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+arXiv:2301.13623v1  [hep-th]  31 Jan 2023
+Unimodular Gravity in Covariant Formalism
+J. Klusoň† and B. Matouš†‡ 1
+† Department of Theoretical Physics and Astrophysics, Faculty of Science,
+Masaryk University, Kotlářská 2, 611 37, Brno, Czech Republic
+‡ North-Bohemian Observatory and Planetarium in Teplice,
+Koperníkova 3062, 415 01, Teplice, Czech Republic
+Abstract
+In this short note we study unimodular gravity in Weyl-De Donder
+formalism. We find corresponding Hamiltonian and study consequence
+of the unimodular constraint on the conjugate covariant momenta. We
+also find covariant Hamiltonian for Henneaux-Teitelboim unimodular
+action and study corresponding equations of motion.
+1
+Introduction and Summary
+Unimodular gravity was firstly introduced by A. Einstein in his paper [3]
+published in 1916. In this work the unimodular constraint √−g = 1 was
+used as gauge fixing condition of general diffeomorphism in order to sim-
+plify calculations.
+Then it was shown in [1, 2] that imposing this condi-
+tion before the variation of Einstein-Hilbert action leads to the traceless
+equations of motion.
+As we review below these equations of motion are
+classically equivalent to the general relativity equations of motion with cru-
+cial difference that the cosmological constant appears as integration con-
+stant rather than true cosmological constant.
+This fact brings new hope
+how to solve cosmological constant problem which was however questioned
+in [4], 2 where it was argued that quantum corrections make the cosmo-
+logical constant ultraviolet sensitive in unimodular gravity as well. On the
+other hand it is important to stress that no definitive conclusions have been
+reached yet regarding this problem and unimodular gravity is still very inten-
+sively studied, for some works devoted to unimodular gravity, see for example
+[7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 22, 23, 24, 25, 26].
+1Email addresses: J. Klusoň: klu@physics.muni.cz, B. Matouš: bmatous@mail.muni.cz
+2For review of unimodular gravity, see for example [5, 6].
+1
+
+One of the most interesting aspects of unimodular gravity is the number
+of physical degrees of freedom. Naively, unimodular constraint √−g = 1 re-
+duces the number of independent components of metric to nine which could
+suggest that the number of physical degrees of freedom is less than in general
+relativity. On the other hand unimodular gravity is invariant under restricted
+diffeomorphism. Taking these two aspects together we find that the num-
+ber of local physical degrees of freedom is the same as in ordinary general
+relativity. This fact was proved with the help of the Hamiltonian analysis
+of unimodular gravity performed in [16, 17, 18, 19, 20, 21]. On the other
+hand as was shown in these papers standard analysis of unimodular gravity
+based on D + 1 splitting of target space-time is rather non-trivial and shown
+complexity of the canonical analysis of systems with constraints.
+Then one could ask the question how unimodular gravity could be de-
+scribed in covariant canonical formalism that is known as Weyl-De Donder
+theory [27, 28]. The key point of this formulation is that we treat all partial
+derivatives as equivalent when we define conjugate momenta. For example,
+if we have scalar field φ with Lagrangian density in D +1 dimensional space-
+time equal to L = −1
+2ηab∂aφ∂bφ − V (φ), we define the conjugate momentum
+as 3
+πa = ∂L
+∂∂aφ = −ηab∂bφ .
+Then covariant canonical Hamiltonian density is defined as
+H = πa∂aφ − L = −1
+2πaηabπb + V (φ) .
+Clearly such a form of Hamiltonian density preserves diffeomorphism invari-
+ance of the theory. This approach is known as multisymplectic field theory,
+see for example [29, 30, 31], for review, see [32] and for recent interesting
+application of this formalism in string theory, see [33, 34].
+It is clear that such covariant canonical formalism is especially suitable
+for manifestly covariant theories as for example general relativity. In fact,
+covariant canonical formalism of general relativity was found long time ago
+by P. Hořava [35]. This analysis was recently generalized to the case of F(R)
+gravity in [37] and further elaborated in [38].
+In this paper we apply this formalism for unimodular theory of gravity in
+D + 1 dimensions. This is non-trivial task due to the well known complex-
+ity of canonical analysis of unimodular gravity in non-covariant formalism.
+3We define ηab = diag(−1, 1, . . ., 1), a, b = 0, 1, . . ., D.
+2
+
+Further, it is also very interesting to study this system since it contains pri-
+mary unimodular constraint and it is non-trivial task how to deal with such
+systems in covariant canonical formalism. In more details, we include this
+primary constraint to the action with corresponding Lagrange multiplier.
+Then we derive corresponding equations of motion. Using these equations
+of motion we find that the unimodular constraint implies another constraint
+on the canonical conjugate momenta. Then we show that this constraint is
+equivalent to the vanishing of the trace of the Christoffel symbols which is
+characteristic property of unimodular theory of gravity [10]. This is nice and
+non-trivial result. On the other hand the Lagrange multiplier corresponding
+to the primary constraint cannot be determined as in non-covariant canon-
+ical formalism by imposing condition of the preservation of the secondary
+constraint due to the fact that the equations of motion for conjugate mo-
+menta are in the form of the divergence of these momenta. For that reason
+we determine this constraint in the same way as in the Lagrangian formalism
+when we calculate the trace of the equations of motion. As a result we obtain
+equations of motion that are traceless and that do not depend on the cos-
+mological constant which is in agreement with the Lagrangian formulation
+of unimodular gravity.
+As the second step in our analysis we find covariant canonical formula-
+tion of Henneaux-Teitelboim formulation of unimodular gravity [16]. In this
+case we again identify covariant Hamiltonian together with set of primary
+constraints. Then we consider canonical form of the action and determine
+corresponding equations of motion. Solving these equations of motion we
+find that Lagrange multiplier is integration constant. In this case we repro-
+duce results well known from Lagrangian analysis. However we mean that
+this is nice and interesting application of the covariant canonical analysis to
+the constraint systems.
+Let us outline our results and suggest possible extension of this work. We
+found covariant Hamiltonian formalism for unimodular gravity. First of all
+we determined covariant Hamiltonian for general relativity action in D + 1
+dimensions where we again introduced variable f ab = √−ggab. At this place
+we would like to stress an importance of this result since it was not apri-
+ori known whether f ab is suitable for formulation of gravity in space-time of
+dimension different from 4. Then we imposed unimodular constraint using
+Lagrange multiplier method and then we studied corresponding equations
+of motion. We found that the consistency of the theory demands that the
+trace of conjugate momenta is zero. Then we showed that this is character-
+3
+
+istic property of unimodular gravity when we pass to Lagrangian formalism.
+Final we found covariant Hamiltonian for Henneaux-Teltelboim formulation
+of unimodular gravity. We identified primary constraints of the theory and
+then we studied equations of motion that follow from canonical form of the
+action. We showed that they precisely reproduce Lagrangian equations of
+motion that is nice consistency check of the covariant canonical formalism.
+We mean that the analysis presented in this paper suggests that covariant
+Hamiltonian formalism is very close to Lagrangian formalism and in some
+situations the covariant Hamiltonian formalism is more suitable than La-
+grangian one, as for example study of thermodynamics properties of horizon
+[36].
+It is also clear that there are more systems that could be analysed with
+the help of covariant canonical formalism. One possibility is to study Weyl
+invariant gravity in this formalism. Another possibility would be to perform
+analysis of theories of gravity with higher derivatives where the classical
+canonical analysis is very complicated, see for example [40].
+We hope to
+return to these problems in future.
+This paper is organized as follows.
+In the next section (2) we review
+properties of unimodular gravity.Then in section (3) we proceed to the co-
+variant canonical formulation of this theory. Finally in section (4) we perform
+covariant canonical formulation of Henneaux-Teltelboim unimodular gravity.
+2
+Brief Review of Unimodular Gravity
+In this section we review basic facts about unimodular gravity. For recent
+very nice and more detailed review, see for example [5, 6].
+Unimodular
+gravity is theory with the constraint √−g = 1. Clearly such a condition
+has a consequence on allowed differomorphism transformation. In fact, let
+us consider general transformation of coordinates
+x′a = xa + ξa(x)
+(1)
+that implies inverse relation
+xa = x′a − ξa(x) ≈ x′a − ξa(x′) + O(ξ2) ,
+(2)
+where a, b, c = 0, 1, . . . , D. Under these transformation the metric gab trans-
+form as
+g′
+ab(x) = gab(x) − ∂cgab(x)ξc(x) − gac(x)∂bξc(x) − ∂aξc(x)gcb(x)
+(3)
+4
+
+that implies following variation of metric
+δgab(x) = g′
+ab(x) − gab(x) = −gac∂bxc − ∂aξcgcb − ∂cgabξc
+so that the variation of the square root of the determinant of metric is equal
+to
+δ
+�
+− det g = −(2∂aξa − ∂cgabgbaξc)
+�
+− det g .
+(4)
+In case of unimodular gravity this variation should vanish and hence we
+obtain following condition on ξa in the form
+∇aξa = ∂aξa + 1
+2gac∂dgcaξd = 0 .
+(5)
+The most straightforward way how to find an action for unimodular gravity is
+to consider standard Einstein-Hilbert action with an unimodular constraint
+added
+S =
+1
+16π
+�
+dD+1x[√−g(R − 2¯Λ) + Λ(√−g − 1)] + Smatt ,
+(6)
+where Λ is Lagrange multiplier whose variation ensures unimodular condition
+and where ¯Λ is constant.
+Performing variation of the action (6) with respect to gab we obtain fol-
+lowing equations of motion
+1
+16π(Rab − 1
+2gab(R − 2¯Λ + Λ)) = Tab ,
+(7)
+where Tab is matter stress energy tensor defined as
+Tab = −
+1
+√−g
+δSmatt
+δgab
+.
+(8)
+The crucial point is that Λ is Lagrange multiplier that should be determined
+as a consequence of the equations of motion. To do this we perform the trace
+of the equation (7) to express Λ as
+Λ = (1 − D)
+1 + D R −
+32π
+D + 1T + 2¯Λ ,
+T ≡ gabTab .
+(9)
+5
+
+Inserting this result into (7) we obtain
+Rab −
+1
+D + 1gabR = 16π(Tab −
+1
+D + 1gabT) .
+(10)
+These equations of motion are trace-free and also most importantly they do
+not contain any information about cosmological constant ¯Λ.
+It is important to stress that even equations of motion of general relativ-
+ity without unimodular constraint imposed split into 9 trace-free equations
+of motion and one additional one.
+To see this consider general relativity
+equations of motion
+Rab − 1
+2gab(R − 2¯Λ) = 16πTab .
+(11)
+Taking the trace of this equation we can express R as
+R =
+2
+1 − D(16πT − (D + 1)¯Λ) .
+(12)
+Note that with the help of this equation we can rewrite (11) into trace-free
+form
+Rab −
+1
+D + 1Rgab = 16π(Tab −
+1
+D + 1Tgab) .
+(13)
+However we should again stress that (12) determines R as function of trace of
+matter stress energy tensor and true cosmological constant term in Einstein-
+Hilbert action while in case of unimodular gravity we express Λ-which is
+Lagrange multiplier and not constant, as function of R, T and ¯Λ, as follows
+from equation (9).
+In order to check equivalence between unimodular gravity and ordinary
+general relativity we should be able to reproduce equation (12) in case of
+unimodular gravity as well. We can do this by following procedure. Consider
+equations of motion (10) and rewrite them into the form
+Rab − 1
+2gabR = 16π(Tab −
+1
+D + 1gabT) +
+1 − D
+2(D + 1)Rgab .
+(14)
+Now we apply covariant derivative on both sides of the equations above
+and using the fact that the covariant derivative of Einstein tensor Gab =
+Rab − 1
+2gabR is zero we get
+1
+D + 1∇b(16πT − 1 − D
+2
+R) = 16π∇aTab .
+(15)
+6
+
+If we consider ordinary form of matter we obtain that divergence of stress
+energy tensor is zero as a consequence of matter equations of motion. Then
+the right side of the equation above is zero and the left side can be easily
+integrated with the result
+R =
+2
+1 − D(16πT + Ω) ,
+(16)
+where Ω now appears as true integration constant rather than the cosmolog-
+ical constant that was imposed in the theory by hand. In other words (16)
+is the last equation of motion of unimodular gravity and we fully recovered
+equivalence with general relativity however keeping in mind that we should
+still have to impose the condition √−g = 1 in the course of calculations.
+Having performed basic review of unimodular gravity we proceed in the
+next section to its formulation in the covariant Hamiltonian formalism.
+3
+Covariant Hamiltonian Formalism For D + 1
+dimensional Unimodular Gravity
+In this section we find covariant Hamiltonian formalism for unimodular grav-
+ity in D + 1 formalism.
+As usual in the covariant formalism we split the Einstein-Hilbert action
+into bulk and boundary terms. Since this procedure is well known, see for
+example [35, 36] and also recent generalization to the case of F(R) gravity
+[37] we write immediately final result
+L = Lbulk + Lsurf ,
+Lbulk =
+1
+16π
+√−g[Γh
+dkΓk
+ghggd − Γf
+fkΓk
+ghggh] +
++ 1
+16π
+¯Λ√−g +
+1
+16πλ(√−g − 1) ≡
+≡ Lquad +
+1
+16π
+¯Λ√−g +
+1
+16πλ(√−g − 1) ,
+Lsurf =
+1
+16π∂j[√−g(gikΓj
+ik − gijΓk
+ik)] ,
+(17)
+where Γa
+bc are Christoffel symbols
+Γa
+bc = 1
+2gad(∂bgdc + ∂cgdb − ∂cgab) ,
+(18)
+7
+
+and where ¯Λ is cosmological constant. Note that the presence of the term
+with Lagrange multiplier allows us to treat all components of metric as in-
+dependent.
+Now we are ready to proceed to the covariant Hamiltonian formulation
+of this theory. The main idea of this formalism is to treat all derivatives
+of dynamical variables on the equal footing [27, 29, 35] which is sharp con-
+trast with the standard canonical formalism where the time coordinate has
+exceptional meaning. This is very attractive idea especially in the context
+of generally covariant theories since sometimes it is very difficult to perform
+D + 1 splitting of targe-space time and corresponding dynamical fields. In
+case of covariant canonical formalism of gravity we define conjugate momenta
+Mcmn to gmn in the following way
+Mcmn = ∂Lbulk
+∂∂cgmn
+.
+(19)
+Note that the momenta are defined by bulk part of the Lagrangian density
+only as follows from the fact that equations of motion are derived by variation
+of the action when we fix metric and its derivative on the boundary, for careful
+discussion see [36].
+Then from (17) we obtain
+Mcmn =
+1
+32π
+√−g[gmkΓc
+kdgdn + gnkΓc
+kdgdm −
+−gmnΓc
+ghggh − Γf
+fk(gkmgcn + gkngcm) + gmngckΓf
+fk]
+(20)
+using
+δΓk
+gh
+δ∂cgmn
+= 1
+4(gksδc
+g(δm
+s δn
+h + δn
+s δm
+h ) +
++gksδc
+h(δm
+s δn
+g + δn
+s δm
+g ) − gksδc
+s(δm
+g δn
+h + δn
+g δm
+h ))
+(21)
+Then we could formulate covariant Hamiltonian formalism using canonical
+variales gab and Mcab. However it turns out that the situation is much simpler
+when we introduce an alternative set of variables [35, 36] that are defined as
+f ab = √−ggab .
+(22)
+8
+
+Then it is easy to see that the conjugate momenta are defined by chain rule
+Nc
+ab = ∂Lquad
+∂∂cf ab =
+∂Lquad
+∂(∂dgmn)
+∂(∂dgmn)
+∂(∂cfab) .
+(23)
+From (22) we see that f ab and gmn are related by point transformations so
+that
+∂dgmn = ∂gmn
+∂f ab ∂df ab .
+(24)
+Then we have
+∂(∂dgmn)
+∂(∂cf ab) = ∂gmn
+∂f ab δc
+d
+(25)
+and finally
+Nc
+ab =
+∂Lquad
+∂(∂cgmn)(−gmkBkl
+abgln) ,
+(26)
+where
+Bkl
+ab = δgkl
+δf ab = (−f)−
+1
+D−1
+�1
+2(δk
+aδl
+b + δl
+aδk
+b ) −
+1
+D − 1f klfab
+�
+,
+(27)
+where we used the fact that
+− det f ≡ −f = (−g)
+D+1
+2 (−g)−1
+(28)
+and consequently
+√−g = (−f)
+1
+D−1 ,
+gab = (−f)−
+1
+D−1f ab .
+(29)
+Then using previous form of Mcmn we obtain
+Nc
+ab =
+∂Lquad
+∂(∂cgmn)(−gmkBkl
+abgln) =
+= − 1
+32π[2Γc
+ab − Γf
+faδc
+b − Γf
+fbδc
+a] .
+(30)
+9
+
+Note that this relation does not depend on the number of space-time di-
+mensions. Then in order to find corresponding Hamiltonian we should find
+inverse relation between Γa
+bc and Na
+bc. Let us presume that it has the form
+Γc
+ab = ANc
+ab + B(Nd
+daδc
+b + Nd
+bdδc
+a) .
+(31)
+Inserting (30) into (31) we obtain
+Nc
+ab = − 1
+32π(2ANc
+ab + 2B(Nd
+daδc
+b + Nd
+bdδc
+a) −
+−(A + B(D + 2))Nf
+faδc
+b − (A + B(D + 2))Nf
+fbδc
+a)
+(32)
+using Γf
+fa = (A + B(D + 2))Nf
+fa. Comparing left and right side we obtain
+that A and B are equal to
+A = −16π ,
+B = −A
+D .
+(33)
+Then it is easy to find kinetic term of covariant Hamiltonian for D + 1
+dimensional unimodular gravity in the form
+Hkin = ∂cf abNc
+ab − Lquad = 16π
+�
+Nb
+cdf daNc
+ab − 1
+DNr
+raf abNs
+sb
+�
+,
+(34)
+where we used the fact that
+∂cf ab = ∂c
+√−ggab + √−g∂cgab = Γd
+dcf ab − Γa
+cdf db − Γb
+dcf da
+(35)
+together with the condition ∇cgab = 0 that implies
+∂c
+√−g = Γd
+dc
+√−g ,
+∂cgab = −(Γa
+cdgdb + Γb
+cdgda) .
