1668.089
+[17]
+4f 116s2
+13/2
+939
+-1668
+937.209
+-1438.065
+[17]
+4f 105d6s2
+(8, 3
+2)
+17/2
+666
+1085
+776.4(4.5)
+608(300)
+[18]
+4f 105d6s2
+15/2
+763
+1127
+783.0(4.5)
+801(300)
+[18]
+4f 116s2
+4Io
+11/2
+1061
+-1315
+1035.140
+-1052.556
+[17]
+4f 105d6s2
+(8, 3
+2)
+13/2
+879
+1829
+916.6(0.5)
+2668(7)
+[18]
+4f 105d6s2
+19/2
+617
+1650
+745.1(1.4)
+1747(78)
+[18]
+4f 116s2
+4Io
+9/2
+1279
+-1174
+1137.700
+-494.482
+[17]
+4f 116s2 4Io
+9/2 state. Here the difference between theory
+and experiment for electric quadrupole HFS constant B is
+significant. In particular, it is 138% for the 4f 116s2 4Io
+9/2
+state. This is the same state which shows poor accuracy
+for the g-factor, which indicates that strong configuration
+mixing affects the HFS as well. It should be mentioned
+that an earlier study performed using the MCDF method
+also found that this state had a low level of accuracy with
+a 117 % deviation from the experimental result [13].
+Note that our investigations of testing the accuracy of
+using the CIPT method on the Er atomic system, which
+has a similar electronic structure, were previously per-
+formed [7]. All the above atomic properties, energies, g-
+factors, and HFS constants A and B for the stable isotope
+with non-zero spin, 167Er, have been calculated. There
+has been a good agreement between measurements and
+our results (see Ref. [7] Tables 1 and 6). In the end, we
+expect that the results for Cf and Es will be accurate as
+well.
+IV.
+IONIZATION POTENTIALS
+Calculating ionization potential (IP) is a good way to
+test the theoretical approach for the ground state. The
+IP is obtained as a difference between the ground state
+energies of the neutral atom and the ion.
+The CIPT
+method, which we use in the present calculations, has a
+feature of having good accuracy for low-lying states, and
+it decreases while going up on the energy scale. The best
+accuracy is expected for the ground state. On the other
+hand, having HFS for the ground state is sufficient to
+extract nuclear parameters µ and Q. Therefore, we cal-
+culate the first ionization potential (IP1) for all atoms
+considered in the present work.
+We calculate ground
+state energies of neutral atoms and corresponding ions
+in the same V N−1 potential and the same single-electron
+basis. This ensures exact cancelation of the energies as-
+sociated with core electrons. The results are presented
+in Table III and compared with available experimental
+data. As can be seen from the table the accuracy of the
+results is 2.7% for Dy, 1.6% for Ho, 0.3% for Cf and 0.8%
+for Es.
+
+5
+TABLE III. Experimental and theoretical values of the first ionization potential IP1 (in cm−1).
+State
+IP1
+Atom
+Initial
+Final
+Presesnt
+Expt.
+Ref.
+Dy
+4f 106s2
+5I8
+4f 106s (8, 1
+2)17/2
+46658
+47901.76(5)
+[25]
+Ho
+4f 116s2
+4Io
+15/2
+4f 116s ( 15
+2 , 1
+2)o
+8
+47819
+48567(5)
+[26]
+Cf
+5f 107s2
+5I8
+5f 107s
+6I17/2
+50821
+50663(2)
+[27]
+Es
+5f 117s2
+4Io
+15/2
+5f 117s
+5Io
+8
+51763
+51358(2)
+[27]
+51364.58(14)stat(50)sys
+[28]
+TABLE IV. Calculated hyperfine structure constants A and
+B (in MHz) for the ground states of Dy, Ho, Cf and Es atoms.
+Atom
+Conf.
+Term
+J
+A
+B
+Dy
+4f 106s2
+5I
+8
+587×gI
+449×Q
+Ho
+4f 116s2
+4Io
+15/2
+661×gI
+-543×Q
+Cf
+5f 107s2
+5I
+8
+608×gI
+477×Q
+Es
+5f 117s2
+4Io
+15/2
+681×gI
+-818×Q
+V.
+RESULTS FOR HFS
+In Table IV, we present the results of our calculations
+of the HFS constants of the ground states of Dy, Ho,
+Cf and Es. We have calculated both, magnetic dipole
+HFS constant A and electric quadrupole HFS constant B,
+which can be used for the extraction of nuclear moments
+for any isotope with non-zero spin. For a better under-
+standing of the accuracy of the calculations for heavy
+actinides, it is instructive to compare electron structure
+factors for the HFS constants with those of lighter atoms,
+Dy, Ho, and Er. The situation is different for the HFS
+constants A and B. The electron structure factor for the
+magnetic dipole constant A is almost the same for heavy
+actinides and their lighter analogs; it varies within 3%.
+The electron structure factors for the HFS constant B are
+also similar, although the variation is larger. It goes from
+about 20% for the Dy, Cf pair to 50% for the Ho, Es pair.
+This justifies using lighter analogs of heavy actinides for
+the estimation of the uncertainty of the calculations. We
+assume 3% uncertainty for the HFS constant A of all con-
+sidered atoms and 16% uncertainty for the HFS constant
+B (as the difference between theory and experiment for
+the ground state of Ho). This latter assumption is rather
+conservative. The difference between theory and experi-
+ment for the HFS constant B of the ground state of Dy
+is about 3% and it is about 10% for the ground state of
+Er [7].
+This high level of accuracy is a bit surprising for atoms
+with open shells. Therefore, it is instructive to see how
+dominating contributions are formed. First, we note that
+according to numerical tests, configuration mixing gives
+a relatively small contribution to the HFS constants.
+About 90% or more comes from leading configurations
+which is 4f n6s2 for Dy and Ho and 5f n7s2 for Cf and
+Es (n=10,11). In these configurations s-electrons form
+a closed shell and do not contribute to the HFS. There-
+fore, all contribution comes from f-electrons. It is well
+known that in the case of excited valence f-states (e.g,
+4f state of Cs or 5f state of Fr) the HF value of the
+energy shift due to HFS operator ⟨4f| ˆf|4f⟩ is small and
+dominating contribution comes from the core polarisa-
+tion correction ⟨4f|δV f
+core|4f⟩ (see, Eq. (9)). The situa-
+tion is different in atoms considered in present work. The
+f electron states are inside the core, localised at about
+the same distances as other states with the same principal
+quantum number, i.e. it is not even the outermost shell.
+For example, ⟨4f|r|4f⟩ < 1aB for Dy, Ho and Er, while
+⟨4f|r|4f⟩ ∼ 20aB for Cs. Being inside the core f-states
+penetrate to short distances near the nucleus making a
+large value of the HF matrix element ⟨4f| ˆf|4f⟩. In con-
+trast, the core polarization correction ⟨4f|δV f
+core|4f⟩ is
+small (∼ 1%). In the end, zero-order matrix elements
+are large while core polarization and configuration mix-
+ing corrections are small.
+This is the key to the high
+accuracy of the results.
+Table V shows the results and analysis of the HFS
+for three isotopes of Es (253−255Es). This table serves
+two purposes. First, this is another confirmation of the
+accuracy of the calculations. However, to compare the
+calculations to the experiment we need to use nuclear
+moments, which are known to have pretty poor accuracy
+(see the table).
+For example, the uncertainty for the
+magnetic moment of the 253Es nucleus is 17%. On the
+other hand, our estimated accuracy for the HFS constant
+A is 3%. This means that we can improve the accuracy
+of the nuclear moments by extracting them from a com-
+parison of the experimental data with our calculations.
+The results are presented in the table. We see that real
+improvement is obtained for µ(253)Es only. For other nu-
+clear moments, the uncertainties are similar but central
+points are shifted.
+New and old values are consistent
+when error bars are taken into account.
+VI.
+CONCLUSIONS
+Magnetic dipole and electric quadrupole HFS con-
+stants A and B were calculated for the ground states of
+heavy actinides Cf and Es. Similar calculations were per-
+formed for the lighter analogs of these atoms, Dy and Ho.
+To establish the accuracy of the results, the comparison
+
+6
+TABLE V. Hyperfine structure constants A and B (in MHz) of the ground state of Es. Nuclear spin I, nuclear magnetic
+moment µ(µN), and nuclear electric quadrupole moment Q(b) values for the isotopes of the 253Es are taken from Ref. [24],
+while 254Es and 255Es parameters are taken from Ref. [3]. gI = µ/I. The last column presents references for experimental data
+on A and B. The values of µ and Q obtained in this work are extracted from comparison of experimental and calculated HFS
+constants assuming 3% uncertainty in calculation of A and 16% uncertainty in calculation of B.
+Isotope
+This work
+Experimental results.
+Nuclear Parameters
+Conf.
+Term
+A
+B
+µ
+Q
+A
+B
+Ref.
+253Es
+µ= 4.1(7), I= 7/2, Q= 6.7(8)
+5f 117s2
+4Io
+15/2
+798
+-5481
+4.12(15)
+4.8(1.0)
+802(18)
+-3916(550)
+[3]
+4.20(13)
+5.3(8)
+817.153(7)
+-4316.254(76)
+[1]
+254Es
+µ= 3.42(7), I= 7, Q= 9.6(1.2)
+5f 117s2
+4Io
+15/2
+333
+-7853
+3.48(10)
+7.6(1.3)
+339(4)
+-6200(300)
+[3]
+255Es
+µ= 4.14(10), I= 7/2, Q= 5.1(1.7)
+5f 117s2
+4Io
+15/2
+806
+-4172
+4.23(26)
+3.7(1.8)
+824(45)
+-3001(1400)
+[3]
+between theory and experiment was done for HFS con-
+stants, energy levels, g-factors and ionization potential,
+everywhere where the experimental data are available.
+We found the uncertainty of 3% for the HFS constant A
+and about 16% uncertainty for the HFS constant B. Us-
+ing the calculated HFS constants of those heavy elements
+considered, nuclear magnetic and electric quadrupole mo-
+ments can be extracted from the measurement data.
+ACKNOWLEDGMENTS
+The authors are grateful to Sebastian Raeder for
+many stimulating discussions. This work was supported
+by the Australian Research Council under Grants No.
+DP190100974 and No. DP200100150. S.O.A. gratefully
+acknowledges the Islamic University of Madinah (Min-
+istry of Education, Kingdom of Saudi Arabia) for funding
+his scholarship.