+(36)
+The final form of the covariant Hamiltonian for unimodular gravity con-
+tains terms with the unimodular constraint and true cosmological constant
+¯Λ. Then the phase-space form of the action has the form
+S =
+�
+dD+1x(Nc
+ab∂cfab−Hkin− 1
+16π(−f)
+1
+D−1 ¯Λ− 1
+16πλ((−f)
+1
+D−1 −1)) , (37)
+10
+
+where λ is Lagrange multiplier corresponding to unimodular constraint. From
+the action above we determine corresponding equations of motion by per-
+forming variation with respect to f ab, Nc
+ab and λ
+δS =
+�
+dD+1x(δNc
+ab∂cfab + Nc
+ab∂cδfab −
+−δHkin
+δNc
+ab
+δNc
+ab − δHkin
+δf ab δf ab −
+−
+1
+16π(D − 1)(λ + ¯Λ)(−f)
+1
+D−1δf abfab − δλ((−f)
+1
+D−1 − 1)) = 0
+(38)
+that implies following equations of motion
+∂cf ab = δH
+δNc
+ab
+,
+(−f)
+1
+D−1 − 1 = 0 ,
+−∂cNc
+ab = δH
+δf ab +
+λ
+16π(D − 1)(−f)
+1
+D−1fab +
+¯Λ
+16π(D − 1)(−f)
+1
+D−1fab ,
+(39)
+or explicitly
+∂cf ab = 16π[Na
+cdf db + Nb
+cdf da − 1
+D(f bdNs
+sdδa
+c + f adNs
+sdδb
+c)] ,
+−∂cNc
+ab = 16π
+2 (Nd
+caNc
+bd + Nd
+cbNc
+ad) −
+−16π
+D Nr
+raNs
+sb +
+λ
+16π(D − 1)(−f)
+1
+D−1fab +
+¯Λ
+16π(D − 1)(−f)
+1
+D−1fab ,
+(−f)
+1
+D−1 − 1 = 0 .
+(40)
+Taking the trace of the second equation we can determine λ as
+λ = 16π(D − 1)
+(D + 1)
+(−∂cNc
+abf ab − 16πNd
+caf abNc
+bd + 16π
+D Nr
+raf abNs
+sb) − ¯Λ ,
+(41)
+where we have took into account the equation on the fourth line in (40).
+11
+
+Then the equations of motion for Nc
+ab have the form
+−∂cNc
+ab = 16π
+2 (Nd
+caNc
+bd + Nd
+cbNc
+ad) − 16π
+D Nr
+raNs
+sb +
++
+1
+(D + 1)(−∂jNj
+ikf ik − 16πNd
+cif ikNc
+kd + 16π
+D Nr
+rif ikNs
+sk)fab .
+(42)
+Clearly this equation is traceless and all dependence on the cosmological
+constant ¯Λ disappears which is an essence of unimodular gravity.
+On the other hand one let us try to calculate the trace of the first equation
+that gives
+∂cf abfab = 16π[Na
+cdf db + Nb
+cdf da − 1
+D(f bdNs
+sdδa
+c + f adNs
+sdδb
+c)]fba
+(43)
+that can be simplified into the form
+∂cf = 32π[D − 1
+D
+]Ns
+sc .
+Now taking into account unimodular constraint we immediately get the con-
+dition
+Ns
+sc = 0
+(44)
+that can be interpreted as secondary constraint.
+On the other hand the
+condition (44) seems to be too strong so that we should discuss it in more
+details.
+We begin with the recapitulation that unimodular gravity in the covariant
+Hamiltonian formalism is described by canonical conjugate variables f ab, Nc
+ab
+that are restricted by unimodular condition together with (44). In order to
+find proper interpretation of the constraint (44) it is instructive to derive
+general relativity variables from f ab, Nc
+ab. As the first step let us consider lin-
+ear combination of Nc
+ab that we denote as Γc
+ab and which is given by following
+prescription
+Γc
+ab = −16πNc
+ab + 16π
+D (Nd
+daδc
+b + Nd
+bdδc
+a) .
+(45)
+This can be always done and we should again stress that Γc
+ab is not related
+to f ab at all. Clearly Γc
+ab = Γc
+ba. Then we define covariant derivative where
+Γc
+ab are coefficients of connection. Let us further define gab and its inverse gab
+in the following way
+gab = f ab(−f)
+1
+1−D ,
+gab = fab(−f)
+1
+D−1 .
+(46)
+12
+
+Let us then define covariant derivative of gab as
+∇cgab = ∂cgab + Γa
+cdgdb + Γb
+cdgda ,
+(47)
+that, using (45), takes the form
+∇cgab = (−f)
+1
+1−D ×
+×[∂cf ab − 16πNa
+cdf db − 16πNb
+cdf da + 16π
+D f bdNr
+drδa
+c + 16π
+D Nr
+drf daδb
+c] = 0 ,
+(48)
+where we used the first equation in (40) that also implies ∂cf mnfmn =
+32π D−1
+D Ns
+sc.
+Now thanks to the equation ∇cgab = 0 we can express Γa
+bc
+in the form of Christoffel symbols
+Γa
+bc = 1
+2gad(∂bgdc + ∂cgdb − ∂dgbc) .
+(49)
+On the other hand let us return to the relation between Γa
+bc and Na
+bc that
+takes the form
+Γf
+fa = −32π
+D Nf
+fa
+(50)
+so that condition that Ns
+sa = 0 implies
+Γs
+sa = 0 .
+(51)
+On the other hand from (49) we obtain
+Γf
+fc = 1
+2gfd∂cgdf = ∂c det g = 0
+(52)
+so that condition Ns
+sc = 0 is equivalent to unimodular condition. It is im-
+portant to stress that the fact that unimodular constraint implies Γs
+sa = 0
+has not been appreciated too much with exception of recent interesting pa-
+per [10] where it was stressed that the equivalence between general relativity
+and unimodular gravity is non-trivial. Rather, it was argued there that the
+natural geometry for unimodular relativity is equiprojective geometry [39].
+We also see that the condition Ns
+sa = 0 emerges naturally in the covariant
+canonical formalism of unimodular gravity.
+13
+
+4
+Covariant Form of Unimodular Gravity
+In this section we perform covariant canonical formalism for Henneaux-
+Teitelboim formulation of unimodular gravity that has the form
+S =
+1
+16π
+�
+dD+1x√−g[R + λ(√−g − ∂aτ a)] ,
+(53)
+where τ a is vector density and λ is Lagrange multiplier. Now the equations
+of motion for λ implies
+√−g − ∂aτ a = 0
+(54)
+while equation of motion for τ a leads to
+∂aλ = 0 .
+(55)
+It is clear that the covariant Hamiltonian formulation of this theory is al-
+most the same as in previous case with difference that there is momentum
+conjugate to τ a. Writting ∂aτ a = ∂bτ aδb
+a we obtain momentum conjugate to
+τ a to be equal to
+pb
+a =
+δL
+δ∂bτ a = − 1
+16πλδb
+a
+(56)
+however this can be interpreted as primary constraints of the theory
+Gb
+a ≡ pb
+a +
+1
+16πλδb
+a .
+(57)
+In fact, the bare Hamiltonian is defined as
+HB = pb
+a∂bτ a + ∂cf abNc
+ab − L =
+= 16π[Nb
+cdf daNc
+ab − 1
+DNr
+raf abNs
+sb] −
+1
+16πλ(−f)
+1
+D−1
+(58)
+and we see that the dependence on momenta pν
+µ is missing. For that reason
+we should consider Hamiltonian with primary constraints included
+HT = 16π[Nb
+cdf daNc
+ab − 1
+DNr
+raf abNs
+sb] −
+1
+16πλ(−f)
+1
+D−1 + Γa
+b(pb
+a +
+1
+16πλδb
+a)
+(59)
+14
+
+and consider corresponding equations of motion that arise from the variation
+of the canonical form of the action
+S =
+�
+dD+1x(∂cf abNc
+ab + pa
+b∂aτ b − 16π[Nb
+cdf daNc
+ab − 1
+DNr
+raf abNs
+sb] +
++ 1
+16πλ(−f)
+1
+D−1 + Γa
+b(pb
+a +
+1
+16πλδb
+a))
+(60)
+so that the equations of motion have the form
+∂cf ab = 16π[Na
+cdf db + Nb
+cdf da − 1
+D(f bdNs
+sdδa
+c + f adNs
+sdδb
+c)] ,
+−∂cNc
+ab = 16π
+2 (Nd
+caNc
+bd + Nd
+cbNc
+ad) − 16π
+D Nr
+raNs
+sb +
+λ
+(D − 1)(−f)
+1
+D−1fab ,
+(−f)
+1
+D−1 + Γa
+a = 0 ,
+∂bτ a + Γa
+b = 0 ,
+∂apa
+b = 0 ,
+pb
+a +
+1
+16πλδb
+a = 0 .
+(61)
+If we combine the first and the second equation on the third line we find
+(−f)
+1
+D−1 = ∂aτ a
+(62)
+that has exactly the same form as equation (54). We further perform partial
+derivative of the fourth equation on the third line and we obtain
+∂bpb
+a = − 1
+16π∂aλ
+(63)
+that using the third equation on the same line implies that
+∂aλ = 0 .
+(64)
+This equation also shows that λ is constant and it can be interpreted as
+integration constant. Then it can be argued in the same way as in the pre-
+vious section that the equations (61) are equivalent to the Lagrangian equa-
+tions of Henneaux-Teitelboim gravity. In other words, covariant Hamiltonian
+description of Henneaux-Teiltelboim gravity is equivalent to corresponding
+Lagrangian description which is nice consistency check.
+Acknowledgement:
+The work of JK is supported by the grant “Dualitites and higher order
+derivatives” (GA23-06498S) from the Czech Science Foundation (GACR).
+15
+
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+19
+

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@@ -0,0 +1,2214 @@
+ DesignCon 2006 
+Time Domain Verification of 
+Differential Transmission Line 
+Modeling Methods 
+Jonathan D. Coker, Mayo Clinic
+Dr. Erik S. Daniel, Mayo Clinic 
+Dr. Barry K. Gilbert, Mayo Clinic 
+gilbert.barry@mayo.edu  507-284-4056
+
+Abstract 
+The advantages and limitations of time-domain pseudo-random binary sequence (PRBS) 
+excitation methods for system identification of individual modes within a multi-
+conductor transmission system are discussed.   We develop the modifications necessary 
+to standard frequency-domain transmission-line models to match time-domain 
+experimental data from several types of transmission systems.  We show a variety of 
+experimental results showing very good to excellent agreement with our model’s 
+predictions, up to approximately 10 GHz.         
+ 
+ 
+ 
+ 
+Author Biographies 
+ 
+Jon Coker is a Principal Project Engineer of the Special Purpose Processor Development 
+Group at the Mayo Clinic.  Mr. Coker graduated from Wheaton College with a Bachelor 
+of Arts degree in 1982 and from the University of Minnesota with a Bachelor of Science 
+degree in electrical engineering in 1984.    He is currently pursuing a Ph.D. degree at the 
+University of Minnesota. 
+ 
+ 
+Erik Daniel received the B. A. degree in physics and mathematics from Rice University, 
+Houston, TX, in 1992.  He received the Ph.D. degree in solid state physics from the 
+California Institute of Technology, Pasadena, CA, in 1997, with thesis research focusing 
+on simulation, fabrication, and characterization of quantum effect semiconductor devices.  
+He currently is a Staff Scientist in the Department of Physiology and Biomedical 
+engineering, Mayo Clinic, Rochester, MN, and the Deputy Director of the Special-
+Purpose Processor Development Group. 
+ 
+ 
+Barry Gilbert received the B.S. degree in electrical engineering from Purdue University, 
+West Lafayette, IN, in 1965 and the Ph.D. degree in physiology and biophysics (with 
+minors in electrical engineering and applied mathematics) from the University of 
+Minnesota in 1972.  He is currently a Staff Scientist in the Department of Physiology and 
+Biomedical engineering, Mayo Clinic, Rochester, MN, and the Director of the Special 
+Purpose Processor Development Group.  
+ 
+ 
+ 
+ 
+ 
+
+Introduction 
+ 
+High-speed digital back-plane communication channels have long since given up their 
+“digital design” status and have come to resemble (architecturally) previously-existing, 
+complex analog communication systems.   Equalization, error correction, modulation 
+codes, or advanced detection schemes are now commonly proposed and implemented for 
+serializer-deserializer (SERDES) channels (see [10], for example). To achieve the 
+maximum benefit of these methods,   we require increasingly accurate system 
+identification of the transmission system.   At the same time,  we find that traditionally 
+excellent transmission-line models begin to diverge from experiment as bandwidths begin 
+to extend into the GHz and tens of GHz range [13]. Thus reliable methods for system 
+identification and model verification for candidate transmission systems supporting such 
+channel implementations are of increasing and fundamental importance. 
+ 
+Several methods exist for accurate system identification of linear systems.   Analysis 
+using network analyzers (including vector network analyzers) is a highly-developed 
+technology using frequency-domain measurements.   Advanced time-domain methods 
+have more recently come on the scene employing TDR and TDT measurements of a step 
+excitation [7-9]. 
+ 
+In this paper, we present a technique to predict and experimentally verify transmission 
+line models purely in the time domain using the method of pseudorandom sequences.   
+Our technique is an extension of a method developed for system identification commonly 
+used in magnetic recording data channels and testers [3,6].     We shall focus on the 
+description of transmission lines which are suitable for a high-speed serial-
+communication link.   We shall compare expected and experimental results from printed 
+circuit board (PCB) transmission lines and to results from cables.    While we shall not 
+argue that the proposed method is experimentally superior to established time-domain 
+techniques for linear systems,  we shall discuss the unique nonlinear-system-
+identification capabilities of a PRBS waveform.  In addition, we shall show a fast 
+algorithm which is simple enough to consider implementing in modern SERDES 
+hardware, thus allowing a wide range of inexpensive and highly capable built-in self-test 
+and board diagnostic capabilities.       
+ 
+We shall describe our specific model of a transmission line.   This section is intended to 
+outline the assumptions and limitations of the model.   We also present our alternative 
+extensions to the standard telegrapher’s model, which we found necessary to adequately 
+describe experiment in some cases. 
+ 
+Transmission Line Modeling 
+In this section, we briefly review a classical transmission-line model, and discuss the 
+parameters which we use in the model. 
+Telegrapher’s Model in the Time Domain 
+
+The basic telegrapher's model for transmission lines is fully worked out in several texts 
+[1,2].    The standard solution gives the voltage transfer function for a wave traveling in 
+the positive z direction as 
+ 
+(
+)
+( )
+z
+H
+e γ ω
+ω
+−
+Δ
+=
+ 
+(1.1) 
+where ω  is the radial frequency, z
+Δ  is the distance down the line, and the complex 
+propagation constant ( )
+γ ω  is: 
+ 
+ 
+( )
+(
+)(
+)
+(
+)(
+)
+R
+j L G
+j C
+R sL G
+sC
+γ ω
+ω
+ω
+=
++
++
+=
++
++
+ 
+(1.2) 
+ 
+If the Fourier transform of an input waveform ( )
+x t is
+( )
+X ω , then the Fourier transform of 
+the waveform at the output of the transmission line, ( )
+y t , is 
+ 
+ 
+( )
+( )
+( )
+Y
+H
+X
+ω
+ω
+ω
+=
+ 
+(1.3) 
+ 
+and the time-domain version of the output waveform is 
+ 
+ 
+1
+( )
+{
+( )
+( )}
+y t
+H
+X
+ω
+ω
+−
+= F
+ 
+(1.4) 
+ 
+Cm
+L
+C
+C
+R
+G
+C
+G
+C
+L + L
+m
+R
+L + L
+m
+Cm
+G
+L
+L
+to
+Infinity
+to
+Infinity
+G
+L
+R
+R
+L
+L
+R
+R
+Gm
+C
+C
+Cm
+G
+G
+Gm
+C
+C
+Cm
+G
+G
+Gm
+Lm
+Lm
+L
+C
+C
+Lm
+(x1 (t) + x
+2(t))
+GENERAL  EQUIVALENT  CIRCUIT  FOR
+SYMMETRIC  SIGNAL  CONDUCTORS
+EVEN-MODE  EQUIVALENT  CIRCUIT
+( Lines  Toggling  in  Same  Direction )
+R
+G + 2G
+m
+G + 2G
+m
+C + 2C
+m
+C + 2C
+m
+L - L
+m
+R
+L - L
+m
+to
+Infinity
+(x1 (t) - x
+2(t))
+x1 (t)
+x2 (t)
+ODD-MODE  EQUIVALENT  CIRCUIT
+( Lines  Toggling  in Opposite  Directions )
+1
+2
+ 
+Figure 1: Simplified Odd Mode and Even Mode Equivalent Circuits for a Symmetric Two-Signal-
+Conductor Generalized Transmission Line (18500). 
+ 
+In Figure 1, we show the standard, generalized equivalent circuit of a symmetric two-
+signal-conductor transmission line, and reduced-complexity, single-ended equivalent 
+circuits for each transmission mode.   Note that the simplified equivalent circuits allow 
+
+direct application of the RLGC model, using the correctly-transformed versions of the 
+RLGC parameters appropriate for the transmission mode. 
+ 
+Variation of RLGC Parameters with Frequency 
+ 
+Having set the stage for the time-domain application of a generic RLGC model, we now 
+focus on the specific forms of each component used our RLGC model.   The following 
+section specifies the formulas used in our basic RLGC model (which are fairly standard) 
+and identifies modifications we thought necessary to adequately explain observed 
+laboratory behavior (which are not always standard). 
+ 
+Series Impedance Variation with Frequency 
+In the case of simple, homogeneous signal conductors, we use the classical result for the 
+series impedance based on surface impedance concepts, which we repeat here:  
+ 
+ 
+(
+)
+AC
+R
+sL
+R
+s
+L s
++
+=
++ ∞  
+(1.5) 
+ 
+where L∞  may be interpreted as the inductance of the system when all currents flow 
+uniformly on the surface of the signal conductors (that is, at moderately high frequency) 
+and 
+AC
+R
+ is a constant which we approximate as 
+ 
+2
+AC
+R
+S
+η
+μ
+σ
+=
+ 
+(1.6) 
+ 
+where 
+ and 
+μ
+σ are the permeability and conductivity of the signal conductor, and S is the 
+length of the effective perimeter of the signal conductor through which the surface 
+currents flow.   The geometry-dependent constant η  represents a factor determining the 
+increase in resistive losses due to currents in the return paths.   In a thin stripline 
+configuration, one might expect the value of η  to be in the neighborhood of 
+2
+η =
+because the widths of the expected return paths are about the same as the 
+circumference of the signal conductor.   In a coaxial cable, one might expect η  to be less 
+than 2, because the return paths in the outer shield are significantly wider than the 
+circumference of the signal conductor.  In practice, any of the parameters of equation 
+(1.6), including 
+AC
+R
+ itself, may be varied in equation (1.5) to match laboratory data.  