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+

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+A Tropical Geometric Approach To Exceptional Points
+Ayan Banerjee,1 Rimika Jaiswal,2, 3 Madhusudan Manjunath,4 and Awadhesh Narayan1, ∗
+1Solid State and Structural Chemistry Unit, Indian Institute of Science, Bengaluru 560012, India
+2Undergraduate Programme, Indian Institute of Science, Bengaluru 560012, India
+3Department of Physics, University of California, Santa Barbara, California 93106-9530, USA
+4Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
+(Dated: February 1, 2023)
+Non-Hermitian systems have been widely explored in platforms ranging from photonics to electric
+circuits.
+A defining feature of non-Hermitian systems is exceptional points (EPs), where both
+eigenvalues and eigenvectors coalesce. Tropical geometry is an emerging field of mathematics at
+the interface between algebraic geometry and polyhedral geometry, with diverse applications to
+science.
+Here, we introduce and develop a unified tropical geometric framework to characterize
+different facets of non-Hermitian systems. We illustrate the versatility of our approach using several
+examples, and demonstrate that it can be used to select from a spectrum of higher-order EPs in
+gain and loss models, predict the skin effect in the non-Hermitian Su-Schrieffer-Heeger model, and
+extract universal properties in the presence of disorder in the Hatano-Nelson model. Our work puts
+forth a new framework for studying non-Hermitian physics and unveils a novel connection of tropical
+geometry to this field.
+INTRODUCTION
+Several branches of mathematics show an unreason-
+able effectiveness in formulating and understanding a
+myriad of physical phenomena [1].
+Striking recent ex-
+amples include the role of topology in condensed matter
+systems [2, 3], advent of knot theory in quantum field
+theory [4], and applications of graph theory in statistical
+mechanics [5].
+Tropical geometry is a branch of modern mathematics
+at the interface between algebraic geometry and polyhe-
+dral geometry [6, 7]. The tropical approach has not only
+had applications to geometry, but also to areas such as
+physics, number theory, genetics, economics, optimiza-
+tion theory, and computational biology [8–11]. Notable
+has been the role of tropical geometry in understanding
+physical systems. Deep connections of tropical geome-
+try to string theory have been discovered [12, 13], while
+tropical algebra has been used to analyze frustrated sys-
+tems such as spin ice and spin glasses [14].
+Another
+recent successful application of tropical ideas has been
+in understanding self-organized criticality in dynamical
+systems [15]. Tropical geometric tools such as the loga-
+rithmic transformation offer drastic computational sim-
+plification, and, interestingly, the low-temperature limit
+of statistical physics can be studied in terms of such a
+tropical mapping [9, 16].
+Hermiticity of operators is a central principle in quan-
+tum mechanics, ensuring that a system has real eigenen-
+ergies and orthogonal eigenstates, and leads to the con-
+servation of probability [17]. In recent decades the no-
+tion of non-Hermiticity has been introduced in a variety
+of physical contexts [18–20]. A unique feature of non-
+Hermitian systems are degeneracies called exceptional
+points (EPs), where both eigenvalues and eigenvectors
+coalesce [21]. The energy level-splitting, ∆λ, upon mov-
+ing away from an EP follows a distinctive fractional de-
+pendence on the perturbation. An N-th order EP [EP-N,
+where two or more eigenvectors (N ≥ 2) coalesce] shows
+a splitting of the form ∆λ ∼ ν1/N, where ν is an external
+perturbation [22, 23]. Recent advances have led to con-
+trollable realization of EPs in a variety of platforms [24–
+30]. Their control has enabled the exploration of novel
+phenomena, such as uni-directional sensitivity [31, 32],
+laser mode selectivity [33, 34], and non-Hermitian skin
+effect (NHSE) [35].
+In this work, we propose and develop a general tropical
+geometric framework for understanding and characteriz-
+ing various facets of non-Hermitian systems. We demon-
+strate that the tropical geometric information encoded
+in the characteristic polynomial of the non-Hermitian
+Hamiltonian can be used to identify and classify EPs us-
+ing valuation and tropical roots – concepts that naturally
+emerge in the tropical setting. We show that EPs of dif-
+ferent orders and their transitions can be captured in an
+elegant manner by amoebas and Newton polygons. We
+illustrate our framework using experimentally-realized
+gain and loss models, and show how it allows obtaining a
+higher-order EP or choosing from a spectrum of EPs. Us-
+ing the paradigmatic non-Hermitian Su-Schrieffer-Heeger
+(SSH) model, we demonstrate how our tropical geomet-
+ric approach can be used to predict the NHSE. Our ap-
+proach naturally allows extracting the universal proper-
+ties of EPs in the presence of disorder, which we high-
+light using the celebrated Hatano-Nelson model.
+Our
+framework allows a unified approach to different facets
+of non-Hermitian phenomena, including EPs, NHSE, and
+holonomy.
+arXiv:2301.13485v1  [quant-ph]  31 Jan 2023
+
+2
+a
+b
+FIG. 1. Illustration of the tropical geometric frame-
+work with two site gain and loss model. (a) Schematic
+of the two site gain and loss model with γ as the gain and
+loss parameter and κ as the coupling between the sites. (b)
+Using tropicalization to find the order of EP. The tropical
+polynomial contains different linear monomials with integer
+coefficients (see Eq. 5).
+The bend locus of the monomials
+(encircled in red) gives the tropical root ω0 = 1/2 implying a
+second order EP.
+Tropical characterization of exceptional points
+Basics of tropical geometry. We begin by briefly
+summarizing the fundamental ideas of tropical geometry
+(see supplemental material [36] for a detailed discussion).
+Broadly speaking, tropical geometry studies solutions of
+systems of polynomials by transforming them into piece-
+wise linear subsets of Euclidean space [37].
+The basic
+algebraic object underlying tropical geometry is the trop-
+ical semiring, (R ∪ {∞}, ⊕, ⊙). This denotes a set that
+is the union of the set of real numbers R, together with
+an element “infinity”, and two operations on it, namely
+tropical addition ⊕ and tropical multiplication ⊙. The
+tropical sum of two numbers is their minimum and the
+tropical product is their usual sum,
+x ⊕ y = min(x, y),
+x ⊙ y = x + y.
+(1)
+Many of the usual axioms of arithmetic remain valid in
+the tropical setting. These operations satisfy all the ring
+axioms except for the existence of an additive inverse and
+thus turn (R ∪ {∞}, ⊕, ⊙) into a semiring.
+Defining order of exceptional points. In the fol-
+lowing, we define the notion of order of an EP of a non-
+Hermitian system. To the best of our knowledge, this
+definition is consistent with the literature on this topic.
+Let H(ν) be the Hamiltonian of a non-Hermitian sys-
+tem in one variable ν with an EP at ν = 0 and let
+p(ν, λ) ∈ C[ν, λ] be its characteristic polynomial. In the
+following, we regard p as a polynomial in one variable
+λ and with coefficients in the field C{{ν}} of Puiseux
+series.
+Definition .1. Let p ∈ C{{ν}}[λ] have at least one non-
+zero root. Suppose that p has a non-trivial Puiseux series
+root, i.e. a root s such that the least exponent of s is
+non-zero. In this case, the order of this EP (at ν = 0)
+is the maximum absolute value of the denominator n of
+m/n ∈ Q (in reduced form) where m/n varies over the
+least exponent in the Puiseux series expansion over all
+the non-trivial roots of p. Otherwise, if all the roots of p
+have zero as their least exponent, then ν = 0 is called a
+degenerate point.
+Consider a system at an EP-N (at ν = 0) given by
+the Hamiltonian H0(x1, x2, ...) where x1, x2... are system-
+dependent parameters.
+When we perturb this system
+around the EP, the eigenvalues of the perturbed Hamil-
+tonian H(ν) = H0 + νH1 follow a Puiseux series in ν,
+λ(ν) = γ1ν1/N + γ2ν2/N + ...,
+(2)
+where ν is the perturbation strength. To leading order,
+the response goes as ∆λEP −N ∝ ν1/N.
+Our tropical
+geometric approach features a characterization of EPs
+by determining such leading order behavior.
+Characterizing exceptional points using tropi-
+cal geometry. Next, we present the tropical geometric
+framework that can be used to reveal the structure of EPs
+and characterize as well as tune them in various physical
+platforms.
+For a field K, a valuation on K is defined as a function
+val : K → R ∪ {∞} such that:
+• val(a) = ∞ if and only if a = 0;
+• val(ab) = val(a) + val(b) ;
+• val(a + b) ≥ min{val(a), val(b)} for all a, b ∈ K.
+In our framework, we primarily deal with the field of
+Puiseux series with coefficients in the complex numbers
+C. This field has a natural valuation which is given by
+taking a non-zero Puiseux series to the lowest exponent
+that appears in its expansion. For example, val(t2 − 2t +
+3) = min{val(t2), val(−2t), val(3)} = min{2, 1, 0} = 0
+and val(t1/2 − t3/4 + t1 + t2 + . . . ) = 1/2.
+In its most basic form, tropical geometry gives a
+method to compute the valuations of the non-zero roots
+of a non-zero polynomial p ∈ K[λ] in terms of the val-
+uations of the coefficients of p. More precisely, given a
+non-zero polynomial p = �d
+i=0 aiλi ∈ K[λ], its tropi-
+calization trop(p) : R → R is defined as trop(p)(ω) =
+mini{val(ai) + i · ω}.
+A real number ω0 is called a tropical root of trop(p) if
+the minimum defining trop(p)(ω0) is attained by at least
+two distinct terms val(aj)+j·ω0 and val(ak)+k·ω0 for j ̸=
+k. Equivalently, the tropical roots of trop(p) precisely
+
+3
+are the real numbers where trop(p) is not differentiable,
+called the bend locus of trop(p).
+The fundamental theorem of tropical geometry asserts
+that the set of tropical roots of trop(p) is precisely the set
+of valuations of the non-zero roots of p [37, Chapter 3,
+Section 2]. This leads us to one of the main propositions
+of our framework.
+For a non-Hermitian Hamiltonian,
+H(ν), with a characteristic polynomial p(ν, λ) ∈ C[ν, λ],
+as described before, p can be regarded as an element in
+C{{ν}}[λ], where C{{ν}} is equipped with its standard
+valuation that takes a non-zero Puiseux series s to the
+exponent of the leading order term of s.
+Proposition .2. Suppose that trop(p(ν, λ)) has a non-
+zero tropical root.