+ 
+However, there exists a common transmission system which does not fit equation (1.5) 
+well.  The Gore EyeOpener ™ cable, for example, uses signal conductors constructed 
+from a heterogeneous combination of metal layers to achieve self-equalizing properties.   
+We now derive an approximation to the surface impedance for a thick bulk material, 
+covered by a relatively thin layer of another conductor, to address this case.   General 
+field solutions to this type of problem were generated by Wait [5], which we here apply 
+to transmission lines. 
+ 
+
+We presume a planar, infinitely-thick conductor of bulk conductivity 
+2
+σ  underneath a 
+thin layer of thickness 
+1τ  and conductivity
+1
+σ .   As in the classical case, the electric field 
+will decay throughout the finite thickness 
+1τ  to the value  
+ 
+1
+1
+1
+1
+0
+( ) |
+z
+y
+j
+y
+E
+E e
+τ
+τ
+δ
+=
++
+−
+=
+ 
+(1.7) 
+ 
+where 
+1δ  is the skin depth in the outer conductor.   At the interface between the two 
+different conductors, the electric field will have a new behavior given by the boundary 
+condition requirements of Maxwell’s equations.  The appropriate constraint is that of 
+continuous tangential electric field across the interface.   Therefore, at the interface, the 
+electric field begins a new exponential behavior in the bulk material with new depth 
+constant
+2
+δ : 
+ 
+ 
+1
+1
+1
+2
+1
+1
+(
+)
+0
+( )
+j
+j y
+z
+E
+y
+E e
+e
+τ
+τ
+δ
+δ
++
++
+−
+−
+−
+=
+ 
+(1.8) 
+  
+Working out the integral for the total current under a width S, the resulting surface 
+impedance is: 
+ 
+ 
+1
+1
+1
+1
+1
+1
+1
+2
+1 1
+1
+2
+1
+2
+1
+1
+{
+}{
+(
+1)
+1}
+{
+}{
+(
+1)
+1}
+j
+z
+s
+AC
+j
+Z
+e
+S
+R
+s
+e
+τ
+δ
+μ
+σ
+η
+σ
+σ δ
+σ
+σ
+σ
++
+−
+−
+−
+−
++
+=
+−
++
+=
+−
++
+ 
+(1.9) 
+ 
+Equation (1.9) adds a new factor to the classical surface impedance.  The factor is a 
+function of the outer-conductor thickness
+1τ  and the ratio of the conductivities of the two 
+materials.  In general, we see that the resistive and reactive portions of the surface 
+impedance are no longer equal when the conductor is composite.    In our physical 
+approximation, we take the conductor width S to be the effective length of the perimeter 
+of the signal conductor.   The overall series impedance is then: 
+ 
+ 
+1
+1
+1
+1
+2
+1
+2
+1
+{
+}{
+(
+1)
+1}
+s
+AC
+R
+sL
+R
+s
+e
+sL
+μ
+ητ
+σ
+σ
+σ
+−
+−
+∞
++
+=
+−
++
++
+ 
+(1.10) 
+ 
+Equation (1.9) is valid when the permeability of the all conductors is that of free space.   
+When the bulk conductor is magnetic, the conductivity of the bulk conductor 
+2
+σ  can be 
+replaced by an effective conductivity 
+ 
+ 
+2
+2'
+R
+σ
+σ
+μ
+=
+ 
+(1.11) 
+ 
+
+where
+R
+μ is the relative permeability of the bulk conductor. 
+ 
+ 
+Shunt Conductance Variation with Frequency 
+Traditionally, the dielectric losses are characterized by a shunt conductance G per unit 
+length: 
+ 
+1
+tan
+tan
+G
+C
+sC
+j
+ω
+δ
+δ
+=
+=
+ 
+(1.12) 
+The “loss tangent”, tanδ , is commonly specified by dielectric manufacturers to be in the 
+0.01 to 0.001 range.   Typically, users are left to presume that the loss tangent is 
+independent of frequency.     In such a case, the total shunt admittance can be written as 
+ 
+ 
+tan
+(1
+tan )
+G
+Cs
+G
+Cs
+j
+Cs
+ω
+δ
+δ
++
+=
++
+=
+−
+ 
+(1.13) 
+ 
+We shall see that this form can give good agreement with laboratory data at frequencies 
+in the few-GHz range (and when dielectric losses are relatively small compared to the 
+series resistance).     At higher frequencies (perhaps in the 10 GHz range) the model’s 
+main deficiency becomes apparent:  the form of equation (1.13) cannot be physically 
+reasonable.   It is well known that this form of dielectric loss is not a physically consistent 
+possibility [14].   In the present case, it is relatively straightforward to see why this is so.  
+If we take the series impedance to be lossless, that is, 
+0
+AC
+R
+=
+, then using equations 
+(1.13),  (1.5),  (1.2), and (1.1),  the transform of the impulse response of the transmission 
+line can be written as: 
+ 
+ 
+(1
+tan
+)
+( ( ))
+j
+LC
+j
+z
+h t
+e
+e
+ω
+δ
+γ
+−
+−
+− Δ
+=
+=
+F
+ 
+(1.14) 
+ 
+Equation (1.14) has a closed-form inverse transform, which is: 
+ 
+ 
+2
+2
+( )
+[(
+)
+]
+h t
+t
+α
+π
+τ
+α
+=
+−
++
+ 
+(1.15) 
+ 
+where the parameters are given as  
+ 
+Re{ 1
+tan }
+Im{ 1
+tan }
+z LC
+j
+z LC
+j
+τ
+δ
+α
+δ
+= Δ
+−
+= Δ
+−
+ 
+(1.16) 
+Therefore, the impulse response of a transmission line with dielectric possessing constant 
+loss tangent is a delayed Lorentzian pulse.  The Lorentzian form gives experimentally 
+plausible insights, such as: the amplitude of the pulse is inversely proportional to the 
+length of the transmission line, with proportionality constant simply related to the loss 
+tangent.   However, we can observe that the impulse response extends back infinitely in 
+time even though its impulse excitation occurred at the time origin.  The model predicts 
+non-causal behavior and is therefore not plausible as a physical model.   We have found 
+that this feature of the constant-loss-tangent model is often the root cause of the failure to 
+match our experimental data at high frequency. 
+
+ 
+Other forms for the variation of the loss tangent with frequency must be applied in these 
+cases.   In Figure 2 we highlight the difference in expected pulse shape between the 
+constant-loss-tangent model and that of a loss tangent varying linearly with frequency.   
+The response in the linear-loss-tangent case is derived numerically using equation (1.4) 
+because the problem does not have a closed-form solution.  We will show later that the 
+linear-variation version can exhibit good fit to experimental data at frequencies in excess 
+of 10 GHz (which is our only justification for using it).  
+ 
+0.8
+0.9
+1.0
+1.1
+1.2
+1.3
+-0.1
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+ Constant  Loss  Tangent
+ Linear  Loss  Tangent
+Modeled  Pulse  Response,  Volts
+Time,  ns
+Constant  Loss  Tangent
+Response  is  Lorentzian
+1
+1 + Dt2
+Linear  Loss  Tangent  Response  Identically  Zero  Outside  These  Limits
+ 
+Figure 2: Comparison of Modeled Time Responses of a Transmission Line with No Series Loss but 
+Finite Dielectric Loss Due to Two Loss Models, Showing Causality Failure of Constant Loss Tangent 
+Assumptions.  (20573) 
+ 
+Time-Domain Laboratory Techniques 
+ 
+Transmission lines (and other linear circuits) are often characterized by s-parameter 
+analysis using a vector network analyzer (VNA).   The methodology for calibration, de-
+embedding, and interpretation of VNA results is a highly-developed specialty, which 
+(when properly applied) will fully characterize the transmission line over a wide 
+bandwidth.     
+ 
+For this work, we have elected to use a time-domain method, for the following reasons.   
+First, most VNAs have two ports and do not simply support differential-mode excitation.     
+
+Second, in many applications, the primary information needed is the classical transfer-
+function of the transmission line system (i.e., the information present in 
+21
+s  in the 
+absence of reflections).   The complexities in testing, calibration, de-embedding, and 
+interpretation for the other s-parameters may not be strictly necessary in some 
+applications.   Third,  a full-fledged experimental characterization of systems utilizing 
+transmission lines is often not limited to pure system-identification techniques, but also 
+may include direct measurements of higher-level system  performance quantities (such as 
+error rate or eye diagrams).   In some applications it is beneficial to integrate as many 
+measurement schemes as feasible into an experimental setup that is as simple (and 
+inexpensive) as possible, so long as the resulting measurements are “good enough” for 
+the application.    
+ 
+Finally, there may exist nonlinearities in the transmission system which will introduce 
+interpretation errors in the s-parameter analysis without warning.   These nonlinearities 
+are not likely to be in the transmission lines and interconnect (which are linear to an 
+excellent approximation) but may exist, for example, in the driver circuitry of a buffer 
+amplifier.   Most (if not all) nonlinearities cannot be characterized by the magnitude and 
+phase of single-sinusoid reception; however, there exist waveforms which do broaden the 
+scope of complete nonlinear characterization.  We shall see that a PRBS is one such 
+waveform. 
+ 
+In the following section, we describe a method which addresses all four of these points.  
+Naturally, the described method will have its own disadvantages, which are generally 
+related to experimental approximations which do not exist (or are unnecessary) in a 
+VNA-type of analysis. 
+ 
+Pseudorandom Sequence Excitation Method 
+ 
+We describe a time-domain system identification method in this section.  We follow the 
+scheme developed in [3,6].    A performance analyzer (‘bit error rate tester’) is used to 
+generate a 127-bit pseudo-random binary sequence (PRBS). The analyzer has true and 
+complement outputs, which are used to drive the differential transmission line.    A digital 
+oscilloscope is used to sample the differential signal at the end of the transmission line.   
+The scope is triggered by the performance-analyzer synch-out pulse, which fires once 
+every PRBS period.  This pulse provides a consistent time datum which is independent of 
+the transmission-line length.   A typical experimental setup (shown for a cable) is as 
+shown in Figure 3. 
+ 
+The upper panel in Figure 4 shows an example of the raw waveform as sampled at the 
+end of a transmission line.   Because the raw waveform is too complicated to interpret 
+easily by eye, the waveform can be processed to generate a discrete-time pulse response.   
+
+15  Meter  Cable  Assembly  Eye  Diagram  Test
+Board  With  Cable  Assembly  Under  Test
+ 
+Figure 3: Test Setup for Measuring Error Rate, Eye Diagrams, and System Transfer Functions in the 
+Time Domain. (18045) 
+0
+10
+20
+30
+40
+50
+0
+.5
+1
+Extracted  Pulse  Response
+Time  from  Signal  Source  Trigger  Datum,  nsec
+-1
+0
+1
+Measured  Voltage
+ 
+Figure 4: Example of a Measured PRBS Waveform and Its Extracted Pulse Response. (18322) 
+
+The response (which we shall call extracted pulse response in keeping with the literature 
+[3,6]), is numerically generated using a two-step process.  First, the raw waveform is re-
+sampled to take care of experimental frequency errors between the PRBS generator and 
+the oscilloscope, and to provide an appropriate sampling rate for subsequent analysis.  
+Second, the re-sampled waveform is mathematically deconvolved with a theoretically 
+perfect version of the same PRBS.   We shall return to describe a very efficient method 
+for this deconvolution. 
+ 
+The extracted pulse response is analogous to the impulse response in continuous time 
+systems.   In this case, the extracted response is the expected response of the transmission 
+line (and measurement system and signal drivers) if the PRBS driver had output a simple 
+base-to-peak digital pulse instead of the PRBS. 
+ 
+The lower panel in Figure 4 shows the extracted response for the given raw waveform.   
+While both waveforms contain exactly the same information, it is clear that the extracted 
+pulse is a short primary response, with some small echoes of interesting shape and 
+location on the time axis.    In Figure 5, the extracted pulse response for two different, but 
+short, PCB transmission lines is shown.   All connectors, test cables, and test conditions 
+are nominally identical, except for the length of the transmission line.    
+5
+0.0
+0.2
+0.4
+0.6
+0.8
+6
+7
+8
+10
+11
+12
+13
+14
+15
+5
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+6
+7
+8
+9
+10
+Time  from  Signal  Source  Trigger  Datum,  nsec
+Extracted  Pulse  Response
+( 6 Inch  Line ),  Volts
+Extracted  Pulse  Response
+( 1.5 Inch  Line ),  Volts
+11
+12
+13
+14
+15
+Nonlinear
+Echo
+Main  Pulse
+First
+Reflection
+Second
+Reflection
+ 
+Figure 5: Extracted Pulse Responses from Two Lengths of Otherwise Identical Lines (1.5” and 6”)  
+(18323) 
+
+ 
+In the upper panel of Figure 5, we zoom in on the interesting sections of the extracted 
+pulse response, for a 1.5 inch PCB line.   As might be expected from the significant 
+impedance mismatches, the signal appears to reflect back and forth between the 
+impedance discontinuities before settling down.     At 2.5 Gbits/second excitation, the 
+pulse response is nominally 400 picoseconds wide, and the first forward-moving 
+reflection should begin to arrive about a half nanosecond after the onset of the main 
+pulse; the second forward-moving reflection arrives a half nanosecond after that, and so 
+forth.  Thus we should not expect to see separation between the main pulse and its 
+subsequent, exponentially decaying reflections. 
+ 
+In the lower panel of Figure 5, we see the effects of quadrupling the transmission-line 
+length.   The now well-resolved reflections occur at approximately 2 nanosecond 
+intervals (as expected).   The main pulse is resolved, and is proportional in magnitude to 
+what the pulse would have been down 6 inches of this transmission line without the 
+reflections due to impedance mismatches, but including any pulse modification due 
+directly to impedance mismatches.   (The interface of two perfect transmission lines of 
+real, but different, impedance at all frequencies will modify the main pulse in amplitude, 
+only). 
+ 
+The main pulse and reflections may also be resolved by increasing the excitation 
+frequency of the PRBS on a given line (instead of changing line lengths, as above).     
+This technique may be limited by the rise time of the drivers, the time span of the impulse 
+response of the transmission lines under test, and the analog bandwidth of the sampling 
+oscilloscope.   
+ 
+These observations suggest a type of de-embedding technique which is very simple, 
+though approximate.   We assume that the combination of the transmission line length, 
+the resolution of the channel response, and the frequency of the PRBS is such that the 
+main pulse is resolvable from reflections or other imperfections.   We then claim, to the 
+extent these assumptions are true, that the resolvable main pulse may be written as 
+ 
+ 
+ 
+ 
+1
+1
+0
+1
+1
+0
+1
+( )
+{
+( )
+( )
+( )
+( )...
+( ,
+)}
+{
+( )
+( )
+( ,
+)}
+driver
+launches
+scope
+tline
+T
+tline
+x t
+F
+X
+H
+H
+H
+H
+z
+F
+X
+H
+H
+z
+ω
+ω
+ω
+ω
+ω
+ω
+ω
+ω
+−
+−
+≅
+Δ
+≅
+Δ
+                 (1.17) 
+     
+where 
+0( )
+X ω  is the Fourier transform of an ideal PRBS sequence, 
+1z
+Δ
+is the length of 
+transmission line, and the total transfer function 
+( )
+T
+H
+ω  is the composite linear response 
+of all non-ideal elements in a practical test.   Note that the interpretation of 
+1( )
+x t is not 
+exactly that of the time response related to the s-parameter 
+21
+s ;   it is only equivalent to s-
+parameter analysis when there are no reflections.  In principle, and with sufficient 
+cleverness, we could also resolve each reflection in the measured waveforms to 
+independently and fully analyze the entire characteristic described by
+21
+s .  However, we 
+
+shall not develop those techniques in this report; instead, we shall focus only on resolving 
+the response that would have occurred without reflections. 
+ 
+If we now consider a transmission line of length 
+2
+1
+z
+z
+Δ
+> Δ
+, then its response can be 
+written as 
+ 
+1
+2
+0
+2
+1
+0
+1
+2
+1
+( )
+{
+( )
+( )
+( ,
+)}
+{
+( )
+( )
+( ,
+)}
+T
+tline
+tline
+x t
+X
+H
+H
+z
+X
+X
+H
+z
+z
+ω
+ω
+ω
+ω
+ω
+ω
+−
+−
+=
+Δ
+=
+Δ
+− Δ
+F
+F
+ 
+(1.18) 
+ 
+where the second line follows from the separability of the transmission line transfer 
+function (equation (1.1)) into component factors, that is, 
+ 
+ 
+2
+1
+2
+1
+2
+1
+2
+1
+(
+)
+(
+)(
+)
+(
+)(
+)
+(
+)(
+)
+2
+1
+2
+1
+( ,
+)
+( ,
+)
+( ,
+)
+z
+z
+z
+z
+z
+z
+z
+z
+tline
+tline
+tline
+H
+z
+e
+e
+e
+e
+H
+z H
+z
+z
+γ
+γ ω
+γ ω
+γ ω
+ω
+ω
+ω
+−
+Δ
+−
+Δ +Δ
+−Δ
+−
+Δ
+−Δ
+−
+Δ
+−Δ
+Δ
+=
+=
+=
+=
+Δ
+Δ
+− Δ
+ 
+(1.19) 
+ 
+Despite its mathematical look, the suggested modeling technique is exceedingly simple: 
+ 
+1. Measure
+1( )
+x t  of a relatively short line and find its extracted pulse response. 
+2. Calculate the expected response of a longer transmission line of length 
+2z
+Δ
+ by 
+applying equation (1.4) for a transmission line of length
+2
+1
+z
+z
+Δ
+− Δ . 
+3. Compare the output of step 2 to an actual measurement of the extracted pulse 
+response of a transmission line of length
+2z
+Δ
+.   Focus on the (by assumption, 
+resolvable) main pulse only; ignoring all other interesting blips. 
+ 
+We have argued that this technique will give an expected waveform, and an actual 
+waveform, to measure the goodness-of-fit of the RLGC model; furthermore this 
+comparison de-embeds all systematic linear effects in the test, to the extent that the main 
+pulse is resolvable in time from all reflections or other phenomenon.   
+ 
+The “resolvability” criterion gives this technique its simplicity, but is its major source of 
+interpretation uncertainty when compared to more accurate experimental methods (such 
+as, a VNA).   A typical “eyeball” comparison is good to perhaps 30-40 dB (a few percent 
+in voltage).   In many applications (such as, the identification of eye diagrams), the 
+“eyeball” criterion is, nearly by definition, good enough.  However, the engineer must 
+judge what kind of accuracy is required for a particular application. 
+ 
+We have argued that the proposed technique effectively de-embeds effects which can be 
+described by a linear transfer function.   We now briefly digress to discuss the effects of 
+stochastic jitter in the trigger waveform and oscilloscope timings.   These effects are not 
+present in frequency-domain measurement techniques, and therefore deserve further 
+discussion.  
+ 
+In general the analysis of the all jitter effects on the waveform captured by the 
+oscilloscope is very difficult, but the following simplifications allow some insight into 
+
+the problem.   If the jitter is small enough that the waveform is well approximated by its 
+first-order Taylor’s series, that is: 
+ 
+ 
+(
+)
+( )
+'( )
+x t
+x t
+x t
+τ
+τ
++ Δ
+≅
++ Δ
+ 
+(1.20) 
+ 
+and all jitter phenomenon are lumped into a single effective jitter at the scope sampler 
+(relative to the signal ( )
+x t ), with jitter variance 
+2
+σ , then the signal can be thought of in 
+two components.   There exists a stochastic portion (whose spectrum is continuous even 
+if ( )
+x t is periodic) whose power is linearly related to 
+2
+σ  and the average square of the 
+derivative of ( )
+x t .  This power comes at the expense of the high frequencies in the 
+deterministic portion of the identified waveform (that is, its averaged sequence value as 
+sampled on the scope).    It can be shown that the jitter produces a stochastic filter whose 
+average transfer characteristic is 
+ 
+ 
+2
+2
+( )
+jitt
+H
+e σ ω
+ω
+−
+=
+%
+ 
+(1.21) 
+ 
+if the jitter is Gaussian.   Therefore, if
+2 /
+f
+π σ
+<<
+ , the jitter effects should be negligible 
+in the sampled waveform ( )
+x t .   If the standard deviation of the jitter is in the few pS 
+range, then frequencies below about 10 GHz are affected by less than 1 percent.  
+ 
+In a sense, 
+( )
+jitt
+H
+ω
+%
+ can be thought of as one of components of the total deterministic 
+transfer function 
+( )
+T
+H
+ω , but only if the various realizations of 
+( )
+jitt
+H
+ω
+%
+ are averaged 
+sufficiently to converge to the average transfer function.  This convergence might 
+happen, for instance, when the waveform reported from the scope is the average of a 
+large number of sample sequences.   In such a case, this portion of the transfer function 
+cancels out in our method (as do the deterministic transfer functions).   However, this is 
+dangerous ground, and the experimenter should take care to ensure that the stochastic 
+conditions of the tests are identical when exploiting this effect.   For example, if trigger 
+jitter is significant, the realization of  
+( )
+jitt
+H
+ω
+%
+ for a waveform derived by “averaging” 
+only one sample waveform is in general quite different from the realization of 
+( )
+jitt
+H
+ω
+%
+ 
+when many averages are taken.  (The frequency response will look better, in general, 
+with only a few averages than it does with long averages).     
+  
+Numerical Methods 
+ 
+In the preceding section,  we developed the concept of the extracted pulse response, 
+which is the circular deconvolution of the measured PRBS sequence with an ideal version 
+of the same sequence.   A good, standard method for deconvolution is to take the Fourier 
+transform of both waveforms, perform a complex division, and then take the inverse 
+Fourier transform.     
+ 
+
+In the present case,  the number of points in the transform is 
+2
+1
+N
+n =
+− , which is very 
+often a prime number, and is never highly composite.   In such cases, the best fast 
+algorithms for discrete Fourier transforms (DFTs) are relatively useless.  In most such 
+cases, the deconvolution will require about 
+2
+n complex multiplication operations and 
+about n complex divisions. 
+ 
+We now show a special deconvolution method which takes optimal advantage of the fact 
+that measured waveform is a filtered version of a PRBS.   This method requires about 
+2
+n  
+real additions, zero multiplications, zero divisions, and zero complex operations of any 
+sort.    
+ 
+We develop the algorithm in the following way.   If the sampled digital-pulse response of 
+the transmission line system is denoted by [ ]
+(
+)
+h n
+h nT
+=
+,  with n an integer and T the bit 
+period of the sequence,  then the expected sampled measured output is giving by the 
+circular convolution of the PRBS and the pulse response: 
+ 
+ 
+[ ]
+[
+] [ ]
+k
+y n
+h k
+n p k
+=
+−
+∑
+ 
+(1.22) 
+where [ ]
+{ 1,1}
+p n ∈ −
+ represents the PRBS and k ranges over the length L of the PRBS, 
+which is always of the form 
+2
+1
+N
+L =
+− .    We shall also call [ ]
+h n  the extracted pulse 
+response, after we have back-calculated it from measured data.   Because this is a circular 
+convolution, the sequences y, p, and h are assumed to be periodic in L; therefore the 
+indices can always be taken modulo L.   
+ 
+We now assume that the mathematical convolution was actually completed in the 
+physical world by playing the ideal PRBS through a linear system, and we are able to 
+measure the analog waveform ( )
+y t directly.   We then resample this waveform to obtain 
+the measured version of [ ]
+y n .  By inspiration, we can generate a new function from [ ]
+y n  
+and note how else it can be expressed: 
+ 
+ 
+1
+[ ]
+[ ] [
+]
+1
+1
+[
+] [ ] [
+]
+1
+[ ]
+l
+l
+k
+x
+Z n
+y n p n
+l
+L
+h k
+n p k p n
+l
+L
+h n
+μ
+=
+−
++
+=
+−
+−
++
+=
++
+∑
+∑∑
+ 
+(1.23) 
+ 
+where 
+x
+μ is the mean of the pulse response [ ]
+x n .  The last line of equation (1.23) is 
+general only when the excitation is a PRBS, and is derived using the properties of a 
+PRBS [4].  Therefore we can recover, within a small DC shift, the extracted pulse 
+response [ ]
+h n  from the measured sequence [ ]
+y n  by a simple transformation using 
+2
+(
+1)
+L L
+L
+−
+≅
+ additions and L trivial multiplications by 1
+± .   Depending on the 
+application, the division by L+1 may or may not be necessary; if it is necessary, 
+implementers should note that the divisor is always exactly a power of two.    With a little 
+
+more complexity in the algorithm it is possible to reconstruct exactly the DC conditions 
+to get a new function
+'[ ]
+[ ]
+Z n
+h n
+≡
+, but in our case we have simply ignored the small DC 
+offset as it makes little practical difference in our analysis.     
+ 
+We have developed this algorithm in a general way when we are interested in analyzing 
+or plotting waveforms with more than one point per PRBS bit.  In such a case, we can 
+decimate a waveform representing M points per bit into M waveforms representing 
+[ ]
+m
+y
+n , that is, the sampled version of ((
+/
+) )
+y n
+m M T
++
+.    We can determine the extracted 
+pulse response for each sub-channel, 
+[ ]
+m
+h
+n , using equation (1.23) and 
+[ ]
+m
+y
+n as input.  
+The resultant response 
+[ ]
+m
+h n  is the sampled version of the analog waveform 
+((
+/
+) )
+h n
+m M T
++
+.   Therefore, we can reconstruct the M-point-per-bit version of the 
+response by re-interleaving the results of each
+[ ]
+m
+h
+n .   Note that the calculations of 
+[ ]
+m
+h n  
+are only dependent on values of y in its own interleave;   whereas in general this property 
+is not true of DFT-type deconvolutions.    This property is another advantage of this type 
+of specialized deconvolution.   
+ 
+Therefore, we have shown a very efficient method for deconvolution when PRBS 
+excitation is used.  Clearly, for offline analysis using relatively small L, there is little 
+practical difference between a few microseconds and a several milliseconds of 
+computation time.   Even so, these differences can become quite significant for large 
+enough L even for offline analysis because L grows exponentially with each PRBS order 
+N.  Perhaps more significantly, the algorithm in equation (1.23) is simple enough to 
+realistically be implemented at-speed in hardware.   For example, the algorithm 
+implementation is simply a single accumulator if we are willing to calculate one
+( )
+Z n  at a 
+time in an operation.       
+ 
+Nonlinear Analysis with PRBS Excitation 
+ 
+Up to this point, we have not described anything of particular value in using a PRBS as 
+the excitation waveform over other types of waveforms (other than its alternative utility 
+as a convenient bit-error-rate test pattern).   Indeed, a TDT-type step waveform gives the 
+same information as does an extracted pulse response for a linear system.  We now 
+digress to explain an interesting and useful feature of this type of PRBS-deconvolution 
+analysis, which allows native nonlinear Volterra analysis of the waveforms. 
+ 
+In each of the extracted pulse waveforms discussed thus far (shown in Figure 4 and 
+Figure 6), a strange blip appears to arrive at the oscilloscope 2.5 nanoseconds before the 
+main pulse.   This blip always shows up in exactly the same place (relative to the main 
+pulse), independent of the transmission-line length or type, or even whether there is a 
+transmission line under test at all.   If the bit excitation frequency is changed, then the 
+time spacing between the blip and the main pulse also changes.    However, if the time 
+spacing is normalized to bit time units, now this number now remains constant, 
+independent of excitation frequency.   Therefore, we have discovered an apparently 
+physical delay effect which seems to know something about the bit times.  No signal 
+
+phenomenon of any kind shows up at these locations when a simple TDR pulse is used as 
+excitation. 
+ 
+The reason for this type of behavior was resolved in the 1980s [3].  The phenomenon 
+came to be understood in the following way.    If a system is linear, then the 
+deconvolution operation, described above, always gives the same results for the extracted 
+pulse response no matter what the underlying data pattern is.    If a system is somewhat 
+nonlinear, then in the general case, the deconvolution operation will contain garbage 
+which will appear as a “noise” everywhere on the extracted pulse response.     However, 
+if the data pattern is a PRBS, then for many types of nonlinearities, the disturbances due 
+to such nonlinearities destructively interfere almost everywhere after the deconvolution 
+operation.    The nonlinearities tend to constructively interfere at specific locations on the 
+extracted pulse response, and with specific shapes and magnitude, all of which vary 
+depending on the nature and severity of the nonlinearity.  This phenomenon is now used 
+universally in magnetic recording to identify various impairments of the magnetic 
+system.   
+ 
+In our case, the blip phenomenon is due to circuit-driver nonlinearity somewhere in the 
+test system.   Analysis shows that this imperfection (if so it may be called) is due to a 
+non-linearity in the driver circuit in the performance analyzer.   Close examination of the 
+output differential pulses show an asymmetry in the rise times (as compared to the fall 
+times) of the output driver.   As a result, a positive-going pulse looks noticeably different 
+than the opposite of a negative-going pulse.   This difference is by definition a nonlinear 
+effect. 
+ 
+It has been shown that the PRBS phenomenon is related to the Volterra-series nonlinear-
+system identification technique for discrete-time systems [4].  This property is another 
+example of the apparently endless set of practical and useful features of a PRBS. 
+ 
+To an excellent approximation, the transmissions lines are natively linear, and (in 
+principle) the choice of excitation waveform is immaterial.    However, circuit drivers, 
+receivers, digital-to-analog converters, equalizers, analog-to-digital converters, and other 
+elements of a complex analog communication system, are not natively linear.  In such a 
+case, analysis using linear assumptions, under conditions of PRBS excitation yields (for 
+no additional work), nonlinear system analysis capability.       
+ 
+Comparison to Theory 
+We now present experimental PRBS results gathered from several different types of 
+transmission lines, and compare the results to expectations from our RLGC model.   We 
+found that a constant loss tangent assumption was adequate at lower frequencies (~2.5 
+Gbits/second excitation) but completely inadequate at 40 Gbits/sec excitation.   We 
+concluded that the series losses were modeled adequately by the standard form at all 
+frequencies tested, except for any frequency using a bilayer-conductor transmission 
+system. 
+ 
+2.5 Gbits/second Excitation Results 
+
+ 
+We applied the preceding techniques to a PCB transmission line.  We designed and 
+procured a passive printed circuit board (PCB) using GETEK ™ dielectric materials.  
+This board was fabricated to verify the accuracy of the RLGC model described above. 
+ 
+Manual RLGC-model fitting procedures produced model results which compared quite 
+favorably to the experimentally-measured results at 2.5 Gbits/second.  These results are 
+compared, for four different lengths of the transmission line, in Figure 6.  All model 
+parameters are frozen at the given values (except, of course, for the transmission-line 
+length) for all model results on this graph.   We used a 6-inch version of the lines as the 
+de-embedding reference.   The important model parameters for the GETEK materials are: 
+ 
+ 
+378 nH/m, 
+117 pF/m, tan
+0.011,
+5.45 7,
+/
+5 mils.
+L
+C
+E
+S
+δ
+σ
+η
+∞ =
+=
+=
+=
+=
+ 
+(1.24) 
+ 
+We tested several lengths of 100 Ohm Skewclear Infiniband  12X cable from 
+Amphenol/SpectraStrip ™.   We merely used the advertised (nominal) odd-mode 
+impedance and velocity numbers to calculate the odd-mode reactive parameters.   
+ 
+The model parameters for the SKEWCLEAR cable are: 
+ 
+ 
+377 nH/m, 
+37.7 pF/m, tan
+0.0001,
+6.0 7,
+/
+17mils.
+L
+C
+E
+S
+δ
+σ
+η
+∞ =
+=
+=
+=
+=
+ (1.25) 
+ 
+ 
+
+10
+12
+14
+0.0
+0.2
+0.4
+0.6
+0.8
+12
+14
+16
+0.0
+0.2
+0.4
+0.6
+0.8
+14
+16
+18
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+18
+20
+22
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Time  from  Signal  Source  Trigger  Datum,  nsec
+Time  from  Signal  Source  Trigger  Datum,  nsec
+Amplitude  of  Pulse  Response,
+Volts
+Amplitude  of  Pulse  Response,
+Volts
+ Measured  12 inch  Trace
+ Modeled  12 inch  Trace
+ Measured  36 inch  Trace
+ Modeled  36 inch  Trace
+ Measured  24 inch  Trace
+ Modeled  24 inch  Trace
+ Measured  60 inch  Trace
+ Modeled  60 inch  Trace
+ 
+Figure 6: Experimental Comparison of GETEK PCB to RLGC Model in Odd Mode with Parameters 
+378 nH/m, 
+117 pF/m, tan
+0.011,
+5.45 7,
+/
+5 mils.
+L
+C
+E
+S
+δ
+σ
+η
+∞ =
+=
+=
+=
+=
+ (18334) 
+ 
+Figure 7 shows the modeled and actual pulse responses for several lengths of cable.   In 
+this case, the cable connectors were clearly a significant factor in the test system response 
+as is seen by the ringing in the response tail.   However, the de-embedding method The 
+reference measurement (shown in the first panel of Figure 7, used as input to the 
+numerical model, was taken from a relatively short, 1 meter cable. 
+ 
+
+12
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+14
+16
+18
+0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+2
+4
+Time  from  Signal  Source  Trigger  Datum,  nsec
+Time  from  Signal  Source  Trigger  Datum,  nsec
+Amplitude  of  Pulse  Response,
+Volts
+Amplitude  of  Pulse  Response,
+Volts
+6
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+22
+24
+26
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+30
+32
+34
+Measured  10 meter  Cable
+Modeled  10 meter  Cable
+Measured  15 meter  Cable
+Modeled  15 meter  Cable
+Measured  5 meter  Cable
+Modeled  5 meter  Cable
+Measured-Data
+Reference  Waveform
+From  1 meter  Cable +
+Connectors
+ 
+Figure 7: Experimental Comparison of SkewClear Cable to RLGC Model using Parameters 
+377 nH/m, 
+37.7 pF/m, tan
+0.0001,
+6.0 7,
+/
+17mils.
+L
+C
+E
+S
+δ
+σ
+η
+∞ =
+=
+=
+=
+=
+ (18324) 
+We also analyzed a Gore EyeOpener Plus cable (26 AWG), which was optimized by 
+Gore for operation at 2.5 Gbits/second.   The experimental data was not fitted well with 
+the traditional model for surface impedance.  We assumed that the cable was built with 
+stratified signal conductors and were able to fit the data much better using equation (1.10)
+for the surface impedance.  The fitted parameters for EyeOpener cable are: 
+ 
+ 
+1
+2
+1
+387 nH/m, 
+37.7 pF/m, tan
+0.0004,
+6 7,
+1 7,
+0.115 mil,
+/
+21 mils.
+L
+C
+E
+E
+S
+δ
+σ
+σ
+τ
+η
+∞ =
+=
+=
+=
+=
+=
+=
+ 
+(1.26) 
+ 
+The effective conductivity of the core conductor is so low, for a metal, that the material is 
+likely to be magnetic.   The effective conductivity of the bulk material is “reduced” by a 
+factor equal to the relative permeability of the material in our model. 
+ 
+Figure 8 shows the modeled and actual pulse responses for two lengths of the Gore 
+EyeOpener Plus cable.   The reference measurement, used as numerical model input, was 
+taken from a relatively short, 1 meter cable.  Because little or no geometry or 
+composition information was available for the cables, we do not claim that the fit 
+physical parameters truly reflect the physical cable characteristics.   We do claim that this 
+
+combination of parameters adequately describes the differential behavior of the cables 
+over a significant range of cable lengths. 
+ 
+0
+1
+2
+3
+4
+5
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Time  from  Signal  Source  Trigger  Datum,  nsec
+32
+33
+34
+35
+36
+37
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Amplitude  of  Pulse  Response,  Volts
+ Measured  5  meter  cable
+ Modeled  5  meter  cable
+ Measured  10  meter  cable
+ Modeled  10  meter  cable
+ 
+Figure 8: Experimental Comparison of EyeOpener Cable to RLGC Model using Parameters 
+1
+2
+1
+387 nH/m, 
+37.7 pF/m, tan
+0.0004,
+6 7,
+1 6,
+0.115 mil,
+/
+21 mils.
+L
+C
+E
+E
+S
+δ
+σ
+σ
+τ
+η
+∞ =
+=
+=
+=
+=
+=
+=
+ 
+(18446) 
+ 
+We conclude that the RLGC model, with appropriate extensions when necessary, 
+accurately describes a variety of cable transmission lines and PCB lines at 2.5 
+Gbits/second excitation.    
+ 
+40 Gbits/second Excitation 
+ 
+We shall now present the time-domain analysis with data taken at 40 Gbits/second 
+excitation.  While (in principle) an infinitely-sharp rise time at 2.5 Gbits/second will also 
+excite very high frequencies which will then be identified with by our method even at 
+low excitation frequency, it’s also true that equipment made to produce 40 Gbits/second 
+data will have significantly smaller rise times and that of equipment made to generate a 
+PRBS at lower frequencies.  Using such equipment is therefore experimentally better if 
+high frequencies are to be accurately modeled by this method. 