+The order of the EP at ν = 0 of
+H(ν) is the maximum absolute value of the denominator
+n of m/n (in reduced form) where m/n varies over all
+the non-zero tropical roots of trop(p(ν, λ)). Otherwise, if
+trop(p(ν, λ)) has no non-zero tropical roots, then ν = 0
+is a degenerate point.
+Proof. By the fundamental theorem of tropical geome-
+try [37, Chapter 3, Section 2], the set of tropical roots
+of trop(p) is precisely the set {val(s)}s where s varies
+over all the non-zero Puiseux series solutions of p(ν, λ) ∈
+C{{ν}}[λ]. With this information at hand, the statement
+follows from the definition of the order of an EP.
+To simply illustrate our framework, we consider an ex-
+perimentally realizable non-Hermitian system consisting
+of two coupled sites with gain and loss (see Fig. 1a). The
+Hamiltonian reads
+H2 =
+�
+α + iγ
+κ
+κ
+−α − iγ
+�
+.
+(3)
+Here α quantifies the onsite energies, γ is the corre-
+sponding gain/loss coefficient, and κ is the coupling be-
+tween the sites.
+This system has an EP at α = 0 if
+γ = κ.
+The characteristic polynomial and the corre-
+sponding tropicalization for γ = κ are
+p(α, λ) = −2iκα − α2 + λ2,
+(4)
+trop (p(α, λ)) (ω) = min (1, 2ω) .
+(5)
+The root of the tropical polynomial is given by the
+bend locus of trop(p(α, λ))(ω) which occurs at ω0 = 1/2,
+as shown in Fig. 1b. Using the fundamental theorem of
+tropical geometry, we then conclude that p(α, λ) has a
+non-zero root with valuation s = 1/2. This implies that
+the roots of p(α, λ), i.e., the eigenvalues of H have the
+form λ ∼ α1/2 near the EP at α = 0. Thus, the EP at
+a
+b
+FIG. 2. Characterization of exceptional points through
+amoebas in a three-site gain and loss model. Realiza-
+tion of Newton polygon (left), amoeba (center), and the spine
+of the amoeba for (a) φ = −π/6 and (b) φ = −π/4. The con-
+vex slope of the Newton polygon defines the order of EPs.
+We obtain third-order EPs in (a) and second-order EPs in
+(b). The interior point in the Newton polygon in (a) results
+in a vacuole in the amoeba. The amoeba structures abruptly
+change from (a) to (b) while transitioning from third-order to
+second-order EPs. We set γ =
+√
+2κ and κ = 1.0.
+α = 0 is a second-order EP (see the supplement [36] for
+a detailed discussion). Further, in the supplement [36],
+we use tropicalization to illustrate how our framework
+provides a natural way to characterize and tune to higher-
+order EPs using companion matrices.
+Relation to amoebas and Newton polygons.
+Above, we saw how tropical geometry can be used to
+determine the order of EPs. A precursor to tropical ge-
+ometry is a construction called the amoeba of a complex
+algebraic variety due to Gelfand, Kapranov and Zelevin-
+sky [38]. Let V ⊆ (C⋆)n be the set of solutions, all of
+whose coordinates are non-zero, of a finite set of Lau-
+rent polynomials in n variables.
+Let Log : (C⋆)n →
+Rn be the logarithmic map that takes (z1, . . . , zn) to
+(log(|z1|), . . . , log(|zn|)). The amoeba of V is the image
+of the logarithmic map restricted to V . A related and im-
+portant notion is the spine of the amoeba and is defined
+as the limit as t → ∞ of the parameterized logarithmic
+map Logt(z1, . . . , zn) = (logt(|z1|), . . . , logt(|zn|)). In the
+Methods section we present the connection of amoeba to
+tropicalization.
+We are primarily concerned with amoebas of polyno-
+mials in two variables with complex coefficients, namely
+characteristic polynomials of a non-Hermitian Hamilo-
+tian in one variable. Typical examples of such amoebas
+are shown in Fig. 2, which will be discussed shortly. The
+amoeba of a typical polynomial contains (unbounded)
+rays that are called its tentacles. We recall that the New-
+ton polygon of p is the convex hull of the exponents of
+the monomials in the support of p. The following propo-
+sition that relates the edges of the Newton polygon of p
+and the amoeba (of the algebraic variety) associated to
+p is of fundamental importance to our framework.
+Proposition .3. The set of directions of the tentacles
+of the amoeba associated to p is precisely the set of outer
+normals of the edges of the Newton polygon of p.
+
+4
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1
+(N-2)⍵
+N⍵
+(N-4)⍵
+N⍵
+1
+a
+d
+b
+c
+e
+FIG. 3.
+Detecting skin effect in the non-Hermitian
+SSH model with higher-order EPs via tropical geom-
+etry (a) Schematic of SSH model with non-reciprocal hop-
+ping and a weak link connecting the last site to the first.
+The inter-unit cell hopping is a constant t2, but the left and
+right intra-unit cell hoppings are given by t ± γ incorporat-
+ing non-Hermiticity in the system. (b) Newton polygons and
+the concomitant amoebas for SSH model with odd number
+of sites.
+Here we choose t1 = 2.0, γ = 1.0 and t2 = 1.0.
+The structure is similar for all values t1 ̸= γ. (c) At t1 = γ
+the Newton polygon abruptly transforms to a single line with
+slope 1/N at t1 = γ, indicative of a higher order EP and the
+skin effect. The amoeba collapses to a single line perpendicu-
+lar to the Newton polygon, characterizing the non-Hermitian
+phase transition. Panel (d) shows the tropicalization for the
+general case corresponding to Eq. 10. Each straight line rep-
+resents a term in Eq. 10. (e) Tropicalization for the case of
+t1 = γ, wherein the coefficients of all the points corresponding
+to λM vanish, other than M = 0, N. The bend locus gives
+the tropical root ω0 = 1/N, which indicates the presence of
+an N-th order EP and, correspondingly, the occurrence of a
+non-Hermitian skin effect. Here we choose N = 5.
+We refer to Proposition 1.9 [38] and Section 1.4 [37]
+for a more general version of this proposition.
+We illustrate this proposition using a three-site non-
+Hermitian trimer model with balanced gain and loss and
+an asymmetric onsite potential. The Hamiltonian for the
+trimer is
+H3 =
+�
+�
+α + iγ κ
+0
+κ
+0
+κ
+0
+κ β − iγ
+�
+� ,
+(6)
+where the different symbols have a meaning analogous
+to the two-site model. We use the transformation β =
+α tan φ to scan all angles in the α-β parameter plane.
+Using the formalism developed above, we can find the
+tropical roots to reveal the nature of EPs.
+In Fig. 2
+we illustrate the amoebas and the concomitant Newton
+polygons for various φ. Note that the steepest slope of
+the Newton polygon ∆ determines the order of the EPs.
+Interestingly, the integer points of the Newton polygon
+correspond to the vacuole in the amoeba (see Fig. 2a).
+The structure of the amoeba drastically transforms from
+φ = −π/4 to π/4 while the tropical roots change from
+1/2 to 1/3 with a transition from second-order to third-
+order EPs. Therefore, the structure of the amoeba can
+be directly used to identify the various non-Hermitian
+phases.
+Tropical analysis of the Su-Schrieffer-Heeger model
+Below we apply the complete tropical approach de-
+veloped in this work to the paradigmatic non-Hermitian
+SSH Hamiltonian with non-reciprocal hopping [39, 40].
+We demonstrate how our tropical geometric approach
+can detect the NHSE, which is a unique feature of non-
+Hermitian systems where a large number of states ac-
+cumulate at boundaries of open systems [35, 41]. The
+Hamiltonian of the non-Hermitian SSH model reads
+HSSH = −
+�
+i
+[t1(c†
+i,Aci,B + h.c.) + t2(c†
+i+1,Aci,B + h.c.)]
++
+�
+i
+γ(c†
+i,Bci,A − c†
+i,Aci,B),
+(7)
+where c†
+i,α(ci,α) is the fermionic creation (annihilation)
+operator at site i for sublattice α = A, B. The intra-
+and inter-unit cell hopping amplitudes are given by t1
+and t2, respectively, and γ introduces a non-reciprocity
+only in the intra-unit cell hopping, resulting in non-
+Hermiticity (see Fig. 3a). We introduce a perturbation,
+σt2, σ ∈ [0, 1], which connects the last and first sites.
+The Hamiltonian takes the matrix form (ϵ = σt2)
+HSSH(ϵ) =
+�
+����
+0
+t − γ
+0 · · ·
+ϵ
+t + γ
+0
+t2 · · · 0
+0
+t2
+0 · · · 0
+...
+...
+...
+...
+...
+�
+����
+N×N
+.
+(8)
+The characteristic equation, in turn, is
+p(ϵ, λ) = mod(N − 1, 2){γ2 − t2
+1}
+N
+2 − t
+N−2
+2
+2
+(t1 + γ)N/2ϵ
++
+�
+M=N,N−2,···
+[zM{γ2 − t2
+1}
+N−M
+2
++ t2OM]λM,
+(9)
+where each zM is a constant, zM ∈ Z. The tropical
+polynomial is calculated to be
+
+3DCoreDiagramforPowerPoint5
+1
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+2⍵ + 1
+4⍵
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+4⍵
+2⍵ + 1
+2
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+4⍵
+2⍵ + 1
+⍵ + 1
+2
+a
+b
+c
+d
+e
+f
+g
+EP
+EP
+FIG. 4.
+Holonomy characterization of disordered Hatano-Nelson model.
+(a) Riemann surface for a quartic root
+in the complex plane. (b) The Hatano-Nelson model exhibits anisotropic exceptional behaviour in the parameter space as
+illustrated by the different projection of eigenbands along different parameter planes (the number of petals representing the
+order of EP) (c), (d) Swapping of eigenmodes arising from Riemann sheet exchange while tracing a loop in parameter space
+given by R = ceiψ, ψ ∈ (0, 2π). We show the holonomy properties when the loop (c) critically touches EP and (d) encloses
+the EP. Note that in (d) the N eigenmodes undergo a cyclic permutation among themselves while in (c) eigenmode evolution
+forms N petals in the complex energy plane where N is the order of EP. Tropicalization and tropical roots showing (e) fourth-
+(θ = 0, φ = π/4), (f) second- (θ = 0, φ = 0), and (g) third-order (θ = π/4, φ = 0) EPs for different values of φ and θ. The
+insets show holonomy characterization in the presence (brown) and absence (blue) of disorder. Disorder preserves the stability
+of EPs but renormalizes the spectral properties.
+trop (p(ϵ, λ)) (ω) = min{m, · · · (N − 2)ω, Nω},
+(10)
+where m = 0 (1) for even (odd) sites. The tropical-
+ization and bend locus for p(ϵ, λ) are shown in Fig. 3d
+and 3e. Strikingly at t1 = γ and t2 → 0, the coefficients
+of all the terms in p(ϵ, λ) vanish other than the ϵ1 and
+the λN terms which lead to the solution λ = ϵ1/N. This
+fractional exponent, in turn, shows that higher-order EPs
+appear for t1 = ±γ with an algebraic multiplicity that
+scales with system size while the geometric multiplicity
+remains unity. This is a signature of the NHSE, wherein
+all the bulk modes collapse to one state and are exponen-
+tially localized at the edge under open boundary condi-
+tions.
+This physics can be beautifully captured by the amoe-
+bas and their corresponding Newton polygons. In Fig. 3b
+and 3c, we present the Newton polygon and associated
+amoeba for this model. The edges of the Newton polygon
+are perpendicular to the tentacles of the amoeba. The
+structure of the amoeba remains invariant unless we have
+t1 = γ, where the amoeba and the corresponding New-
+ton polygon strikingly collapse to a single straight line,
+as shown in Fig. 3c. The Newton polygon in the latter
+case has a slope of 1/N, establishing the presence of an
+EP-N.
+Application to disorder and holonomy
+Our tropical geometric framework can be used to ex-
+tract universal properties of EPs even in the presence of
+disorder as we show next. To illustrate, we consider the
+celebrated Hatano-Nelson model [42] under open bound-
+ary conditions with N sites, along with upper corner per-
+turbations, i.e., additional couplings in the (1, j)-th en-
+tries of the Hamiltonian, where j = N, N − 2 · · · . For 4
+sites, the Hamiltonian reads
+H4 =
+�
+�
+�
+�
+0
+δ
+η
+∆
+(2 + δ)
+0
+δ
+0
+0
+(2 + δ)
+0
+δ
+0
+0
+(2 + δ) 0
+�
+�
+�
+� .
+(11)
+The Hamiltonian can be written as a sum of two com-
+panion matrices indicating that it features different ex-
+ceptional behaviour along different sections of the param-
+eter space. To study the structure of EPs in the parame-
+ter space of δ, ∆ and η, we shift to generalized spherical
+coordinates δ = r cos θ cos φ, ∆ = r cos θ sin φ, η = r sin θ
+and study the tropicalization of the characteristic equa-
+tion p(r, λ). We find highly anisotropic behaviour result-
+ing in EP-2, EP-3 or EP-4 along various directions, as
+summarized in Fig. 4. Please refer to the supplement for
+
+6
+a more detailed analysis [36].
+Here, we will use the N = 4 case as an example to show
+that the exceptional behaviour in Hatano-Nelson model
+remains universal even in the presence of certain kinds of
+disorder, and the disordered Hamiltonians are homotopic
+to each other with respect to the tropicalization. The
+Hamiltonian, H4, in the presence of a general form of
+scaling disorder reads
+Hdis =
+�
+�
+�
+�
+0
+aδ
+η
+∆
+(2 + δ)c
+0
+δb
+0
+0
+(2 + δ)d
+0
+δm
+0
+0
+(2 + δ)n
+0
+�
+�
+�
+� ,
+(12)
+where a, b, c, m and n are arbitrary real numbers that
+introduce disorder in the asymmetric hopping terms.
+Such models are well-studied and can be experimentally
+realized in different physical settings [43]. The form of
+the characteristic equation now changes, but remarkably,
+its tropicalization remains the same as for H4.
+trop (p(r, λ)(ω)) = min(4ω, 2ω + 1, ω + 1, 1),
+(13)
+for cos θ, sin θ, cos φ, sin φ ̸= 0.
+The tropical polyno-
+mial remains invariant to the values of disorder scaling
+parameters suggesting the exceptional behaviour remains
+invariant, or is universal in the presence of disorder. Our
+framework makes this apparent through the tropicaliza-
+tion. A complementary view is to analyze the holonomy
+around the EPs. Consider varying some system param-
+eters to form a loop in the parameter space while si-
+multaneously tracing the evolution of the complex eigen-
+modes.
+If the loop encloses an EP-N, N eigenmodes
+would undergo a cyclic permutation, which can be un-
+derstood using holonomy matrices [44, 45].
+Whereas,
+if the loop marginally touches an EP-N, the projection
+of the eigenmode evolution forms N petals in the com-
+plex energy plane, as shown in Fig. 4c. We used such a
+marginally touching loops to study the holonomy prop-
+erties for Hdis. We find that in the presence of disor-
+der, the eigenvalues get scaled, however their holonomy
+properties do not change, as shown in insets of Fig. 4e-g.
+As the tropicalization remains invariant, the set of dis-
+ordered Hamiltonians are homotopically connected and
+the EPs are universal.
+OUTLOOK
+Our work opens up several avenues for exploration.
+While we have formulated the tropical geometric frame-
+work for a single variable, we envisage that it should be
+possible to generalize this to several variables – this will
+allow treating multiple perturbations on the same foot-
+ing. It will be interesting to use our approach to clas-
+sify the different non-Hermitian symmetry classes, and
+explore potential connections of tropical geometry to K
+theory [46]. Since our approach allows treating disorder
+in a natural way, it could be interesting to connect trop-
+ical geometry and random matrices, which have appli-
+cations in many different fields of physics [47]. We also
+expect our analytical approach to be practically useful
+for tuning to EPs and identifying conditions for NHSE in
+various experimental arenas. Finally, we note that, very
+recently, amoebas have been used to determine the gen-
+eralized Brillouin zone for non-Hermitian systems [48].
+In summary, we have introduced and developed a new
+framework to characterize EPs using tropical geometry.
+We have illustrated its implications using paradigmatic
+SSH and Hatano-Nelson models.
+Our work, bridging
+the fields of tropical geometry and non-Hermitian phe-
+nomena, is particularly timely given the surge of interest
+in non-Hermitian systems.
+We hope that our findings
+motivate further synergy between mathematics and non-
+Hermitian physics.
+Acknowledgments
+A.B. thanks the Prime Minister’s Research Fellowship.
+R.J. thanks the Kishore Vaigyanik Protsahan Yojana fel-
+lowship. M.M was supported by a MATRICS grant from
+the Department of Science and Technology (DST) In-
+dia. A.N. is supported by the startup grant of the Indian
+Institute of Science (SG/MHRD-19-0001) and by DST-
+SERB (project number SRG/2020/000153). A part of
+this work was carried out during the program “Combi-
+natorial Algebraic Geometry: Tropical and Real” held at
+the International Centre for Theoretical Sciences, Ban-
+galore (ICTS) in June-July, 2022. We thank ICTS for
+their warm hospitality. We thank Vamsi P. Pingali for
+bringing us together on this project. We thank G. Reddy
+and N. Aetukuri for feedback on the manuscript. MM
+also thanks Indian Institute of Science, Bangalore and
+Waseda University, Tokyo for their hospitality during the
+visits in May, 2022 and January, 2023, respectively, in
+which a part of the work was carried out.
+Author contributions
+A.B. and R.J. carried out the calculations in consul-
+tation with M.M. and A.N. All authors analyzed the re-
+sults, developed the theory, and wrote the manuscript.
+
+7
+METHODS
+Fundamentals of tropical geometry.
+Here we
+summarize some of the fundamentals of tropical geom-
+etry. The algebraic structure of tropical geometry is also
+known as the min-plus algebra. Many of the usual ax-
+ioms of arithmetic remain valid in the tropical setting.
+For instance, addition and multiplication are commuta-
+tive
+x ⊕ y = y ⊕ x,
+x ⊙ y = y ⊙ x.
+(14)
+Associative property also holds, as does the distribu-
+tive law
+x ⊙ (y ⊕ z) = x ⊙ y ⊕ x ⊙ z.
+(15)
+Both tropical operations have an identity element –
+infinity for addition and zero for multiplication.
+x ⊕ ∞ = x,
+x ⊙ 0 = x.
+(16)
+A distinct feature of tropical arithmetic is the absence
+of subtraction operation.
+On the other hand, tropical
+division is the classical subtraction. So, (R ∪ {∞}, ⊕, ⊙)
+satisfies all the ring axioms except for the existence of
+additive inverse – such algebraic structures are termed
+semirings.
+Newton polygon formalism. We briefly discuss the
+Newton polygon formalism which is dual to amoebas. Let
+us start with the Puiseux series solution of the equation
+f(x, y) = 0 in a suitable neighbourhood of the origin (in
+our case an EP (ν = 0)). Any polynomial f(x, y) with
+the form
+f(x, y) =
+�
+η,ζ
+aηζxηyζ,
+(17)
+admits a solution y = txµ, where t is a complex number
+and µ = p/q is a positive rational number. One can find
+a solution by substituting y = txµ in Eq. 17, to obtain
+f(x, txµ) =
+�
+η,ζ
+aηζxη+µζtζ = xξ �
+η,ζ
+aηζtζ.
+(18)
+The above equation puts a constraint that f(x, y) con-
+tains only monomials xηyζ for which η + µζ = ξ, which
+is the essential feature of the Newton polygon. The ge-
+ometric interpretation of Newton polygon is embedded
+in the following mapping. Each monomial xηyζ maps to
+the pair (η, ζ) of natural numbers comprising a set of N2
+lattice points with integer coordinates for non-zero co-
+efficients aηζ ̸= 0. This set of lattice points forms the
+carrier ∆(f) of f, thus
+∆(f) = {(η, ζ) ∈ N2|aηζ ̸= 0}.
+(19)
+For a convergent power series f(x, y) with a carrier
+∆ (f), one can define a convex hull from each point of
+the carrier ∆ (f). The boundary of the convex hull, de-
+lineating a compact polygonal path, gives the Newton
+polygon of f. The steepest segment of the Newton poly-
+gon gives the lowest order term for the Puiseux series
+solution, thus defining the order of EP [49, 50]. More
+concretely, the condition η + µζ = ξ for all (η, ζ) ∈ ∆(f)
+indicates that all points of ∆(f) lie on a line, with a slope
+− 1
+µ, and the line meets the α−axis at η = ξ.