+ 
+In Figure 9, we show comparisons for model versus actual pulse responses for various 
+lengths of a PCB trace made using Nelco 1300 ™ material.   In this case we found much 
+
+improved experimental fit using a linear loss tangent model.   The graphs show quite a bit 
+of attenuation for this type of transmission line at these frequencies (which is not at all 
+surprising as this material is not intended for use up at these frequencies over any 
+reasonable length of trace).   We find the fit of these curves to be fairly good, but clearly 
+there exist differences between expected and actual traces which do not exist in the 2.5 
+Gbit/second data.   The frequency content of these waveforms is only significant up to 
+approximately 10 GHz of bandwidth, so we do not claim to have a model which matches 
+data above this bandwidth. 
+ 
+Model parameters for the 40 Gbits/second Nelco 1300 PCB traces are: 
+ 
+ 
+327 nH/m, 
+125 pF/m, /
+5.75 mils, tan =2E-13 , =5E7 S/m 
+L
+C
+S η
+δ
+ω σ
+∞ =
+=
+=
+(1.27) 
+ 
+Note that this model uses a loss tangent which is proportional to frequency. 
+ 
+Measured  Reference  (3"  Trace)
+20"  Trace
+34"  Trace
+42"  Trace
+Measured
+Modeled
+Measured
+Modeled
+Measured
+Modeled
+2.4
+0.0
+0.1
+0.2
+0.3
+2.5
+2.6
+2.7
+Time,  ns
+Pulse  Response,  Volts
+2.8
+2.9
+3.0
+1.1
+0.0
+0.1
+0.2
+0.3
+1.2
+1.3
+1.4
+Time,  ns
+Pulse  Response,  Volts
+1.5
+1.6
+1.7
+2.0
+0.0
+0.1
+0.2
+0.3
+2.1
+2.2
+2.3
+Time,  ns
+Pulse  Response,  Volts
+2.4
+2.5
+2.6
+2.4
+0.0
+0.1
+0.2
+0.3
+2.5
+2.6
+2.7
+Time,  ns
+Pulse  Response,  Volts
+2.8
+2.9
+3.0
+ 
+Figure 9: Experimental Comparison of Nelco 1300 PCB to RLGC Model at 40 Gbits/s using 
+Parameters 
+327 nH/m, 
+125 pF/m, /
+5.75 mils, tan =2E-13 , =5E7 S/m
+L
+C
+S η
+δ
+ω σ
+∞ =
+=
+=
+(20568) 
+Finally, in Figure 10 we show the results from data taken at 40 Gbits/second from an 
+advanced version of the SkewClear cable.   This version is designed for higher 
+frequencies and includes a modified ground conductor. 
+ 
+ 
+
+The model parameters for the upgraded SkewClear cable are: 
+ 
+ 
+451 nH/m, 
+40 pF/m, /
+20 mils, tan =1E-13 ,
+=5E7 S/m 
+L
+C
+S η
+δ
+ω σ
+∞ =
+=
+=
+ (1.28) 
+ 
+Note that again the loss tangent model is linear in frequency. 
+0.9
+1.0
+1.1
+1.2
+1.3
+1.4
+1.5
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Pulse  Response,  Volts
+Time, ns
+2.5
+2.6
+2.7
+2.8
+2.9
+3.0
+3.1
+0.0
+0.1
+0.2
+0.3
+0.4
+Pulse  Response,  Volts
+Time, ns
+ Measured
+ Modeled
+1 Meter  Reference,
+Differentially  Excited  and  Received
+1 Meter  Reference,
+Singly  Excited,  Differentially  Received
+5 Meter  Cable,
+Differentially  Excited  and  Received
+5 Meter  Cable,
+Singly  Excited,  Differentially  Received
+0.9
+1.0
+1.1
+1.2
+1.3
+1.4
+1.5
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Pulse  Response,  Volts
+Time, ns
+2.5
+2.6
+2.7
+2.8
+2.9
+3.0
+3.1
+0.0
+0.1
+0.2
+0.3
+0.4
+Pulse  Response, Volts
+Time, ns
+ Measured
+ Modeled
+Mismatch  Indicates
+Even  Mode  Leakage
+ 
+Figure 10: Experimental Comparison of Upgraded SkewClear Cable to RLGC model at 40Gb/s 
+under Single Ended and Differential Excitation Conditions using Model Parameters 
+451 nH/m, 
+40 pF/m, /
+20 mils, tan =1E-13 ,
+=5E7 S/m 
+L
+C
+S η
+δ
+ω σ
+∞ =
+=
+=
+ (20571) 
+In this case, the fit is nearly as excellent as for the 2.5 GHz case, even though the 
+significant frequencies in this waveform extend up to approximately 20 GHz.   The usual 
+model comparison (in the upper right graph) shows excellent fit except for the area 
+around 100 pS before the main pulse, which shows a relatively small blip of apparently 
+unknown origin.    
+ 
+We shall take this opportunity to show the types of conclusions one may draw from 
+investigations using time domain techniques.  We have shown similar data on the lower 
+portion of the figure except in this case the excitation is provided only on one side of the 
+differential cable, with the other input terminated.   Theoretically, or ideally, this 
+condition should produce exactly half the differential signal at the output terminals (and 
+indeed this is very well approximated for the 1 meter reference signals).   However,  the 
+singly-excited model comparison is rather weak, though (evidently) the errors apparently 
+nearly cancel when the system is both differentially excited and differentially received 
+(the upper right case).   The simplest explanation for this behavior is a mode leakage 
+between even and odd modes down the length of the line;  and indeed, we have shown by 
+
+similar methods that the even mode signal has a slightly higher velocity than does the odd 
+mode, such that the even mode shows up about 100 pS before the odd mode at a distance 
+of 5 meters.  
+ 
+Conclusions 
+We have developed a time-domain method for experimental verification of an RLGC 
+model, and showed adequate experimental fit over a wide variety of transmission line 
+types up to approximately 10 GHz in bandwidth.   We found that, depending on the 
+frequency and transmission line type, specific extensions were required to adequately 
+explain laboratory behavior.   We found that modifications to both the dielectric (shunt) 
+loss model and the resistive (series) loss models were necessary to adequately cover all 
+cases. 
+ 
+We have shown that, despite a fairly substantial list of assumptions, a PRBS time-domain 
+method can be used in a de-embedding fashion to remove many experimental 
+impediments.  Indeed, the PRBS technique can be used to easily identify and quantify 
+nonlinear effects. 
+ 
+We proposed an extremely efficient method for implementing the PRBS identification 
+process, which can either speed up offline computation when the waveform is large, or, 
+can be used in hardware to directly implement the method in real-time so long as an 
+appropriately complex ADC is available. 
+ 
+We note that the proposed technique is similar to (in fact, it is a simplification of) a 
+generalized TDT method, possessing a similar set of advantages and disadvantages when 
+compared to frequency-domain methods.   For the price of software processing and a real 
+time scope, this method can be used to turn a bit-error-rate test system into a reasonable 
+system identification station.  
+ 
+While we have shown laboratory data indicating the utility of this measurement method 
+using high-quality lab equipment,  we believe that this type of analysis will become most 
+attractive in self-test applications in product hardware.   As the analog-to-digital 
+converters (ADCs) in a SERDES migrate from traditional 1-bit comparators to those 
+required to support advanced signaling and detection schemes, we expect that methods 
+such as those described in this paper will be used to advance the operational and 
+diagnostic capabilities of the overall system, as they have in other systems [12]. 
+Acknowledgements 
+ 
+The authors would like to thank Eric Hanlon and Devon Post for providing test boards 
+and cables for this activity;  Jason Prairie for his data-taking expertise; Patrick Zabinski 
+for many instructive and seminal conversations; and Steve Richardson, Elaine Doherty, 
+and Deanna Jensen for generating the graphics for this report. 
+ 
+References 
+ 
+
+[1] Richard E. Matick.  Transmission Lines for Digital and Communication Networks. 
+IEEE Press, 1999. 
+ 
+[2] Simon Rano, John R. Whinnery, and Theodore Van Duzer.  Fields and Waves in 
+Communication Electronics.  John Wiley and Sons, second edition, 1984. 
+ 
+[3] Dean Palmer, Jon Coker, Michael Meyer, and Pablo Ziperovich.  Overwrite in thin 
+media measured by the method of pseudorandom sequences.  IEEE Transactions on 
+Magnetics, 24(6), November 1988. 
+ 
+[4] Reto Hermann.  Volterra Modeling of digital magnetic saturation recording channels.  
+IEEE Transactions on Magnetics, September 1990. 
+ 
+[5] James R. Wait. Electromagnetic Waves in Stratified Media.  IEEE Press, 1995. 
+ 
+[6] Dean Palmer, Pablo Ziperovich, Roger Wood, and Thomas Howell.  Identification of 
+nonlinear write effects using pseudorandom sequences.  IEEE Transactions on 
+Magnetics, volume 24, November 1988. 
+ 
+[7] D.A Smolyansky and S.D Corey.   Computing self and mutual capacitance and 
+inductance using even and odd TDR measurements. Electrical Performance of Electronic 
+Packaging, 2002. 
+ 
+[8]  Cherry Wakayama and Jeff Loyer.  Correlation between VNA and TDR/TDT 
+extracted s-parameters up to 20 GHz.   TDA Systems customer note. 
+ 
+[9] James R. Andrews.  Time domain spectrum analysis and s-parameter vector network 
+analysis.  Picosecond Labs app note AN-16A, November 2004.  
+ 
+[10]  Martin Schmatz et al.  A 22 Gbit/s PAM-4 receiver in 90 nm CMOS-SOI 
+technology.  Symposium on VLSI Circuits, 2005. 
+ 
+[11] Fidanboylu, K.M.; Riad, S.M. and A. Elshabini-Riad.   An enhanced time-domain 
+approach for dielectric characterization using stripeline geometry.  IEEE Transactions on 
+Instrumentation and Measurement, Volume 41,  Issue 1,  Feb. 1992 
+ 
+[12] United States Patent 5392295.  Error measurement circuit.  February 21, 1995. 
+ 
+[13] Engin, A.E.; Mathis, W.; John, W.; Sommer, G.; and Reichl, H.  Time-domain 
+modeling of lossy substrates with constant loss tangent.  Proceeding of  the 8th IEEE 
+Workshop on Signal Propagation on Interconnects,  May 2004. 
+ 
+[14] Coperich Branch, K.M et al. Physically consistent transmission line models for high-
+speed interconnects in lossy dielectrics.  Advanced Packaging, IEEE Transactions on  
+Volume 25,  Issue 2,  May 2002 Page(s):129 - 135 
+

4NE1T4oBgHgl3EQfmASv/content/tmp_files/2301.03293v1.pdf.txt

@@ -0,0 +1,957 @@
+Distributed Multirobot Control for
+Non-Cooperative Herding
+Nishant Mohanty∗, Jaskaran Grover∗, Changliu Liu, Katia Sycara
+The Robotics Institute, Carnegie Mellon University, Pittsburgh, USA
+{nishantm,jaskarag,cliu6,sycara}@andrew.cmu.edu
+Abstract. In this paper, we consider the problem of protecting a high-
+value area from being breached by sheep agents by crafting motions for
+dog robots. We use control barrier functions to pose constraints on the
+dogs’ velocities that induce repulsions in the sheep relative to the high-
+value area. This paper extends the results developed in our prior work
+on the same topic in three ways. Firstly, we implement and validate our
+previously developed centralized herding algorithm on many robots. We
+show herding of up to five sheep agents using three dog robots. Secondly,
+as an extension to the centralized approach, we develop two distributed
+herding algorithms, one favoring feasibility while the other favoring opti-
+mality. In the first algorithm, we allocate a unique sheep to a unique dog,
+making that dog responsible for herding its allocated sheep away from the
+protected zone. We provide feasibility proof for this approach, along with
+numerical simulations. In the second algorithm, we develop an iterative
+distributed reformulation of the centralized algorithm, which inherits the
+optimality (i.e. budget efficiency) from the centralized approach. Lastly,
+we conduct real-world experiments of these distributed algorithms and
+demonstrate herding of up to five sheep agents using five dog robots.
+Videos of these results are available at https://bit.ly/3bZq0dB.
+Keywords: Herding, Barrier Functions, Quadratic Programming
+1
+Introduction
+Recent developments in robotics and sensing have created significant interest
+among researchers to deploy multiple robots to operate cooperatively towards
+achieving a common goal. Many works have developed techniques to tackle real-
+world problems using multi-robot systems (MRS), like conducting surveys or
+automating warehouses [1,2], [3]. The major developments in MRS for enabling
+multiple robots to behave cooperatively have been based on interactions within
+a single team, i.e., a robot interacts with other robots in its group to achieve a
+given objective [4,5]. The main features of these types of algorithms are a) local
+∗ These authors contributed equally to this work.
+This research is supported by AFOSR FA9550-18-1-0097 and AFRL/AFOSR
+FA9550-18-1-0251
+arXiv:2301.03293v1  [cs.RO]  9 Jan 2023
+
+2
+Nishant Mohanty∗, Jaskaran Grover∗, Changliu Liu, Katia Sycara
+interaction, b) collision-free motion within the group, and c) achieving collective
+behavior using local interaction [6].
+In literature, there are studies on MRS that involve interaction between mul-
+tiple groups of agents. Here, along with the local interaction with group members,
+the individuals also interact with an external agent from another group. An ex-
+ample of this is a scenario where a group of adversarial robots has a goal of their
+own that might damage a given high-value unit. Here, a group of defenders must
+interact with the adversarial robots to ensure the safety of the unit. [7,8]. In this
+paper, we propose a provably correct controller for the group of defenders (“dog
+robots”) to prevent an adversarial group (the “sheep robots”) from breaching a
+protected zone. This is challenging because dog robots do not control the sheep
+robots directly; rather have to rely on the interaction dynamics between the dogs
+and sheep to influence the sheep’s behavior.
+In our prior work [9], we developed a centralized algorithm to solve this
+problem using control barrier functions. In this work, a) we provide more ex-
+perimental validation of the centralized algorithm, b) propose two distributed
+algorithms, and c) provide simulations and experiments to validate these algo-
+rithms. Our formulation computes the velocity of each dog locally to prevent
+sheep from breaching the protected zone(s). In the first distributed algorithm,
+we allocate each sheep to a unique dog and pose a constraint on that dog’s
+velocity to herd its allocated sheep away from the protected zone. We provide
+proof of feasibility of this approach, thus showing that whenever the number
+of sheep and dogs are equal, the herding problem is well-posed. Our previously
+proposed centralized algorithm lacked this feasibility guarantee. However, it did
+not necessitate equal numbers of dogs and sheep; in fact, in many experiments,
+fewer dogs than sheep were sufficient to herd all the sheep away. This obser-
+vation led us to develop the second algorithm. In this algorithm, we construct
+an iterative distributed approach that asymptotically attains the same veloci-
+ties as computed by the centralized approach, thereby attaining the same total
+optimality (measured in terms of the total movement the dogs exhibit) as the
+centralized approach and obviating the need to have equal numbers of dogs and
+sheep. We build on the dual-decomposition algorithms proposed in [10,11] for de-
+veloping this distributed algorithm. Both of our proposed distributed algorithms
+are compositional in nature i.e., we can protect multiple zones by including more
+constraints, as shown in figure 1(c). To highlight the performance of our formu-
+lation, we provide results from numerical simulations showing the success of our
+approach for multiple dogs against multiple sheep. Finally, we demonstrate our
+algorithm on real robots and show multiple dog robots successfully preventing
+the breaching of protected zones against multiple sheep robots.
+The outline of this paper is as follows: in section 2, we give a brief review of
+the prior work in this area. In section 3, we provide a mathematical formulation
+of the problem statement. In section 4, we show how to use control barrier
+functions to pose constraints on dog velocities. Section 5 provides simulations
+accepted in IEEE Conference on Decision and Control 2022
+
+Distributed Herding
+3
+and experimental results to demonstrate the proposed approach. Finally, we
+summarize our work in section 6 along with our directions for future work.
+2
+Prior Work
+The framework of multi-group interaction within MRS has many applications
+beyond the adversarial problem statements. The shepherding problem is an ex-
+ample of such a category. In [12,13], the authors have proposed methods to
+enable multiple shepherd agents to influence a flock of sheep by modeling the
+interaction as repulsion forces. The Robot Sheepdog Project [14,15] conducted
+a real-world demonstration of a shepherding algorithm where a group of mobile
+ground robots cooperatively herded a flock of ducks to a given goal location.
+In the literature, there are several works on non-cooperative shepherding as
+an example of a multi-group interaction type problem. The works like [13], [16],
+[17], [18], [19], [20]. deal with a problem where the sheep robots do not exhibit
+adversarial behavior. They do not have any goals of their own. However, they
+experience a repulsive force from the dog robots, which is exploited to produce
+the desired behavior in the sheep robots. For example, collecting all the sheep
+at some location and then driving them to a target goal.
+Differently from prior work, our sheep may or may not be adversarial. We call
+them adversarial if their goal lies inside the protected zone and non-adversarial
+otherwise. Our safe control synthesis approach remains the same regardless. The
+dog robots observe and generate their control commands considering the cohesion
+between the sheep robots, the attraction to their goal location, and the repulsion
+experienced by them from the dog robots. And as we use control barrier functions
+to generate the constraints on the velocity of the dog robots, it only requires the
+dynamics of the sheep to be represented as a symbolic function. Thus allowing
+for the sheep to experience any kind of attractive or repulsive forces.
+3
+Problem Formulation
+Consider a scenario with n sheep agents flocking towards a common goal loca-
+tion. One commonly assumed model for flocking is the Reynolds-Boids dynamics
+[21] that considers inter-sheep cohesive forces, inter-sheep repulsive forces, and
+attraction to a common goal. In the presence of dog agents, each sheep’s dy-
+namics would include repulsive forces from each dog robot. While en route to
+their goal, the sheep, having no knowledge about high-value regions in workspace
+(protected zones), pose a risk of breaching them. Thus, our problem is to orches-
+trate the motions of dog robots by capitalizing on the repulsions that the sheep
+experience from the dogs to prevent this breaching. Next, we pose this problem
+in formal terms.