+Amoeba and tropicalization. We next present the
+connection between amoeba and tropicalization as used
+in the main text. The absolute value |.| over the complex
+numbers satisfies the archimedean property [51, Chapter
+9, page 313]. Any field F has an absolute value |.|t that is
+non-archimedean, i.e., does not satisfy the archimedean
+property: |0F |t = 0 and |c|t = 1 for all c ̸= 0F in F,
+where 0F is the additive identity of F.
+This is usu-
+ally called the trivial absolute value on F. Otherwise,
+the non-archimedean absolute value is called non-trivial.
+Fields such as the rational numbers Q, the field C((t)) of
+formal Laurent power series in one variable (with com-
+plex coefficients) are naturally equipped with non-trivial
+(non-archimedean) absolute values.
+More explicitly, i.
+|n|p := e−valp(n) for any n ∈ Q where p is a prime, valp(n)
+is ordp(r) − ordp(s) where n = r/s such that r, s ∈ Z,
+s ̸= 0 and ordp(i), for an integer i, is the largest power
+of p that divides i, ii. |ℓ(z)| for a Laurent power series
+ℓ(z) is defined as e−ord(ℓ(z)) where ord(ℓ(z)) is the least
+exponent of z in the support of ℓ. The rational num-
+ber ord(ℓ(z)) is also called the valuation of ℓ(z) and is
+denoted by val(ℓ(z)).
+Suppose that K is an algebraically closed field (every
+polynomial of degree at least one in K[t] has a root)
+equipped with a non-trivial, non-archimedean absolute
+value |.|K. Our primary example of such a field is the
+field C{{t}} of Puiseux series is one variable. The notion
+of amoeba that we defined over the complex numbers
+can also be mimicked over K as follows. Suppose that
+V ⊆ (K⋆)n is the set of solutions, all of whose coordinates
+are non-zero, to a finite set of Laurent polynomials in n-
+variables with coefficients in K. Let LogK : (K⋆)n → Rn
+be the map LogK(s) = − log |s|K (note that s ̸= 0K) [52].
+The tropicalization of V is defined as the image of the
+map LogK restricted to V . Hence, the tropicalization of
+V is a non-archimedean analogue of an amoeba.
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+

QNAzT4oBgHgl3EQflf3i/content/tmp_files/2301.01550v1.pdf.txt

@@ -0,0 +1,977 @@
+arXiv:2301.01550v1  [math.GM]  4 Jan 2023
+Logarithmic Integration Method for Solving of First and Second
+Order Differential Equations
+A. Ponomarenko
+In this article we present logarithmic methods for solving first order and second order ordinary
+differential equations. The essence of the method is that we apply the basic properties derivatives
+and logarithms to reduce the number of terms in the equation. Here we carry out this only for
+equations of the first and second order. Similar methods can also be used to obtain solutions to
+higher order equations.
+Keywords: differential equations, Riemann integrable functions, logarithmic methods.
+The main methods for solving ordinary differential equations have long been studied and are
+known to everyone. 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. It
+describes homogeneous linear equations with constant coefficients. 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. This article discusses a new approach to solving ordinary differential equa-
+tions using the simplest elementary operations. The calculations can be cumbersome, but we
+do not lose particular solutions to differential equations.
+Let f(x), g(x) be Riemann integrable functions; y = y(x), y′(x) = dy(x)
+dx , y′′(x) = d2y(x)
+dx2 ,
+log y = ln y = loge y; C, C1, C2, C1,1, ... , C1,7, C2,1, C2,2 is an integration constant. The symbol
+⇒ between two formulas will mean that the second formula follows from the first one.
+1
+First order differential equations
+1.1. Linear inhomogeneous first order differential equation:
+y′(x) + f(x)y(x) = g(x).
+(1)
+Logarithmic integration method. In equation (1) the function g(x) is not identically zero. Then
+y(x) be not identifically zero. Then with equations (1) we get
+y′(x)
+y(x) + f(x) = g(x)
+y(x),
+⇒
+(log |y(x)|)′ + f(x) = g(x)
+y(x),
+⇒
+(log |y(x)|)′ +
+��
+f(x)dx
+�′
+= g(x)
+y(x),
+⇒
+(2)
+(log |y(x)|)′ +
+�
+log e
+�
+f(x)dx�′
+= g(x)
+y(x),
+⇒
+�
+log |y(x)| + log e
+� f(x)dx�′
+= g(x)
+y(x),
+⇒
+1
+
+�
+log
+�
+|y(x)|e
+� f(x)dx��′
+= g(x)
+y(x),
+⇒
+�
+y(x)e
+�
+f(x)dx�′
+y(x)e
+� f(x)dx
+= g(x)
+y(x),
+⇒
+�
+y(x)e
+� f(x)dx�′
+= g(x)e
+� f(x)dx,
+⇒
+y(x)e
+� f(x)dx =
+�
+g(x)e
+� f(x)dxdx + C,
+⇒
+y(x) = e−
+�
+f(x)dx
+��
+g(x)e
+�
+f(x)dxdx + C
+�
+.
+(3)
+Remark 1.1.1. 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. 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.
+Let C1 = −F(x0), where F(x) is a function that has
+property F ′(x) = f(x). Then the equation (5) can be represented as
+(log |y(x)|)′ +
+�� x
+x0
+f(t)dt
+�′
+= g(x)
+y(x),
+⇒
+(log |y(x)|)′ +
+�
+log e
+� x
+x0 f(t)dt�′
+= g(x)
+y(x),
+⇒
+�
+log |y(x)| + log e
+� x
+x0 f(t)dt�′
+= g(x)
+y(x),
+⇒
+�
+log
+�
+|y(x)|e
+� x
+x0 f(t)dt��′
+= g(x)
+y(x),
+⇒
+�
+y(x)e
+� x
+x0 f(t)dt�′
+y(x)e
+� x
+x0 f(t)dt
+= g(x)
+y(x),
+⇒
+�
+y(x)e
+� x
+x0 f(t)dt�′
+= g(x)e
+� x
+x0 f(σ)dσ,
+⇒
+y(x)e
+� x
+x0 f(t)dt =
+� x
+x0
+g(τ)e
+� τ
+x0 f(σ)dσdτ + C,
+⇒
+y(x) = e− � x
+x0 f(t)dt
+�� x
+x0
+g(τ)e
+� τ
+x0 f(σ)dσdτ + C
+�
+.
+(6)
+If in the equation (6) we let C = y(x0), then we have the formula (4).
+2
+
+1.2. Bernoulli Differential equation:
+y′ + f(x)y = g(x)yα,
+(7)
+where α ∈ R\{0, 1}.
+Logarithmic integration method. Let y is not identically zero. Then from the equations (7)
+we obtain
+y′
+y + f(x) = g(x)
+y
+yα,
+⇒
+(log |y|)′ + f(x) = g(x)
+y
+yα,
+⇒
+(log |y|)′ +
+��
+f(x)dx
+�′
+= g(x)
+y
+yα,
+⇒
+(log |y|)′ +
+�
+log e
+�
+f(x)dx�′
+= g(x)
+y
+yα,
+⇒
+�
+log |y| + log e
+� f(x)dx�′
+= g(x)
+y
+yα,
+⇒
+�
+log
+�
+|y|e
+� f(x)dx��′
+= g(x)
+y
+yα,
+⇒
+(8)
+�
+ye
+� f(x)dx�′
+ye
+� f(x)dx
+= g(x)
+y
+yα,
+⇒
+�
+ye
+�
+f(x)dx�′
+= g(x)e
+�
+f(x)dxyα,
+⇒
+�
+ye
+� f(x)dx�′
+= g(x)e(1−α) � f(x)dxyαeα � f(x)dx,
+⇒
+�
+ye
+� f(x)dx�′
+= g(x)e(1−α) � f(x)dx �
+ye
+� f(x)dx�α
+,
+⇒
+�
+ye
+� f(x)dx�′
+�
+ye
+�
+f(x)dx�α = g(x)e(1−α) � f(x)dx,
+⇒
+�
+1
+1 − α
+�
+ye
+�
+f(x)dx�1−α�′
+= g(x)e(1−α)
+�
+f(x)dx,
+⇒
+1
+1 − α
+��
+ye
+�
+f(x)dx�1−α�′
+= g(x)e(1−α)
+�
+f(x)dx,
+⇒
+��
+ye
+� f(x)dx�1−α�′
+= (1 − α) g(x)e(1−α) � f(x)dx,
+⇒
+(9)
+�
+ye
+�
+f(x)dx�1��α
+= (1 − α)
+�
+g(x)e(1−α)
+�
+f(x)dxdx + C,
+⇒
+ye
+�
+f(x)dx =
+�
+(1 − α)
+�
+g(x)e(1−α)
+�
+f(x)dxdx + C
+�
+1
+1−α
+,
+⇒
+3
+
+y = e− � f(x)dx
+�
+(1 − α)
+�
+g(x)e(1−α) � f(x)dxdx + C
+�
+1
+1−α
+.
+(10)
+Remark 1.2.1. At the beginning of the course of the method, we assumed that y be not
+identically zero 0. It follows that the equation (7) has a particular solution y = 0, if α ∈ (0, 1).
+Remark 1.2.2. (The second version of the logarithmic method.)
+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).
+(11)
+The equation (11) is a linear inhomogeneous first order differential equation, with respect to the
+function y1−α. 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
+�
+.
+(12)
+The formula (12) implies the solution (10).
+Remark 1.2.3. (The third version of the logarithmic method.)
+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.
+(13)
+The equation (13) is similar to the equation (9).
+1.3. The equation of the form:
+y′ + f(x)eβy = g(x),
+(14)
+4
+
+where β ∈ R\{0}.
+Logarithmic integration method. 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).
+(15)
+The equation (15) is a linear inhomogeneous first order differential equation, with respect to the
+function e−βy. Its solution, by the formula (3), has the form
+e−βy = e−β � g(x)dx
+�
+β
+�
+f(x)eβ � g(x)dxdx + C
+�
+.
+(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
+�
+.
+(17)
+2
+Second order differential equations
+2.1. Linear homogeneous second order differential equation:
+y′′ + by′ + cy = 0,
+(18)
+where b ∈ R, c ∈ R.
+Let y is not identically zero. 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. Let in the equation (19):
+(log |y|)′ = z.
+Then we have equation (19) in the form
+z′ + z2 + bz + c = 0,
+⇒
+(20)
+z′
+z2 + bz + c = −1.
+(21)
+Case 1. b2 − 4c > 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
+

QNAzT4oBgHgl3EQflf3i/content/tmp_files/load_file.txt

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+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'}

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-
+∗[email protected],[email protected]
+†[email protected], [email protected]
+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
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+

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@@ -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: [email protected]
+† Corresponding author: [email protected]
+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<pch,jet
+T
+<80 GeV
+R=0.4, |ηch,jet|<0.5
+80<pch,jet
+T
+<100 GeV
+ALICE [PRL 128, no.10, 102001 (2022)]
+JETSCAPE [MATTER (vacuum)]
+R=0.4, |ηch,jet|<0.5
+60<pch,jet
+T
+<80 GeV
+0.2
+0.3
+0.4
+zg
+0.5
+1.0
+1.5
+MC/Exp.
+0.2
+0.3
+0.4
+zg
+0.2
+0.3
+0.4
+0.5
+zg
+FIG. 1. (Color online) Distributions of jet splitting momentum fraction zg for charged jets in p+p collisions at √s = 5.02 TeV
+and the ratios for different jet cone size R, and pch,jet
+T
+range. The Soft Drop parameters are zcut = 0.2 and β = 0. The
+solid lines and circles with statistical error bars show the results from JETSCAPE and the experimental data from ALICE
+Collaboration [57], respectively. The bands indicate the systematic uncertainties of the experimental data.
+0.2
+0.3
+0.4
+zg
+0.0
+0.5
+1.0
+1.5
+2.0
+PbPb
+pp
+�
+1
+σjet
+dσSD,jet
+dzg
+�
+PbPb, √sNN = 5.02 TeV
+Charged Jets, anti-kt
+Soft Drop zcut = 0.2, β = 0
+0-10%
+R=0.2, |ηch,jet|<0.7
+60<pch,jet
+T
+<80 GeV
+0.2
+0.3
+0.4
+zg
+0-10%
+R=0.4, |ηch,jet|<0.5
+80<pch,jet
+T
+<100 GeV
+ALICE [PRL 128, no.10, 102001 (2022)]
+JETSCAPE [MATTER+LBT (w/ coherence)]
+JETSCAPE [MATTER+LBT (w/o coherence)]
+0.2
+0.3
+0.4
+0.5
+zg
+30-50%
+R=0.4, |ηch,jet|<0.5
+60<pch,jet
+T
+<80 GeV
+FIG. 2. (Color online) Ratios of zg distributions for charged jets between Pb+Pb and p+p collisions at √sNN = 5.02 TeV
+for different centrality, jet cone size R, and pch,jet
+T
+range. 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 ALICE Collaboration [57] are shown by squares with
+statistical errors (bars) and systematic uncertainties (bands).
+for each condition.
+The denominator is calculated for
+p+p collisions, and the numerator is for a given central-
+ity class of A+A collisions, where ⟨Ncoll⟩ is the average
+number of binary nucleon-nucleon collisions in the given
+centrality class.
+Figure 6 shows jet RAA as a function of pjet
+T
+for dif-
+ferent rg intervals. As already described in Ref. [1], for
+the case of inclusive jets (top left plot in Fig. 6), no clear
+differences due to coherence effects are observed in the
+jet RAA. Note that the overall medium coupling param-
+eter αfix
+s
+is adjusted separately for each setup (αfix
+s
+= 0.3
+for the case with coherence effects, and 0.25 for the case
+without coherence effects). It should also be noted that
+our simulations, which do not contain any nuclear shad-
+owing effects, have a somewhat sharper rise than the AT-
+LAS data and are somewhat consistent with the data
+from CMS.
+Moving to the case of Soft Drop fail (top middle plot
+in Fig. 6), one notices that the data clearly prefer the
+simulation with coherence as opposed to that without
+coherence.
+The reduced suppression for the case with
+coherence can be understood under the assumption that
+the prong structure is established in the high virtuality or
+MATTER stage. In this stage, the effective jet quench-
+
+000000
+582000000
+5828
+0.2
+0.3
+0.4
+zg
+0.0
+0.5
+1.0
+1.5
+2.0
+AuAu
+pp
+�
+1
+σjet
+dσSD,jet
+dzg
+�
+AuAu 0-10%, √sNN = 200 GeV
+Charged Jets, anti-kt
+Soft Drop zcut = 0.2, β = 0
+R=0.2, |ηch,jet|<0.7
+0.2
+0.3
+0.4
+0.5
+zg
+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. 3. (Color online) Ratios of zg 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 of JETSCAPE with 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.
+0
+1
+2
+3
+4
+5
+6
+1
+σjet
+dσSD,jet
+d(rg/R)
+pp, √s = 5.02 TeV
+Charged Jets, anti-kt
+Soft Drop zcut = 0.2, β = 0
+R=0.2, |ηch,jet|<0.7
+60<pch,jet
+T
+<80 GeV
+Soft Drop zcut = 0.2, β = 0
+R=0.4, |ηch,jet|<0.5
+60<pch,jet
+T
+<80 GeV
+ALICE [PRL 128, no.10, 102001 (2022)]
+JETSCAPE [MATTER (vacuum)]
+Soft Drop zcut = 0.4, β = 0
+R=0.4, |ηch,jet|<0.5
+60<pch,jet
+T
+<80 GeV
+0.00
+0.05
+0.10
+0.15
+rg
+0.5
+1.0
+1.5
+MC/Exp.
+0.0
+0.1
+0.2
+0.3
+rg
+0.00
+0.05
+0.10
+0.15
+0.20
+rg
+FIG. 4. (Color online) Distributions of jet splitting radius rg for charged jets in p+p collisions at √s = 5.02 TeV and the ratios
+for different jet cone size R, and pch,jet
+T
+range. The Soft Drop parameters are zcut = 0.2 and β = 0. The solid lines and circles
+with statistical error bars show the results from JETSCAPE and the experimental data from the ALICE Collaboration [57],
+respectively. The bands indicate the systematic uncertainties of the experimental data.
+ing strength with the virtuality dependence ˆq · f(Q2) is
+smaller for the case with coherence effects compared to
+that without. For the case without coherence, the larger
+value of ˆq · f(Q2) = ˆq in the MATTER stage leads to the
+formation of wider prongs, leading to a reduction in the
+number of jets that fail the Soft Drop condition.
+It bears repeating yet again: The comparisons of simu-
+lations to data presented in this paper do not include any
+parameter tuning to fit any substructure data. All pa-
+rameter tuning was carried out in the calculation of the
+single high-pT particle and jet suppressions in Ref. [1].
+All simulation results presented in this paper are predic-
+tions.
+Figure 7 shows jet RAA as a function of rg for different
+pjet
+T
+intervals.
+The yellow-shaded regions in the figure
+indicate the areas of bins containing contributions from
+jets with a transverse scale µ⊥ ≊ pjet
+T rg ⪅ 1 GeV, where
+the perturbative description of parton splitting in the
+model is not valid. To regulate the infra-red singularity
+in the splitting function, the model needs to specify a
+minimum cut-off scale for resolvable splitting [172], which
+here is Qmin = 1 GeV.
+In other words, the jet structure of the yellow-shaded
+region is governed by the effects from non-perturbative
+dynamics, namely hydrodynamic evolution of the soft
+
+000000
+582000000
+5829
+0.00
+0.05
+0.10
+0.15
+rg
+0.0
+0.5
+1.0
+1.5
+2.0
+PbPb
+pp
+�
+1
+σjet
+dσSD,jet
+d(rg/R)
+�
+PbPb, √sNN = 5.02 TeV
+Charged Jets, anti-kt
+0-10%
+Soft Drop zcut = 0.2, β = 0
+R=0.2, |ηch,jet|<0.7
+60<pch,jet
+T
+<80 GeV
+0.0
+0.1
+0.2
+0.3
+rg
+30-50%
+Soft Drop zcut = 0.2, β = 0
+R=0.4, |ηch,jet|<0.5
+60<pch,jet
+T
+<80 GeV
+ALICE [PRL 128, no.10, 102001 (2022)]
+JETSCAPE [MATTER+LBT (w/ coherence)]
+JETSCAPE [MATTER+LBT (w/o coherence)]
+0.00
+0.05
+0.10
+0.15
+0.20
+rg
+30-50%
+Soft Drop zcut = 0.4, β = 0
+R=0.4, |ηch,jet|<0.5
+60<pch,jet
+T
+<80 GeV
+FIG. 5. (Color online) Ratios of rg distributions for charged jets between Pb+Pb and p+p collisions at √sNN = 5.02 TeV for
+different centrality, jet cone size R, soft drop parameter zcut, and pch,jet
+T
+range. 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 ALICE Collaboration [57] are shown by squares with statistical errors (bars) and systematic
+uncertainties (bands).
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+RAA
+Inclusive
+PbPb, 0-10%, √sNN = 5.02 TeV
+anti-kt, R = 0.4, |yjet| < 2.1
+Soft Drop, zcut = 0.2, β = 0.0
+rg = 0 (Soft Drop Fail)
+0 < rg < 0.022
+200
+400
+600
+800 1000
+pjet
+T (GeV)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+RAA
+0.022 < rg < 0.1153
+JETSCAPE
+[MATTER+LBT (w/ coherence)]
+JETSCAPE
+[MATTER+LBT (w/o coherence)]
+200
+400
+600
+800 1000
+pjet
+T (GeV)
+0.1153 < rg < 0.2642
+ATLAS [arXiv:2211.11470]
+CMS (|ηjet| < 2.0) [JHEP 05, 284 (2021)]
+200
+400
+600
+800 1000
+pjet
+T (GeV)
+0.2642 < rg < 0.4
+FIG. 6.
+(Color online) Nuclear modification factor RAA as a function of pjet
+T
+for inclusive jets, jets failing the Soft Drop
+condition (rg = 0), and groomed jets with different splitting radius rg 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. The results are compared to ATLAS data [58] (squares) and CMS data for |ηjet| < 2.0 [39] (triangles) are
+shown with statistical errors (bars) and systematic uncertainties (bands).
+thermalized portion of jets (not modeled in this study),
+hadronization and subsequent dynamics, rather than the
+
+000000
+582000000
+58210
+perturbative parton-level dynamics. Note that one needs
+to examine the results shown in Figs. 5 and 6 with the
+same considerations for regions with small rg or small
+pjet
+T . In this regard, it should also be noted that the re-
+sults in Fig. 5 are for charged jets.