+Consider the protected zone P ⊂ R2 as a disc centered at xP with radius
+Rp, i.e., P := {x ∈ R2| ∥x − xP ∥ ≤ Rp}. We denote the flock of sheep as S and
+the position of the ith sheep as xSi ∈ R2. The collective positions of all sheep
+is denoted as xall
+S := (xS1, xS2, ..., xSm). Similarly, we denote the set of all dogs
+
+4
+Nishant Mohanty∗, Jaskaran Grover∗, Changliu Liu, Katia Sycara
+using D. The position of the kth dog is xDk ∈ R2 and the positions of all dogs
+collectively is xall
+D := (xD1, xD2, ..., xDn). Each sheep follows single integrator
+dynamics ˙xSi := f i(xS1, ..., xSn, xD1, ..., xDn), given by
+˙xSi = uSi = kS
+�
+j∈S\i
+�
+1 −
+R3
+S
+∥xSj − xSi∥3
+�
+(xSj − xSi)
+�
+��
+�
+inter-sheep cohesion and repulsion
++
+kG (xG − xSi)
+�
+��
+�
+attraction to goal
++ kD
+�
+l∈D
+xSi − xDl
+∥xSi − xDl∥3
+�
+��
+�
+repulsion from dogs
+(1)
+Here, RS is a safety margin that each sheep tends to maintain with every other
+sheep, xG is the sheep’s desired goal and kS, kG and kD are proportional gains
+corresponding to the attractive and repulsive forces. We model each dog as a
+velocity controlled robot with the following dynamics:
+˙xDk = uDk ∀k ∈ {1, 2, · · · , n}
+(2)
+Before posing the problem, we state some assumptions on the dogs’ knowledge:
+Assumption 1. The dog robots have knowledge about the sheep’s dynamics i.e.
+(1) and can measure the sheep’s positions accurately.
+Assumption 2. Each dog robot can measure the velocities of other dog robots
+(by using numerical differentiation, for example).
+Problem 1. Assuming that the initial positions of the sheep xSi(0) /∈ P ∀i ∈
+S, the dog robots’ problem is to synthesize controls {uD1, · · · , uDn} such that
+xSi(t) /∈ P ∀t ≥ 0 ∀i ∈ S.
+4
+Controller Design
+In this section, we show two approaches to solve Problem 1, building on our
+previously proposed centralized algorithm [9]. Define a safety index h(·) : R2 −→
+R that quantifies the distance of Si from P:
+h(xSi) = ∥xSi − xP ∥2 − (r + Rp)2
+(3)
+Here r is a safety buffer distance. Thus, we require h(xSi(t)) ≥ 0 ∀t ≥ 0. We
+define x = (xall
+S , xall
+D ) as the aggregated state of all sheep and all dogs. To
+ensure, h(xSi(t)) ≥ 0 ∀t ≥ 0, we treat h(·) as a control barrier function require
+its derivative to satisfy
+˙h(x) + p1h(xSi) ≥ 0.
+(4)
+
+Distributed Herding
+5
+Here p1 is a design parameter and is chosen based to satisfy
+p1 > 0
+and
+p1 > −
+˙h(x(0))
+h(xSi(0)).
+(5)
+The first condition on p1 requires that the pole is real and negative. The second
+depends on the initial positions x(0) of all the sheep and dogs relative to the
+protected zone. Note that the constraint in (4) does not contain any dog velocity
+terms, which is what we require to control each dog. Therefore, we define the
+LHS of (4) as another control barrier function v(x) : R4n −→ R:
+v = ˙h + p1h,
+(6)
+and require its derivative to satisfy the constraint: ˙v(x) + p2v(x) ≥ 0. Here p2
+is another design parameter which must satisfy
+p2 > 0
+and
+p2 > −
+¨h(x(0)) + p1 ˙h(x(0))
+˙h(x(0)) + p1h(xSi(0))
+(7)
+Using (3), (6) and the constraint on the derivative, we get
+¨h(x) + α˙h(x) + βh(xSi) ≥ 0
+(8)
+where α := p1 + p2 and β := p1p2. The derivatives of h(·) are:
+˙h(x) = 2(xSi − xP )T ˙xSi = 2(xSi − xP )T f i(x)
+(9)
+¨h(x) = 2f T
+i f i + 2(xSi − xP )T
+� �
+j∈S
+JS
+jif i +
+�
+l∈D
+JD
+li uDl
+�
+(10)
+where the jacobians are defined as JS
+ji := ∇xSj f i(x) and JD
+li := ∇xDl f i(x)
+Note that (10) contains the velocity terms of all dogs. In [9], we leveraged this
+observation to obtain a linear constraint on the velocity of all dogs collectively
+for preventing sheep Si from breaching P:
+AH
+i uall
+D ≤ bH
+i ,
+where
+(11)
+AH
+i := (xP − xSi)T �
+JD
+1i, JD
+2i, · · · , JD
+ni
+�
+bH
+i := f T
+i f i + (xSi − xP )T �
+j∈S
+JS
+jif j + α(xSi − xP )T f i + β h
+2
+A centralized algorithm was developed that collectively computes the velocities
+of all dogs using the following QP
+uall
+D = arg min
+uall
+D
+∥uall
+D ∥2
+subject to
+AH
+i uall
+D ≤ bH
+i ∀i ∈ S.
+(12)
+Building on this centralized approach, in this paper, we develop two distributed
+approaches wherein we allow each dog to compute its velocity locally such that
+the computed velocities will make the dog herd the sheep away from P.
+
+6
+Nishant Mohanty∗, Jaskaran Grover∗, Changliu Liu, Katia Sycara
+4.1
+Approach 1: One dog to one sheep allocation based approach
+In this approach, we assume that we have an equal number of dogs and sheep. By
+exploiting this equality, we assign a unique sheep Si for i ∈ {1, · · · , n} to a unique
+dog Dk for k ∈ {1, · · · , n} and make Dk responsible for herding Si away from
+P. In other words, Dk computes a velocity uDk that repels Si from P thereby
+ensuring that xSi(t) /∈ P ∀t ≥ 0. The premise is that owing to the equality,
+each sheep will end up being herded by a unique dog, therefore, no sheep will
+breach the protected zone . Now while this strategy necessitates having an equal
+number of dogs and sheep, the benefit of this approach stems from the feasibility
+guarantee (that we prove shortly), which the centralized approach lacks. Simple
+algebraic manipulation of constraint (11) yields a constraint on the velocity of
+Dk as follows
+AH
+i uDk ≤ bH
+i ,
+where
+(13)
+AH
+i := (xP − xSi)T JD
+ki
+bH
+i := f T
+i f i + (xSi − xP )T ��
+j∈S
+JS
+jif j + αf i + β h
+2 +
+�
+l∈D\k
+JD
+li uDl
+�
+Here AH
+i ∈ R1×2 and bH
+i ∈ R. The term uDl in the expression of bH
+i is computed
+by using numerical differentiation of the positions xDl. We pose a QP to obtain
+the min-norm velocity for Dk as follows
+u∗
+Dk = arg min
+uDk
+∥uDk∥2
+subject to
+AH
+i uDk ≤ bH
+i
+(14)
+The obtained velocity u∗
+Dk guarantees that the protected zone P will not be
+breached by sheep Si by ensuring that h(xSi(t)) ≥ 0 ∀t ≥ 0. Since each dog
+in D is in-charge of herding exactly one sheep in S, feasibility of (13) ∀k ∈ D
+would ensure no sheep breaches P. Next, we show the conditions under which
+(14) remains feasible but first state some assumptions.
+Assumption 3. We make the following assumptions on the distances between
+pairs of agents:
+1. There exists a lower bound and upper bound on the distance between any pair
+of sheep, i.e, LS ⩽
+��xSi − xSj
+�� ⩽ MS, ∀i, j ∈ S and i ̸= j.
+2. There exists a lower bound on the distance between every sheep and dog, i.e.,
+∥xSi − xDk∥ ≥ LD ∀i ∈ S and k ∈ D.
+3. There exists a upper bound on the distance between each sheep and its goal
+i.e., ∥xSi − xG∥ ⩽ MG and between the sheep and the center of the protected
+zone i.e., ∥xSi − xP ∥ ⩽ MP .
+Note that although Si is assigned to Dk, the position of the remaining dogs
+{1, · · · , n}\k and the remaining sheep {1, · · · , n}\i do influence Dk’s constraint pa-
+rameters (AH
+i , bH
+i ), and in turn, its computed velocity u∗
+Dk.
+
+Distributed Herding
+7
+Theorem 1. In a scenario with ‘n’ dogs and ‘n’ sheep, with each dog assigned
+a unique sheep, the herding constraint (13) for a given dog is always feasible,
+provided assumptions 3 are met.
+Proof. See appendix (section 7).
+4.2
+Approach 2: Iterative distributed reformulation of (12)
+The distributed formulation proposed in (14) comes with a feasibility guarantee
+ensuring that all sheep will be herded away from P. While vital, this comes at
+the cost of requiring as many dog robots as the number of sheep agents. This
+is because, in a way, this equality ensures that controlling the sheep from the
+perspective of dog robots is not an underactuated problem. Be that as it may,
+in our simulations and experiments involving the centralized approach with an
+equal number of dogs and sheep, we frequently observed that not all dog robots
+needed to move to repel the sheep away from P i.e., equality may have been an
+overkill. Thus, in terms of budget efficiency, at least empirically, the centralized
+approach outweighs the distributed approach.
+This raises the question, can we convert the centralized algorithm of (12) into
+a distributed version that inherits the budget efficiency (optimality) promised by
+(12)? Indeed, we found out that [10,11] propose algorithms to convert constrained-
+coupled convex optimization problems (such as (12)) into distributed counter-
+parts. They combine techniques called dual decomposition and proximal min-
+imization and develop iterative distributed schemes which consist of local op-
+timization problems. The solutions to these optimization problems asymptoti-
+cally converge to the solution of centralized optimization under mild convexity
+assumptions and connectivity properties of the communication network. In our
+case, this network refers to the communication between dog robots. Below, we
+present the distributed dual sub-gradient method of [10,11] adapted to the costs
+and constraints of (12). This algorithm calculates an estimate of dog Dk’s ve-
+locity ˆuDk which, given large enough iterations Kmax, matches with the kth
+velocity component in the optimal velocities u∗all
+D
+returned by (12). Ak ∈ RnS×2
+refers to those columns of AH that correspond to uDk in uall
+D .
+Algorithm 1 Distributed Dual Subgradient for (12) (based on sec. 3.4.2 in [11])
+Initialize Lagrange Multiplier: µ0
+k = 0 ∈ RnS
+Evolution: t = 1, 2, · · · , Kmax
+Gather Multipliers µt
+r from Dr ∀r ∈ {1, · · · , nD}\k
+Average Multipliers: vt+1
+k
+=
+1
+nD
+�
+r∈{1,··· ,nD}\k µt
+r
+Local Solution: ut+1
+Dk = arg min
+u
+∥u∥2 + (vt+1
+k
+)T (Aku −
+1
+nD bH) = − 1
+2AT
+k vt+1
+k
+Update Multiplier: µt+1
+k
+=
+�
+vt+1
+k
++ γt
+�
+Akut+1
+Dk −
+1
+nD b
+��
++
+Return Average: ˆuDk = (1/Kmax) �Kmax
+t=1
+ut
+Dk
+
+8
+Nishant Mohanty∗, Jaskaran Grover∗, Changliu Liu, Katia Sycara
+5
+Results
+In this section, we provide simulation and real-world experimental results demon-
+strating our proposed distributed algorithms.
+5.1
+Simulation Results
+We first validate the first distributed algorithm and the feasibility proof given
+in 4.1. For this, we model the sheep with the Reynolds-Boids dynamics (1) with
+gains kS = 0.5, kG = 1 and kD = 0.1. The dogs use (14) to compute their
+velocities, where hyperparameters α and β are computed following (5) and (7).
+We chose a circular protected zone of radius Rp = 0.6m and center xP at origin.
+The sheep are initialized outside of the protected zone, and their goal location xG
+is chosen such that their nominal trajectory would make them breach the zone,
+thus necessitating intervention from dogs. The positions of dogs are initialized
+randomly within a certain range of the protected zone. In figures 1(a) and 1(b),
+we show two examples involving a) two dog robots vs. two sheep robots and b)
+three dog robots vs. three sheep robots. To demonstrate the compositionality of
+our approach, we consider two protected zones in figure 1(c) where we have four
+dogs defending both zones from four sheep. In all these simulations, none of the
+sheep breach any zone, thus demonstrating the correctness of our approach. In
+the interest of space, we skip the simulation results for the algorithm in 4.2 but
+do provide experimental results.
+(a) Two dogs v. two sheep. (b) Three dogs v. three sheep (c) Four dogs v. four sheep.
+Fig. 1: Preventing the breaching of the protected zone using our proposed dis-
+tributed algorithm in section 4.1. Here dogs are shown in blue and sheep in red.
+The green disc represents the protected zone. The nominal task of the sheep is
+to go straight towards goal xG. However, since this would result in infiltration
+of the protected zone, the dog intervenes using the control algorithm presented
+in (14). In Fig. 1(c), we defend two protected zones from four sheep.
+
+1.5
+1
+0.5
+0
+-0.5
+-1
+-1.5
+1
+01.5
+1
+0.5
+0
+-0.5
+1
+-1.5
+1
+01.5
+1
+0.5
+0
+-0.5
+1
+-1.5
+0
+-1Distributed Herding
+9
+5.2
+Robot Experiments
+In this section, we show the results obtained by performing robot experiments
+by implementing the distributed algorithms of section 4.1 and section 4.2. Ad-
+ditionally, we also present more experimental results for our prior centralized
+algorithm from [9] (because at the time, we did not have as many robots). We
+conduct these experiments in our lab’s multi-robot arena, which consists of a
+14ft × 7ft platform with multiple Khepera IV robots and eight Vicon cameras
+for motion tracking. Although Khepera robots have unicycle dynamics, [9] con-
+sists of a technique to convert the single-integrator dynamics (assumed for dogs
+and sheep) to linear and angular velocity commands for the robots.
+First of all, to build upon our previous work, we show additional experiments
+using centralized velocity computation of the dog robots (12). Figure 2 shows a
+case with 2 dog and 4 sheep robots. The dog robots have a green tail, and the
+sheep robots have an orange tail. The tails are pointing in the opposite direction
+of the robot’s heading angle. The protected zone is the green-colored circular re-
+gion. This figure shows the performance in the case of an underactuated system,
+i.e, there are more sheep against less number of dogs. Another example is shown
+in figure 3 where 3 dogs successfully prevent breaching against 5 sheep robots.
+Following that, multiple experiments were conducted using the distributed
+algorithm presented in section 4.1, which requires equal numbers of dogs and
+sheep. Figure 4 shows 4 dog robots against 4 sheep robots scenario. Here we
+take two protected zones and show that the dogs can protect both of them. This
+highlights the compositional nature of our proposed algorithm. We conducted
+experiments with 5 dog robots and 5 sheep robots, as shown in Figure 5. Here we
+can see some dog robots did not require to move as the assigned sheep were being
+prevented from entering the protected zone due to the configuration of the flock
+itself. Finally, we test our distributed algorithm presented in section 4.2. Figure 6
+shows a case where 2 dogs prevent the breaching of protected zone against three
+dogs. This highlights that our distributed approach can handle under-actuated
+scenarios. Figure 7 and figure 2 can be compared to see both centralized and
+distributed algorithm handling a similar scenario of 2 dogs against 4 sheep.
+6
+Conclusions
+In this paper, we developed a novel optimization-based distributed control tech-
+niques to enable multiple dog robots to prevent the breaching of protected zones
+by sheep agents. We provided proof of feasibility of the controller when n dog
+robots face an equal number of sheep robots. Additionally, we developed another
+distributed algorithm that iteratively computes a solution that agrees with the
+solution returned by the centralized problem without requiring equal number of
+dogs and sheep. We experimentally validated both distributed algorithms in ad-
+dition to validating our previously developed centralized control. We show that
+multiple dog robots can prevent breaching of protected zone in both simulation
+and real-world experiments. In future work, we aim for the dog robots to learn
+the dynamics of the sheep robots online while preventing them from breaching.
+
+10
+Nishant Mohanty∗, Jaskaran Grover∗, Changliu Liu, Katia Sycara
+(a) t = 0s
+(b) t = 5s
+(c) t = 12s
+(d) t = 30s
+Fig. 2: Experiments for Centralized Control: Two dogs defending the pro-
+tected zone from four sheep using centralized control algorithm (12) from our
+prior work [9]. Video at https://bit.ly/3OTAnOu.
+(a) t = 0s
+(b) t = 5s
+(c) t = 30s
+(d) t = 50s
+Fig. 3: Experiment for Centralized Control: Three dogs (green-tailed
+robots) defending a protected zone from five sheep (orange-tailed robots) us-
+ing centralized control (12) from our prior work [9]. Video at https://youtu.be/
+2 Xuxnd9jZw.
+
+leepGoaleepGoalleepGoaleepGoaleepGoaleepooaeepGoaieep GoalDistributed Herding
+11
+(a) t = 0s
+(b) t = 6s
+(c) t = 12s
+(d) t = 20s
+Fig. 4: Experiment for the distributed algorithm in section 4.1 : Four
+dogs (green-tailed robots) defending two protected zone from four sheep (orange-
+tailed robots). The goal position xG (red disc) is in extreme left that would
+encourage sheep to breach both zones. However, our proposed algorithm moves
+the dogs so that none of the zones get breached. Video at https://bit.ly/3yo9ziC.
+(a) t = 0s
+(b) t = 12s
+(c) t = 25s
+(d) t = 40s
+Fig. 5: Experiment for the distributed algorithm in section 4.1) : Five
+dogs (green-tailed robots) defending the protected zone from five sheep (orange-
+tailed robots). The sheep’s goal (red disc) is in the center of the protected zone.
+Eventually, in this scenario a deadlock occurs where all sheep come to a stop
+outside the protected zone. Video at https://bit.ly/3o51Cu1.
+
+leenGnaeereepGoal12
+Nishant Mohanty∗, Jaskaran Grover∗, Changliu Liu, Katia Sycara
+(a) t = 0s
+(b) t = 4s
+(c) t = 15s
+(d) t = 30s
+Fig. 6: Experiment for distributed algorithm in section 4.2) : Two dogs
+(green-tailed robots) defending the protected zone from three sheep (orange-
+tailed robots). The goal position xG (red disc) is at the center of the zone. Video
+at https://youtu.be/IbCjkR1ye0c.
+(a) t = 0s
+(b) t = 4s
+(c) t = 15s
+(d) t = 30s
+Fig. 7: Experiment for distributed algorithm in section 4.2) : Two dogs
+(green-tailed robots) defending the protected zone from four sheep (orange-tailed
+robots). This case is similar to the one shown in fig. 2. Video at https://youtu.
+be/51FoHZWFYC4.