+Given that the calculation with coherence (solid red
+lines in the Figs. 6 and 7) have a larger αfix
+s
+than calcu-
+lations without coherence, there are a larger number of
+softer near collinear partons branched in the later low-
+virtuality stage, which leads to an enhancement of the
+RAA at non-perturbatively low rg, indicated by the yel-
+low band in Fig. 7. As a result, in this region, the solid
+red line (RAA with coherence) will always exceed the
+dashed green line (RAA calculated without coherence).
+This excess at very low rg, which emanates from the
+lack of non-perturbative modification of the jets in the
+simulation, also strongly affects the RAA as a function
+of pjet
+T
+for 0 < rg < 0.022, which is the top right plot
+in Fig. 6. As pjet
+T
+increases, the deviation of the curves
+from the data increases as more soft partons pile up at
+low momentum around the jet. This deviation will be
+addressed when a non-perturbative modification for soft
+partons radiated from the jet is included in the simula-
+tions.
+At very large rg, with rg > 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)
+10<pch,jet
+T
+<30 GeV
+30<pch,jet
+T
+<50 GeV
+FIG. 11. (Color online) Ratios of jet fragmentation functions for charged jets with R = 0.4 and |ηch,jet| < 1.0 between 0-10%
+Au+Au and p+p collisions at √sNN = 200 GeV as a function of z (left) and ptrk
+T
+(right) from MATTER+LBT of JETSCAPE
+with coherence effects. 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 fact, no new simulations were run for this paper. The
+presented results were calculated from the simulations
+carried out for Ref. [1].
+To investigate the contribution of coherence effects
+based on the ability of the medium to resolve the par-
+tons radiated from splits at high energy and virtuality,
+we performed numerical simulations for two cases, with
+and without coherence effects. These coherence effects
+are implemented as the Q2-dependent modulation fac-
+tor in the medium-modified splitting function and give
+a drastic reduction of the interaction with the medium
+with increasing parton virtuality.
+The distribution of jet splitting momentum fraction
+(zg) shows almost no visible modification due to the
+
+000000
+582000000
+582000000
+58214
+medium effects for any kinematic configurations in both
+cases with and without coherence effects. This extremely
+small sensitivity to the medium effects is consistent with
+the experimental data taken by ALICE at the LHC. Our
+predictions for future RHIC measurements also show no
+significant modification.
+Then, we presented the observables related to the jet
+splitting radius (rg).
+In comparison with the ALICE
+data, both results with and without coherence effects sat-
+isfactorily capture the monotonically decreasing behavior
+with increasing radius and give good agreement. Here, no
+conclusions about coherent effects could be drawn from
+this analysis in comparison with the data from ALICE.
+We reiterate again that our simulations reduce to and re-
+produce the zg and rg distributions in the absence of the
+medium, in comparison with data from p+p collisions.
+In comparison with data from ATLAS [58], we demon-
+strated that coherence effects manifest, even at the qual-
+itative behavior level, in rg-dependent RAA with finer
+binning. In both the RAA as a function of pjet
+T
+for dif-
+ferent bins of the angle rg as well as the RAA as a func-
+tion of rg in different pjet
+T
+bins, there is a clear difference
+between simulations with and without coherence effects.
+The experimental data clearly prefer simulations with co-
+herence effects. This indicates that the scattering with
+the medium constituents at high virtuality is reduced due
+to the finer scale of the medium probed by the jet parton.
+Finally, we found that coherence effects may also be
+visible as a more prominent enhancement at large z in
+the modification pattern of the jet fragmentation func-
+tions.
+The energy loss of hard leading partons, which
+form the jet core components with large transverse mo-
+mentum, is highly suppressed by coherence effects due to
+their large virtualities. The data have a slight preference
+for simulations with coherence if one restricts attention
+to particles with z ⪆ 0.1. For both the case with and
+without coherence, the simulations produce fewer parti-
+cles at very small z (z ⪅ 0.02), with the case without
+coherence performing marginally better.
+This paper constitutes the third installment of jet and
+hadron-based observables from the MATTER+LBT sim-
+ulations in the JETSCAPE framework [1, 124]. In all
+three of these papers, including the current effort, we
+have demonstrated wide-ranging agreement for the hard
+sector of jets, between simulations, typically with coher-
+ence and experimental data. The only remaining issues
+within the hard sector of the jet are related to coinci-
+dence measurements. These will be presented in a future
+effort.
+In terms of physics included within these simulations,
+the one remaining component is the very soft sector of
+jets. In the current effort, this was pointed out in the
+discussion of the low rg section of the rg dependent RAA,
+and the low-z and low-pT sector of the jet fragmenta-
+tion function. This requires incorporating an energy de-
+position scheme in which partons with energy compara-
+ble to the ambient temperature are converted into an
+energy-momentum source term and then included back
+in the hydrodynamic calculation.
+As may be obvious,
+these simulations require close to a single hydro run per
+hard event and, as such, are very computationally de-
+manding. Various schemes to approximately incorporate
+soft physics without the need for full hydrodynamic sim-
+ulation are currently underway. The analysis of certain
+jet-based observables predominantly sensitive to the soft
+sector of jets will be carried out after these efforts are
+complete.
+ACKNOWLEDGMENTS
+This work was supported in part by the National Sci-
+ence Foundation (NSF) within the framework of the
+JETSCAPE collaboration, under grant number OAC-
+2004571 (CSSI:X-SCAPE). It was also supported un-
+der ACI-1550172 (Y.C. and G.R.), ACI-1550221 (R.J.F.,
+F.G., and M.K.), ACI-1550223 (U.H., L.D., and D.L.),
+ACI-1550225 (S.A.B., T.D., W.F., R.W.), ACI-1550228
+(J.M., B.J., P.J., X.-N.W.), and ACI-1550300 (S.C.,
+A.K., J.L., A.M., H.M., C.N., A.S., J.P., L.S., C.Si., I.S.,
+R.A.S. and G.V.); by PHY-1516590 and PHY-1812431
+(R.J.F., M.K. and A.S.); it was supported in part by
+NSF CSSI grant number OAC-2004601 (BAND; D.L. and
+U.H.); it was supported in part by the US Department of
+Energy, Office of Science, Office of Nuclear Physics un-
+der grant numbers DE-AC02-05CH11231 (X.-N.W.), DE-
+FG02-00ER41132 (D.O), DE-AC52-07NA27344 (A.A.,
+R.A.S.), DE-SC0013460 (S.C., A.K., A.M., C.S., I.S.
+and
+C.Si.),
+DE-SC0021969
+(C.S.
+and
+W.Z.),
+DE-
+SC0004286 (L.D., U.H. and D.L.), DE-SC0012704 (B.S.),
+DE-FG02-92ER40713 (J.P.) and DE-FG02-05ER41367
+(T.D., W.F., J.-F.P., D.S. and S.A.B.). The work was
+also supported in part by the National Science Founda-
+tion of China (NSFC) under grant numbers 11935007,
+11861131009 and 11890714 (Y.H. and X.-N.W.), un-
+der grant numbers 12175122 and 2021-867 (S.C.), by
+the Natural Sciences and Engineering Research Coun-
+cil of Canada (C.G., M.H., S.J., and G.V.), by the
+Office of the Vice President for Research (OVPR) at
+Wayne State University (Y.T.), by JSPS KAKENHI
+Grant No. 22K14041 (Y.T.), by the S˜ao Paulo Research
+Foundation (FAPESP) under projects 2016/24029-6,
+2017/05685-2 and 2018/24720-6 (A. L. and M.L.), and
+by the University of California, Berkeley - Central China
+Normal University Collaboration Grant (W.K.).
+U.H.
+would like to acknowledge support by the Alexander
+von Humboldt Foundation through a Humboldt Research
+Award. C.S. acknowledges a DOE Office of Science Early
+Career Award. Computations were carried out on the
+Wayne State Grid funded by the Wayne State OVPR.
+The bulk medium simulations were done using resources
+provided by the Open Science Grid (OSG) [174, 175],
+which is supported by the National Science Foundation
+award #2030508. Data storage was provided in part by
+the OSIRIS project supported by the National Science
+Foundation under grant number OAC-1541335.
+
+15
+0
+10
+20
+30
+40
+pjet
+T (GeV)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+RAA
+AuAu 0-10%, √sNN = 200 GeV
+anti-kt
+Full Jet
+|ηjet|<1
+0
+10
+20
+30
+40
+50
+pch,jet
+T
+(GeV)
+JETSCAPE
+MATTER+LBT (w/ coherence)
+Charged Jet
+pch,lead
+T
+>5 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. The solid, dashed, and dash-dotted lines with statistical error bars
+show the results for R = 0.2, R = 0.4, and R = 0.6, respectively.
+Appendix: Jets suppression at RHIC
+For the benchmarking purposes for our jet substruc-
+ture results in Au+Au collisions at √sNN = 200 GeV
+presented in the main body of the paper, we also show
+the predictions of RAA for inclusive full and charged jets
+from the same event generation by MATTER+LBT with
+coherence effects in Fig. 12.
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+arXiv:2301.04074v1  [quant-ph]  10 Jan 2023
+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
+Microcavities have been shown to influence the reactivity of molecular ensembles by strong cou-
+pling of molecular vibrations to quantized cavity modes. 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. 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. 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. 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. It is argued that the cavity “catalyzes” the reaction by transfer of virtual photons to
+the molecule. 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. The latter is traced back to complex entanglement dynamics of the
+ensemble, which we quantify by means of von Neumann–entropies. 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.
+I.
+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].
+In one class of promising ex-
+periments, molecular vibrations interact strongly with
+the ground state of infrared active cavity modes.
+In
+this vibrational strong coupling (VSC) regime, signifi-
+cantly altered thermal ground state chemistry has been
+observed[8–12]. While in early examples of VSC–altered
+reactivity, rates usually were retarded, meanwhile also
+experiments exist with accelerated rates[13].
+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]. 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]. At least in quantum me-
+chanical treatments, typically single molecules are con-
+sidered and the transition to ensembles of N molecules
+∗ ericwfi[email protected]
+is mimicked via effective single–molecule models with
+scaled molecule–cavity couplings[25]. 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.
+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.
+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.
+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.
+We extract approximate H–transfer rates from
+the numerically obtained dynamics, which increase with
+light–matter coupling and ensemble size.
+The cavity
+“catalytic��� effect in the single–molecule limit, i.e., 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. 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]. 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. 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.[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.