+
+leepGoalheepGoalheepGoalheepGoalheepGoalheepGoalheepGoalDistributed Herding
+13
+7
+Appendix: Proof of feasibility for Approach 1
+Theorem 1. In a scenario with ‘n’ dogs and ‘n’ sheep, with each dog assigned
+a unique sheep, the herding constraint (13) for a given dog is always feasible,
+provided assumptions 3 are met.
+Proof. Our strategy to guarantee feasibility of constraint (13) relies on ruling
+out situations in which it is infeasible. (13) can become infeasible
+– either when AH
+i = 0 and bH
+i < 0 (possibility 1)
+– or when bH
+i = −∞ (possibility 2).
+To determine the conditions in which possibility 1 occurs, we calculate the de-
+terminant of JD
+ki as
+det(JD
+ki) =
+−2k2
+D
+∥xDk − xSi∥3
+The determinant det(JD
+ki) is non-zero as long as the distance between dog Dk and
+sheep Si is finite. Therefore, JD
+ki will have no null space, implying that AH
+i ̸= 0
+∀xSi ∈ R2, xDk ∈ R2. This rules out possibility 1 for infeasibility. To rule out
+possibility 2, we need to check for condition when bH
+i −→ −∞. Given bH
+i in (13),
+we find its worst case lower bound. Here f T
+i f i ≥ 0 and as we assume that at
+the current time step, the sheep is outside P, this ensures β h
+2 ≥ 0. By removing
+these terms, the lower bound of bH
+i
+can be given as
+bH
+i ≥
+�
+j∈S\i
+(xSi − xP )T JS
+jif j + (xSi − xP )T JS
+iif i +
+�
+l∈D\k
+(xSi − xP )T JD
+li uDl
++ α(xSi − xP )T f i
+(1)
+Using the triangle inequality on the RHS and Cauchy-Schwarz inequality on
+individual terms, we get
+bH
+i ≥
+�
+j∈S\i
+�
+−σmax
+�
+JS
+ji
+�
+∥xSi − xP ∥ ∥f j∥
+�
+− σmax
+�
+JS
+ii
+�
+∥xSi − xP ∥ ∥f i∥
+(2)
++
+�
+l∈D\k
+�
+−σmax
+�
+JD
+li
+�
+∥xSi − xP ∥ ∥uDl∥
+�
+− α∥xSi − xP ∥∥f i∥
+where σmax is the largest singular value of a matrix. Further, using the fact that
+the largest singular value of a matrix (σmax) is upper bounded by its Frobenius
+norm (σF ), we obtain
+bH
+i ≥
+�
+j∈S\i
+�
+−σF
+�
+JS
+ji
+�
+∥xSi − xP ∥ ∥f j∥
+�
+− σF
+�
+JS
+ii
+�
+∥xSi − xP ∥ ∥f i∥
+(3)
+�
+l∈D\k
+�
+−σF
+�
+JD
+ki
+�
+∥xSi − xP ∥ ∥uDl∥
+�
+− α∥xSi − xP ∥∥f i∥
+Now to compute this lower bound we make use of assumption 3. We use the
+dynamics in (1) to compute JS
+ii and obtain the upper bound on σF
+�
+JS
+ii
+�
+and use
+the bounds on distances from assumption 3 to get following upper bound:
+
+14
+Nishant Mohanty∗, Jaskaran Grover∗, Changliu Liu, Katia Sycara
+σF
+�
+JS
+ii
+�
+⩽
+�
+j∈S\i
+kS
+�
+√
+2 + (3 +
+√
+2)R3
+∥xSi − xSj∥3
+�
++
+√
+2kG +
+�
+l∈D\k
+�
+3 +
+√
+2
+�
+kD
+∥xSi − xDl∥3
+⩽ (n − 1)
+�
+√
+2kS + (3 +
+√
+2)kSR3
+L3
+S
+�
++
+√
+2kG + n
+��
+3 +
+√
+2
+�
+kD
+L3
+D
+�
+:= λM
+We omit the proof of this computation in the interest of space. Similarly, using
+the dynamics in (1), we compute an expression for JS
+ji and obtain an upper
+bound on σF
+�
+JS
+ji
+�
+as follows:
+σF
+�
+JS
+ji
+�
+⩽
+√
+2kS + (3 +
+√
+2)kSR3
+∥xS1 − xSj∥3 ⩽
+√
+2kS + (3 +
+√
+2)kSR3
+L3
+S
+:= λS
+Likewise, an upper bound of σF
+�
+JS
+li
+�
+, is given by
+σF
+�
+JS
+li
+�
+⩽ (3 +
+√
+2)kSR3
+∥xS1 − xDl∥3 ⩽ (3 +
+√
+2)kSR3
+L3
+D
+:= λD
+Lastly, we use obtain an upper bound on the dynamics of each sheep f i as:
+∥f i∥ ⩽
+�
+j∈S\i
+kS
+�
+∥xSi − xSj∥ +
+R3
+∥xSi − xSj∥2
+�
++ kG∥xG − xSi∥
++
+�
+l∈D
+kD
+∥xSi − xDl∥
+∥xSi − xDl∥3
+(4)
+Now we need to compute the maximum possible value of the RHS to get the
+upper bound of the sheep dynamics. The first term has a local minima at ∥xSi −
+xSj∥ = (2)1/3R. Therefore the maximum value can occur at either the lower
+bound or upper bound of ∥xSi − xSj∥. Thus the maximum value of the first
+term can be given as Fmax := max(kSLS + kS R3
+L2
+S , kSMS + kS R3
+M 2
+S ). Second term
+is maximum when ∥xG − xSi∥ = MG. The last term is maximum when distance
+of the sheep to the dogs are minimum, ∥xSi −xDk∥ = LD. Using these the upper
+bound on the sheep dynamics is computed as:
+∥f i∥ ⩽ (n − 1)Fmax + kGMG + nkD
+� 1
+L2
+D
+�
+Assuming that the velocity of the dog robots have an upper bound, and by
+taking the upper bound on the dynamics of all the sheep to be equal, the lower
+bound on bH
+i
+from 3 is (taking γ = −(α + λM + (n − 1)λS)Mp)
+bH
+i ⩾ γ
+�
+(n − 1)Fmax + kGMG + nkD
+L2
+D
+�
+− (n − 1)λDMP ∥uD∥max
+This shows that bH
+i has a finite lower bound, thus ruling out possibility 2. Thus,
+the herding constraint (13) for a one dog to repel one sheep from the protected
+zone is always feasible. Since each sheep in S is allocated to one unique dog in
+D, extension of this feasibility result to all sheep ensures that none of them will
+breach the protected zone.
+
+Distributed Herding
+15
+References
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+kiva systems and the grand challenges ahead,” IEEE Transactions on Automation
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+adversarial swarms with parameter uncertainty,” arXiv preprint arXiv:2108.04205,
+2021.
+8. T. Tsatsanifos, A. H. Clark, C. Walton, I. Kaminer, and Q. Gong, “Model-
+ing and control of large-scale adversarial swarm engagements,” arXiv preprint
+arXiv:2108.02311, 2021.
+9. J. Grover, N. Mohanty, W. Luo, C. Liu, and K. Sycara, “Noncooperative herd-
+ing with control barrier functions: Theory and experiments,” arXiv preprint
+arXiv:2204.10945, 2022.
+10. A. Falsone, K. Margellos, S. Garatti, and M. Prandini, “Dual decomposition
+for multi-agent distributed optimization with coupling constraints,” Automatica,
+vol. 84, pp. 149–158, 2017.
+11. G. Notarstefano, I. Notarnicola, A. Camisa et al., “Distributed optimization for
+smart cyber-physical networks,” Foundations and Trends® in Systems and Con-
+trol, vol. 7, no. 3, pp. 253–383, 2019.
+12. J.-M. Lien, O. B. Bayazit, R. T. Sowell, S. Rodriguez, and N. M. Amato, “Shep-
+herding behaviors,” in IEEE International Conference on Robotics and Automa-
+tion, 2004. Proceedings. ICRA’04. 2004, vol. 4.
+IEEE, 2004, pp. 4159–4164.
+13. A. Pierson and M. Schwager, “Controlling noncooperative herds with robotic
+herders,” IEEE Transactions on Robotics, vol. 34, no. 2, pp. 517–525, 2017.
+14. R. Vaughan, N. Sumpter, J. Henderson, A. Frost, and S. Cameron, “Robot con-
+trol of animal flocks,” in Proceedings of the 1998 IEEE International Symposium
+on Intelligent Control (ISIC) held jointly with IEEE International Symposium on
+Computational Intelligence in Robotics and Automation (CIRA) Intell.
+IEEE,
+1998, pp. 277–282.
+15. ——, “Experiments in automatic flock control,” Robotics and autonomous systems,
+vol. 31, no. 1-2, pp. 109–117, 2000.
+16. A. Pierson and M. Schwager, “Bio-inspired non-cooperative multi-robot herding,”
+in 2015 IEEE International Conference on Robotics and Automation (ICRA).
+IEEE, 2015, pp. 1843–1849.
+
+16
+Nishant Mohanty∗, Jaskaran Grover∗, Changliu Liu, Katia Sycara
+17. R. A. Licitra, Z. I. Bell, E. A. Doucette, and W. E. Dixon, “Single agent indirect
+herding of multiple targets: A switched adaptive control approach,” IEEE Control
+Systems Letters, vol. 2, no. 1, pp. 127–132, 2017.
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+herding of n-agents: A switched systems approach,” IFAC-PapersOnLine, vol. 50,
+no. 1, pp. 14 374–14 379, 2017.
+19. E. Sebasti´an and E. Montijano, “Multi-robot implicit control of herds,” in 2021
+IEEE International Conference on Robotics and Automation (ICRA). IEEE, 2021,
+pp. 1601–1607.
+20. M. Bacon and N. Olgac, “Swarm herding using a region holding sliding mode
+controller,” Journal of Vibration and Control, vol. 18, no. 7, pp. 1056–1066, 2012.
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+New York, NY, USA: Association for Computing Machinery,
+1987, p. 25–34.
+

69E1T4oBgHgl3EQfTgNE/content/tmp_files/2301.03078v1.pdf.txt

@@ -0,0 +1,822 @@
+Structural tuning magnetism and topology in a magnetic
+topological insulator
+Christopher Eckberg,1, 2, 3, 4, ∗ Gang Qiu,4, ∗ Tao Qu,4 Sohee Kwon,5 Yuhang
+Liu,5 Lixuan Tai,4 David Graf,6 Su Kong Chong,4 Peng Zhang,4 Kin L.
+Wong,4 Roger K. Lake,5 Mahesh R. Neupane,2, 5, 7 and Kang L. Wang4, †
+1Fibertek Inc., Herndon, Virginia 20171, USA
+2DEVCOM Army Research Laboratory, Adelphi, Maryland 20783, USA
+3DEVCOM Army Research Laboratory,
+Playa Vista, California 90094, USA
+4Department of Electrical and Computer Engineering,
+University of California, Los Angeles, California 90095, USA
+5Department of Electrical and Computer Engineering,
+University of California, Riverside, CA, 92521, US
+6National High Magnetic Field Laboratory,
+Florida State University, Tallahassee, Florida, 32310, USA.
+7Materials Science and Engineering Program,
+University of California, Riverside, CA, 92521, US
+(Dated: January 10, 2023)
+∗ These two authors contributed equally
+† wangkl@ucla.edu
+1
+arXiv:2301.03078v1  [cond-mat.mes-hall]  8 Jan 2023
+
+To date, the most widely-studied quantum anomalous Hall insulator (QAHI)
+platform is achieved by dilute doping of magnetic ions into thin films of the al-
+loyed tetradymite topological insulator (TI) (Bi1−xSbx)2Te3 (BST) [1–4]. In these
+films, long-range magnetic ordering of the transition metal substituants opens
+an exchange gap ∆ in the topological surface states, stabilizing spin-polarized,
+dissipationless edge channels with a nonzero Chern number C. The long-range
+ordering of the spatially separated magnetic ions is itself mediated by electronic
+states in the host TI, leading to a sophisticated feedback between magnetic and
+electronic properties. Here we present a study of the electronic and magnetic
+response of a BST-based QAHI system to structural tuning via hydrostatic pres-
+sure. We identify a systematic closure of the topological gap under compressive
+strain accompanied by a simultaneous enhancement in the magnetic ordering
+strength. Combining these experimental results with first-principle calculations
+we identify structural deformation as a strong tuning parameter to traverse a
+rich topological phase space and modify magnetism in the magnetically doped
+BST system.
+Time-reversal invariant Z2 TIs feature gapless edge and surface states protected by time-
+reversal symmetry. Due to this time-reversal symmetry requirement, electronic band struc-
+tures of Z2 TIs respond strongly to magnetic perturbation [5–7]. This relationship is notably
+exemplified in the realization of the quantum anomalous Hall effect in magnetic TI systems.
+In QAHIs, long-range magnetic order gaps the otherwise mass-less topological surface state.
+When the chemical potential is positioned within the exchange gap, the 2D density of states
+vanishes, and a chiral edge state represents the lone channel for electrical transport. The
+resulting phase is characterized by a combination of long-range magnetic order, quantized
+Hall conductivity, and vanishing longitudinal resistance; all of which persist in the ab-
+sence of an applied magnetic field. The promise of technologically significant phenomena in
+QAHI materials including dissipationless, non-reciprocal electrical transport [8, 9], quantized
+magneto-electric dynamics [10, 11], and exotic quasiparticle excitations [12, 13] to name a
+few, has stimulated a tremendous research effort in QAHI systems and magnetic topological
+matter in general.
+Though recent discoveries have notably expanded the landscape of known QAHI hosts
+[14–16], the most mature and widely studied QAHI platform is the Cr substituted
+2
+
+(Bi1−x−ySbxCry)2Te3 (CBST) system grown using molecular beam epitaxy (MBE). Due to
+the greatly reduced spin-orbit coupling strength of Cr compared with the Bi/Sb atoms it
+substitutes, in quantized CBST the concentration of magnetic dopants must be left rela-
+tively dilute lest they promote a topological phase transition to a trivial insulating state
+[6, 17]. The dilute magnetism and disordered, dual-doped crystal structure of CBST im-
+bues an unfortunate fragility onto the quantum anomalous Hall effect in CBST at elevated
+temperatures [18–21]. This fragility of the quantum anomalous Hall state presents a major
+hurdle limiting the technical applicability of these compounds. Operational temperatures
+may be improved to a degree by varying dopant concentrations and profiles [22, 23]. How-
+ever, in practice chemical optimization of these materials is a delicate and imperfect process,
+as chemical composition simultaneously impacts the positioning of the chemical potential,
+electronic band structure, disorder profile, and magnetic ordering strength. A cleaner tuning
+parameter to more directly engineer CBST band structures is therefore essential to improve
+CBST QAHI operating temperatures.
+Here we report the magnetic and electronic evolution of CBST in response to a continuous
+deformation of the crystal lattice via hydrostatic pressure. Pressure dependent experiments
+were performed on gated Hall bar devices at pressures up to 1.6 GPa and temperatures
+as low as 20 mK. Our experiments demonstrate the electronic and magnetic properties
+of CBST to be highly responsive to strain, with lattice compression both suppressing the
+QAHI phase and enhancing the magnetic order. First-principle calculations confirm these
+effects emerge from a structural driven evolution of the CBST band structure, and indicate
+a rich topological phase space may be addressed through the application of even larger
+pressures. Together, these experimental and theoretical results demonstrate crystal strain
+as an effective tuning parameter to selectively modify the low energy electronic structure
+in BST based magnetic topological matter, establishing structural engineering as a viable
+pathway to control critical material properties in the future.
+Experiments are conducted on 6 quintuple layer (QL) thick MBE grown CBST films with
+a magnetic Curie temperature TC of roughly 20 K. Data are presented for three different
+photolithographically defined Hall bar devices, labelled P1, P2, and P3. Samples P2 and
+P3 were fabricated simultaneously from the same wafer in field-effect transistor geometries,
+where an approximately 20 nm thick HfOx layer serves as the gate dielectric. These two
+devices display virtually identical behaviors over a wide range of temperature and magnetic
+3
+
+field [24], and, in the following, their properties will frequently be compared directly. P1
+meanwhile was fabricated from a separate wafer. The impact of pressure on the topological
+transport signatures and magnetism of these devices was studied using a piston cell equipped
+for electrical transport experiments. During experiment, P1 was measured in a dilution
+refrigerator while P2 and P3 were primarily studied in a 3He sorption cryostat.
+Taken
+together, data gathered on these different devices span nearly three decades in temperature
+ranging in regime from kbT << ∆ to kbT ≈ TC.
+We begin by presenting the ambient pressure properties of device P2 (Fig. 1). At TC,
+the systems develops a magnetization when subjected to a small external field.
+When
+cooled to lower temperatures, this magnetic order manifests a rapidly increasing anomalous
+Hall signal, and at temperatures well below TC ρxx begins to rapidly decrease as seen in
+Fig. 1 (a). At dilution refrigerator conditions, ρxx approaches zero while ρyx approaches the
+quantized value of h/e2 ≈ 25.8 kΩ. Field dependent hysteresis loops in a dilution refrigerator
+environment are presented in Fig. 1 (b). In these data, transitions between ρyx plateaus
+accompany the switching of the magnetic order in the system between the down and up
+states and mark a topological transition between C = 1 and C = −1. The ρxx peaks and ρyx
+zero crossings observed in the magnetic hysteresis loops occur at the magnetic coercive field
+µ0Hc and correspond with an Mz = 0 condition (Fig. 1 (b)).
+Figure 1 (c) demonstrates the gate response of device P2 at a temperature of T = 500 mK.
+At this relatively elevated temperature thermal excitations into dissipative states precludes
+the high quality quantization observed in the dilution cooled regime. Nevertheless, a clear
+ρxx (ρyx) minimum (maximum) is observed at an optimized gate voltage of roughly −1.5 V;
+indicative of the incipient QAHI phase. The magnetic coercive field µ0Hc, a rough avatar
+for the magnetic ordering strength, is also measured as a function of the gate voltage. We
+observe an enhanced magnetic order when the system is driven away from the charge neutral
+point. Such an enhancement in magnetism with the addition of carriers into the system has
+been previously reported, and is commonly attributed to itinerant carrier mediated RKKY
+interactions strengthening the coupling between Cr-ions [25–27]. In a narrow range near V c
+g ,
+however, µ0Hc exhibits very little if any response to the changing gate voltage (gray regime
+in Fig. 1 (c)). In this region the carrier concentration is minimized, and the Cr-Cr magnetic
+coupling is sustained by the van Vleck mechanism [5, 28, 29].