+The paper is organized as follows. In Sec.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. In Sec.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.
+Finally, Sec.IV summarizes this work.
+Most
+of the numerical details and parameters as well as some
+further results can be found in the Supplementary Infor-
+mation (SI).
+II.
+THEORY AND MODEL
+A.
+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. Fig.1a). 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
+.
+(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. SI, Sec.I). As shown in the SI, Sec.I,
+this Hamiltonian is a one–dimensional approximation ob-
+tained from a two–dimensional Hamiltonian developed
+originally in Ref.[31].
+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 = −0.572 a0,
+which relates to the enol (OH) configuration, and a local
+minimum at q = 0.947 a0 corresponding to the enethiol
+(SH) configuration of TAA. 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 = 0.0, i.e.,
+the classical energy difference between the two isomers is
+517 cm−1. 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 = 966.3 cm−1 and ε0 = 1092.8 cm−1, respec-
+tively, giving a corresponding quantum mechanical en-
+ergy difference of ∆ε10 = 126.5 cm−1.
+Details on the
+transfer potential, V (q), with all numerical parameters
+are provided in the SI, Sec.I.
+The second term in Eq.(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.e., ℏωc
+=
+∆ε10
+=
+126.5 cm−1. Further, the light–matter interaction term,
+ˆHSC, is given by
+ˆHSC =
+�
+2ωc
+ℏ gN d(q) xc
+,
+(4)
+with collective transfer coordinate, q = (q1 . . . qN). 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. 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. The latter is a lin-
+ear approximation to the dipole function given in Ref.[31]
+and reads
+d(qi) = d0 + dS (qi − q0)
+,
+(5)
+with parameters d0, dS and q0 given in the SI, Sec. I.
+We note that the dipole moment takes positive values
+at both the enol minimum (dOH = 1.678 ea0) and the
+enethiol minimum (dSH = 1.482 ea0), i.e., d(qi) changes
+smoothly without sign change along the (classical) reac-
+tion path. Note further that in Eq.(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. The single–molecule coupling
+
+3
+FIG. 1.
+(a) Schematic setup for thioacetylacetone (TAA) isomerization in a single–mode cavity model.
+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. 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. The
+single–molecule cPES is a function of the H–transfer coordinate, q, and a cavity displacement coordinate, xc. The uncoupled
+case with η = 0.0 is shown with lowest–state energies, E0 = ε0 + ℏωc
+2 , and, E1 = ε1 + ℏωc
+2 , where ε0 = 966.3 cm−1 and
+ε1 = 1092.8 cm−1 are the lowest two eigenenergies of the one–dimensional H–transfer Hamiltonian, ˆHS, with potential, V (q).
+We set the cavity frequency to be resonant with the lowest molecular vibrational transition, i.e., ℏωc = ∆ε10 = 126.5 cm−1.
+constant, g, in Eq.(4) has dimension of an electric field
+strength and is modeled here as, g = ℏωc
+d10 η[34], where,
+d10 = ⟨ψ0|d(q)|ψ1⟩ = 0.042 ea0, is the fundamental tran-
+sition dipole moment of the H–transfer system. 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 = 0.0 (no molecule–cavity coupling) and η = 0.09. The
+vibrational strong coupling (VSC) regime is determined
+by 0 < η < 0.1.[35] Note, we do not refer to alterna-
+tive definitions of VSC related to dissipation strengths
+here.[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. 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
+.
+(7)
+For the uncoupled, single–molecule case (η = 0.0, N =
+1), the cPES, V (q, xc), is shown in Figs.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. These states are simple product states for η = 0.0 and
+therefore also correspond to the enol and enethiol forms,
+weakly delocalized with small contributions at the other
+minimum as indicated.
+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.
+SI, Sec. II for details) as imple-
+mented in the Heidelberg MCTDH package[41]. 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).
+The latter corresponds to the ground state of the bare
+cavity with zero physical photons.
+We will discuss
+differences to photon number expectation values for a
+molecule–cavity system at finite light–matter interaction
+strength (η > 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.
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+
+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)
+∗ ericwfi[email protected]
+
+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 = (õS q, õB 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 = õS
+,
+s′′
+1 = 0
+,
+s′
+2(q) = õB
+�
+2 aS q + 3 bS q2�
+,
+s′′
+2(q) = õB (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,
+õS 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).
+

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+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.
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+Royal
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+of
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+

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@@ -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)<p
+�
+rp
+�
+(k − 1)2�
++ rp (k + 1 − 3n)
+�
+,
+=
+−p|JΓ| +
+p
+�
+k=2
+rp
+�
+(k − 1)2�
++
+3n−2
+�
+k=2
+rp (k + 1 − 3n) +
+4n−1
+�
+k=3n−1
+rp (k + 1 − 3n) ,
+=
+−p(2n − 2) +
+p−1
+�
+k=1
+rp
+�
+k2�
++
+3n−2
+�
+k=2
+(k + 1 − 3n + p) +
+4n−1
+�
+k=3n−1
+(k + 1 − 3n),
+=
+p−1
+�
+k=1
+rp
+�
+k2�
++
+4n−1
+�
+k=2
+(k + 1 − 3n) + p(2 − 2n + 3n − 3),
+=
+p−1
+�
+k=1
+rp
+�
+k2�
++ 3n − 2.
+Since rp (k2 − k + 2 − 3n) = 2−3n+p for k = 0, p the result follows. Observe
+that
+p−1
+�
+k=0
+rp
+�
+k2 − k + 2 − 3n
+�
+=
+p−1
+�
+k=1
+rp
+�
+k2�
++ p.
+The other two formulas are done analogously.
+□
+Compare this result with Theorem 5.1 in [4] when p = 4n + 1,
+p−1
+�
+k=0
+rp
+�
+k2 + k + 1 − n
+�
+=
+p−1
+�
+k=1
+rp
+�
+k2�
+− p = p(2n − 1).
+Remark 5.2 The first two formulas in Theorem 5.1 are actually valid for
+any n but the third one is only valid for n > 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
+

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+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. Neural net-
+works have extensively been used before as approximators; in this work, we make
+a step further and use them for the first time as abstractions. 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. For this purpose, we employ a counterexample-guided inductive synthesis
+procedure. We show that this produces a neural ODE with non-deterministic distur-
+bances that constitutes a formal abstraction of the concrete model under analysis.
+This guarantees a fundamental property: if the abstract model is safe, i.e., free from
+any initialised trajectory that reaches an undesirable state, then the concrete model
+is also safe. 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. We
+demonstrate that our approach performs comparably to the mature tool Flow* on
+existing benchmark nonlinear models. 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.
+1
+Introduction
+Dynamical models describe processes that are ubiquitous in science and engineering. 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]. 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. 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]. 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.
+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]. For most practical models that contain nonlinear terms [81, 105],
+∗The authors are listed alphabetically
+36th Conference on Neural Information Processing Systems (NeurIPS 2022).
+arXiv:2301.11683v1  [cs.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. 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]. 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]. In addition, existing formal approaches rely on symbolic reasoning
+techniques that explicitly leverage the structure of the dynamics. This results in verification procedures
+that are bespoke to restricted classes of models. For example, it is common for formal verification
+procedures to require the input model to be Lipschitz continuous. 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]. 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.
+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]. 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. Our method follows an approach that
+constructs abstract, finite-state machines whose states are augmented with continuous linear dynamics
+and non-deterministic drifts. 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; this process has been widely applied in formal
+verification [8,15,16,21,46,58,65,73,91,94,99,100,102].
+Hybridising involves partitioning the state space and computing a local overapproximation of the
+concrete model within each region of the partition. Common approaches for hybridisation partition the
+state space by tuning the granularity of rectangular or simplicial meshes, until a desired approximation
+error is attained. This may yield abstract hybrid automata that are too large in the number of discrete
+states to be effectively verified. 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. 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].
+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. 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. We show that this ultimately enables
+verifying nonlinear dynamical models using efficient safety verifiers for hybrid automata with affine
+dynamics (cf. Figure 1).
+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]. 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; 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.
+If the latter disproves the bound, then it produces a counterexample which its added to the set of
+samples and the loop is repeated. If it certifies the bound, then the procedure returns a neural network
+approximation and a sound upper-bound on the error. 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.
+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
+...
+˙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. Section 5, NL2). The
+concrete dynamics are abstracted by a neural ODE with ReLU activation functions and a certified
+upper-bound on the approximation error. This characterises a polyhedral partitioning and defines a
+hybrid automaton with affine dynamics and additive non-deterministic drift. Flowpipe propagation is
+finally performed through a region of non-Lipschitz continuity.
+models, and compare our approach with Flow*, the state-of-the-art verification tool for nonlinear
+models [34,35,37]. 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]. 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. 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. 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].
+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;
+• 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;
+• we define a translation from neural abstractions defined using ReLU activation functions to
+hybrid automata with affine dynamics and additive non-determinism;
+• we implement our approach2 and demonstrate its comparable performance w.r.t. 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.
+We consider there to be no significant negative societal impact of our work.
+2The code is available at https://github.com/aleccedwards/neural-abstractions-nips22.
+3
+
+0.5
+0.0
+-0.5
+0.5
+0.0
+0.52
+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.
+Definition 1 (Dynamical Model). 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. 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). 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. 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
+
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+16
+
+Checklist
+1. For all authors...
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s
+contributions and scope? [Yes]
+(b) Did you describe the limitations of your work? [Yes] Please see Section 5.2
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] See 1
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to
+them? [Yes]
+2. If you are including theoretical results...
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] See 2
+(b) Did you include complete proofs of all theoretical results? [Yes] See 2
+3. 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. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+(a) If your work uses existing assets, did you cite the creators? [Yes] We have cited all
+used tools.
+(b) Did you mention the license of the assets? [Yes] See Section 5
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes]
+The code will be included in the supplementary material.
+(d) Did you discuss whether and how consent was obtained from people whose data you’re
+using/curating? [N/A]
+(e) Did you discuss whether the data you are using/curating contains personally identifiable
+information or offensive content? [N/A]
+5. If you used crowdsourcing or conducted research with human subjects...
+(a) Did you include the full text of instructions given to participants and screenshots, if
+applicable? [N/A]
+(b) Did you describe any potential participant risks, with links to Institutional Review
+Board (IRB) approvals, if applicable? [N/A]
+(c) Did you include the estimated hourly wage paid to participants and the total amount
+spent on participant compensation? [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
+