+Following ambient pressure characterization, QAHI devices were loaded into a piston
+4
+
+pressure cell (Fig. 2(a)) and studied at hydrostatic pressures up to 1.6 GPa (Fig. 2 (b)).
+Comparison to the related Sb2Te3 system suggests a nearly isotropic compression of roughly
+1% may be expected in our CBST films at the maximal pressure [30, 31], though clamping
+by the more rigid GaAs substrate may slightly mute the lattice compression experienced by
+our thin film devices [31, 32]. Despite the modest compression that may be anticipated in
+these experiments, our QAHI devices are nevertheless quite responsive to lattice tuning in
+the range of pressure studied. Figure 2 presents a summary of basic transport data collected
+at 1.5 K, indicating that the system trends away from quantized transport with shrinking
+unit cell size. This is evidenced by an increase in ρxx and concomitant reduction in ρyx
+seen in both gate voltage traces (Fig. 2 (c), 2 (d)) and field hysteresis loops (Fig. 2 (e), 2
+(f)). Despite the trend away from quantized transport, a ρyx (ρxx) maxima (minima) is still
+seen near V c
+g at all pressures. That the gate cannot recover the same degree of transport
+quantization at all pressures suggests the flow away from idealized QAHI behavior is due to
+a pressure driven modification of the electronic band structure rather than a rigid shift of the
+chemical potential, as the latter scenario could possibly be compensated for by sweeping out
+carriers using the gate. In fact, the observation of a consistent V c
+g at all pressures suggests
+any shifting of the Fermi energy within the pressure range explored is minimal.
+Comparison of the temperature and voltage dependent ρxx and ρyx at pressures of 0,
+0.7, and 1.6 GPa are shown as two-dimensional color plots in Figs. 3 (a-f), providing a
+qualitative visualization of a closing topological gap with increasing pressure. By fitting the
+temperature dependent ρxx at Vg = 0 V and µ0H = 2 T to an Arrhenius model (Fig. 3 g) the
+value of this gap is quantified at all pressures. The pressure dependence of the topological
+gap is summarized in Fig. 3 (i), indicating the gap remains intact, but decreases in a linear
+fashion by almost exactly a factor of 2 from 1.2 K to 0.6 K over the pressure range explored.
+Extrapolating the linear trend to zero suggests a critical pressure PC of approximately 3.3
+GPa, at which point we anticipate a topological phase transition away from the QAHI
+state to occur. To demonstrate that the QAHI phase does persist to the highest pressures
+measured in this study, in Fig. 3 (h) we present data collected on another device, sample P1,
+at a temperature of 20 mK and pressure of 1.6 GPa. At these conditions, we still observe
+conductivity values within 3% of the quantized expectation of e2/h, confirming that 1.6 GPa
+is insufficient to drive these samples out of the QAHI ground state.
+Intriguingly, while the transport gap closes in pressurized QAHI samples, we observe
+5
+
+a pressure driven enhancement in the magnetic order as demonstrated by an increasing
+coercive field. This enhancement is visible in Fig. 4, where the field dependent ρxx hysteresis
+loops collected from device P1 at a temperature of 20 mK are presented. The ρxx peaks
+are clearly pushed to higher fields with increasing pressure, reflecting the growth in µ0Hc
+from a value of 133 mT at 0.1 GPa to a notably larger 144 mT at 1.6 GPa. At the 20
+mK temperature where these data are collected kbT << ∆. Consequently, the strength of
+magnetic interactions dependent upon itinerant charge carriers are vanishingly weak and
+this enhanced magnetic order is presumably sustained through the van Vleck mechanism.
+In addition to the magnetic enhancement observed at 20 mK, an increasing coercive field
+under pressure is also observed in devices P2/P3 between 280 mK and 15 K, demonstrating
+this effect to be consistent between samples and persistent over a wide temperature range
+[24]. Gate-dependent data collected in P2/P3 demonstrate the magnetic enhancement can
+be maximized by gate-tuning towards the valence band; indicating that pressure likely also
+enhances the hole-mediated RKKY interaction in a manner consistent with previous reports
+in more traditional dilute magnetic semiconductors [33].
+The strengthened magnetic ordering indicates that it is unlikely that the pressure driven
+reduction in the transport gap is a consequence of a reduced exchange field at the Dirac sur-
+face states. Alternative sources that may account for the suppressed gap include increasing
+surface hybridization or occupation of delocalized, dissipative states. To understand what
+effects are dominant in our system, we perform first principle band structure calculations
+for a 6 QL thick slab of CBST host compound Sb2Te3. As Bi primarily functions as a
+counter-dopant in CBST [34], and Cr d-electrons do not contribute to the density of states
+at the Fermi energy [5], Sb2Te3 calculations capture the principal details of the CBST band
+structure [29] while notably excluding the material magnetism and resulting exchange gap.
+Thus, in the presented calculations, trends in the surface hybridization are observed directly
+and are not obfuscated by magnetic gapping of the surface band structure. Following pre-
+vious example [30, 35], pressure is simulated by isotropically compressing the boundaries
+of the computational unit and allowing all interior atoms to relax to their lowest energy
+configuration.
+Band structure calculations were performed in 1% strain increments between 0% and 4%.
+Calculations at 0% and -4% compressive strains are presented in Fig. 5 (a-d). Consistent
+with previous reports, we find that increasing pressure widens the direct, bulk band gap at
+6
+
+Γ, while simultaneously raising the energy of the valence band valley Evb between Γ and M
+[30]. At the computational thickness of 6 QL, we observe a small hybridization gap m in the
+surface bands even in the zero-compression limit (Fig. 5 (e)). This feature is amplified as
+the unit cell size is decreased, indicating increasingly pronounced hybridization between top
+and bottom surfaces of the TI under pressure. Finally, using the calculated band structures,
+we extract the van Vleck susceptibility χV V according to the relationship:
+χV V = 1
+N
+�
+k
+�
+Enk<µ<Emk
+4µ0µ2
+B
+⟨nk| ˆSz |mk⟩ ⟨mk| ˆSz |nk⟩
+Emk − Enk
+(1)
+Here µ0 is the vacuum permeability, µB is the Bohr magneton, ˆSz is the spin operator.
+n, |nk⟩, and Enk represent the band index, wave function and eigenvalue of nth band in
+valence bands at momentum k, while m, |mk⟩ and Emk correspond to conduction band
+states. Based upon these calculations we observe a clear though modest enhancement in
+χV V with increasing pressure (Fig. 5 (f)). This enhancement emanates primarily from states
+near Γ, suggesting it is born from increasing mixing between the inverted Sb p−
+1z and Te p−
+2z
+states in the compressed lattice.
+The above observations have significant implications to the QAH state. The pressure
+driven enhancement in m biases the system towards a trivial insulator phase, with the
+electronic phase transition occurring once the magnitude of m grows larger than that of
+the exchange gap ∆ex. Meanwhile, once the energy of the valence band valley exceeds the
+Fermi energy (i.e. Evb > 0) delocalized states in the valence band will populate down to
+lowest temperatures and the system will behave as a metal. In the case when Evb > 0 and
+∆ex > m carrier conduction in the chiral channels will relax internally through the dissipative
+valence band states, precluding transport quantization. The calculated evolutions of m, χV V ,
+and Evb with increasing pressure (Fig. 5 (e-g)) therefore imply a coalescence of metallic,
+insulating, and QAHI electronic ground states in the pressure tuned CBST system. Based
+upon these calculated band structures and the known evolutions of the hybridization gap,
+bulk state quantum confinement, and exchange energy with decreasing CBST thickness
+[18, 29, 36], we present a proposed topological phase diagram in layer thickness and pressure
+dependent parameter space in Fig. 5(h). These phase boundaries could be further adjusted
+by tuning along other axes such as external magnetic field [18, 29] or applied gate voltage.
+We will also note that the QAHI/insulator phase boundary presented here assumes that
+the hybridization gap is more responsive to pressure than the exchange gap, an assumption
+7
+
+supported by the relatively weak pressure effect on χvv (Fig. 5 (f)) compared with m (Fig. 5
+(e)). It is possible, however, that the exchange gap may feature its own pressure dependence,
+altering the trajectory of the ∆ex = m boundary.
+Having discussed the calculated pressure dependent band structure for this system, we
+now consider how these calculations comport with the experimentally determined results.
+Notably, our band structure calculations unambiguously indicate a trend away from the
+QAHI state in compressed tetradymite TIs, consistent with the experimental reality. While
+the calculations indicate that the electronic state beyond Pc may be either trivially insulat-
+ing or metallic depending upon the details of the material, in our samples we believe the
+topological phase transition occurring at Pc to be towards the metallic regime. We come to
+this conclusion through the observations of reduced ρxx at temperatures above TC, as well
+as reductions in the ρxx peaks at µ0Hc with increasing pressure; both of which suggest an
+increasing density of states near the Fermi energy. Meanwhile, the enhanced χvv observed in
+calculation captures the increasing magnetic ordering strength observed in our pressurized
+QAHI films.
+To conclude, these results establish lattice deformation as an effective, clean tuning pa-
+rameter for modifying the electronic and magnetic properties of alloyed QAHI materials.
+Though a significant material response is observed in the pressure range explored in this
+study, we believe increasing pressure may evoke even more dramatic electronic and mag-
+netic responses. Crucially, Pc is well below the 9+ GPa threshold at which a structural
+phase transition from rhombehedral to monoclinic crystallographic point groups has been
+previously reported in tetradymite TI systems [30, 37, 38], indicating future experiments
+may explore a significantly larger pressure range without concern of interference from ad-
+ditional structural phases. Finally, on the basis of these results, we propose that tensile
+strain, as opposed to its compressive counterpart studied here, may present an exciting
+tuning parameter to explore in future efforts to enhance QAHI behavior.
+8
+
+I.
+METHODS
+A.
+Material Growth
+All CBST films were grown in an ultra-high vacuum, Perkin-Elmer molecular beam epi-
+taxy (MBE) system. Epi-ready semi-insulating GaAs (111)B substrates were used for the
+growth. Before growth, the substrates were loaded into the MBE chamber and pre-annealed
+at the temperature of 630 °C in a Te-rich environment in order to desorb the oxide on the
+surface. During growth, the substrate was kept at 190 °C. High-purity Bi, Sb, Cr and Te
+sources were evaporated simultaneously from standard Knudsen cells. The growth process
+was monitored by the reflection high-energy electron diffraction (RHEED) in-situ, and the
+digital RHEED images were captured using a KSA400 system built by K-space Associates,
+Inc.
+Sharp and streaky lines in the RHEED pattern indicate good epitaxial crystalline
+quality.
+B.
+High pressure experiments
+Pressure was applied using a standard piston pressure-cell. To fit within the active area
+of the pressure cell, single devices were cut from pre-patterned wafers to dimensions of less
+than 3.0 mm, and were fixed to a fiber optic using epoxy to orient the sample within the
+pressure cell. Thin platinum wires were attached by hand to the contact pads of the device
+under test with silver paint. A small ruby chip was fixed to the tip of the fiber optic, which
+was used to calibrate the pressure at room temperature and again at low temperature. A
+PTFE cup was filled with Daphne 7575 oil and fixed in place over the sample platform so
+that the device was surrounded by the hydrostatic fluid. Once assembled, the cell was placed
+in a hydraulic press where a piston fed through a hole in the threaded top screw of the cell
+was used to add pressure. When the appropriate pressure was reached, the top screw was
+clamped, locking in the pressure.
+Transport measurements were collected using a low-frequency (< 10 Hz) lockin technique
+with ac excitations of 10 nA. Gate swept data display a small hysteresis based upon the
+gate history. To compensate for this effect and ensure consistency all data presented were
+taken during sweeps from +3.25 to −5 V.
+Cryogenic sample environments for ambient pressure experiments were maintained us-
+9
+
+ing a Quantum Design Physical Property Measurement System equipped with a dilution
+refrigerator insert. High pressure measurements were conducted in high magnetic field cells
+SCM-1 and SCM-2 at the National High Magnetic Field Laboratory in Tallahassee. SCM-1
+is equipped with a dilution refrigerator cryogenic environment while SCM-2 was operated
+with pure 3He cooled using a sorption pump. Both SCM-1 and SCM-2 are equipped with
+18 T superconducting magnets. To compensate for the remnant field of the superconduct-
+ing magnet, the field dependent data were calibrated using a Hall sensor. Additionally, to
+account for the magnetoresistance of the SCM-2 thermometers, the temperatures used in
+Figs. 3 and 4 were calibrated using the strong temperature dependence of the QAHI mate-
+rial itself. For details of the field and temperature calibration processes please refer to the
+Supplemental Information [24].
+C.
+First principle calculations
+We perform first-principles calculations as implemented in the Vienna Ab Initio Simu-
+lation Package (VASP) [39].The Perdew, Burke, Ernzerhof (PBE) form of the generalized
+gradient approximation is used as the exchange-correlation functional [40]. The computa-
+tional cell employed is constructed from six QL Sb2Te3 slabs stacked with a 40 ˚A thick
+vacuum region. We apply an energy cutoff of 500 eV and a 8x8x1 Γ-centered k-grid to
+optimize cell structure and atomic positions. The optimized lattice constant of the slab is
+a = b = 4.3307 ˚A and c = 31.09 ˚A. Tri-axial compressive strains between -4.0% and 0.0%
+are applied by shrinking the perimeter of the Sb2Te3 slab. At each strain, atomic positions
+inside the unit cell are allowed to relax in all directions. Spin-orbit coupling is included
+during the charge density relaxation for electronic band structures.
+Based upon the band structure, van Vleck susceptibilities were calculated according to
+Eq. 2 [5].
+χV V = 1
+N
+�
+k
+�
+Enk<µ<Emk
+4µ0µ2
+B
+⟨nk| ˆSz |mk⟩ ⟨mk| ˆSz |nk⟩
+Emk − Enk
+(2)
+where, as described in the main text, µ0 is the vacuum permeability, µB is the Bohr magne-
+ton, ˆSz is the spin operator. n, |nk⟩, and Enk represent the band index, wave function and
+eigenvalue of nth band in valence bands at momentum k, while m, |mk⟩ and Emk correspond
+to conduction band states. The susceptibility averages over k points in the first Brillouin
+10
+
+zone, where N is the number of k points. We focus on the k symmetric lines (K − Γ − M)
+around the Dirac point which make the susceptibility calculation feasible.
+The wave functions |nk⟩ is a spinor, with both spin up and spin down components due
+to spin-orbit coupling, as shown in Eq. 3.
+|nk⟩ =
+�
+�ψ↑
+nk(r)
+ψ↓
+nk(r)
+�
+�
+(3)
+Here ψ↑
+nk(r) and ψ↓
+nk(r) are spin up and down components of a real space wave function.
+The real space wave functions are found through summing over plane-wave vectors G and
+their associated plane-wave coefficients. The wavefunctions are read from WAVECAR files,
+produced by VASP [41], with a cut-off plane-wave energy of 500 eV.
+II.
+ACKNOWLEDGMENTS
+C.E. is an employee of Fibertek, Inc. and performs in support of Contract No.W15P7T19D0038,
+Delivery Order W911-QX-20-F-0023. The views expressed are those of the authors and do
+not reflect the official policy or position of the Department of Defense or the US government.
+The identification of any commercial product or tradename does not imply endorsement or
+recommendation by Fibertek Inc. This work was supported by the NSF under Grants No.
+1936383 and No. 2040737, the U.S. Army Research Office MURI program under Grants No.
+W911NF-20-2-0166 and No. W911NF-16-1-0472. A portion of this work was performed at
+the National High Magnetic Field Laboratory, which is supported by the National Science
+Foundation Cooperative Agreement No.
+DMR-1644779 and the State of Florida.
+This
+work was supported in part by the DEVCOM Army Research Laboratory (ARL) Research
+Associateship Program (RAP) Cooperative Agreement(CA) W911NF-16-2-0008. This work
+used the Extreme Science and Engineering Discovery Environment (XSEDE) [42], which
+is supported by National Science Foundation Grant No. ACI-1548562 and allocation ID
+TG-DMR130081.
+11
+
+III.
+AUTHOR CONTRIBUTIONS
+C.E., G.Q., L.T., and K.L.W. conceived of and designed the experiments. QAHI materials
+were grown by P.Z. and L.T.. Devices were fabricated by G.Q., S.K.C., and K.W.. Pressure
+dependent transport measurements were performed by C.E., G.Q., and D.G.. First principle
+calculations were performed by S.K., Y.L., and M.N., and T.Q. calculated the van Vleck
+susceptibilities. C.E., G.Q., and K.L.W. wrote the manuscript with contributions from all
+authors.
+IV.
+DATA AVAILABILITY
+The data represented in the figures are available with the online version of this paper.
+All other data that supports the plots within this paper and other findings of this study are
+available from the corresponding author upon reasonable request.
+V.
+COMPETING INTERESTS
+The authors declare no competing interests.
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+15
+
+0.1
+1
+10
+0
+10
+20
+30
+ 
+T (K)
+�  (kΩ)
+TC
+(a)
+� xx
+� yx
+0
+10
+20
+30
+M (emu/cc)
+-0.5
+0.0
+0.5
+-1.0
+-0.5
+0.0
+0.5
+1.0
+1.5
+1 mT
+� xx
+ �  (h/e
+2)
+ � 0H (T)
+� yx
+50 mK
+(b)
+-5
+0
+3
+135
+140
+145
+� 0Hc (mT)
+Vg
+� xx
+� yx
+(c)
+500 mK
+0.0
+0.5
+1.0
+ �  (h/e
+2)
+FIG. 1.
+Summary of magnetic and electrical properties in a QAHI. (a) Temperature
+dependent ρxx (red) and ρyx (black) are presented for a QAHI device P2. Temperature dependent
+sample magnetization acquired using a piece of unpatterned film is included for comparison (blue).
+All data was collected at a magnetic field of 1 mT. Curie temperature denoted in the figure is
+determined by Arrott analysis [24]. (b) Field dependent ρxx
+and ρyx
+data collected at 50mK
+demonstrating well-quantized transport behavior. (c) Gate dependent ρxx (red), ρyx (black), and
+magnetic coercive field µ0Hc (blue) at a temperature of 500 mK are displayed. In these data, µ0Hc
+was determined by the location of the zero crossing points of the Hall bar’s two ρyx channels. The
+error bars, meanwhile, represent the standard deviation of the four separate transitions measured
+(i.e. up-down and down-up transitions in each channel). Grey shading denotes the center of the
+emerging topological gap.
+16
+
+H
+𝜀zz
+𝜀xx
+𝜀yy
+Cr
+Te
+Bi/Sb
+(a)
+(b)
+12
+14
+16
+18
+2 T
+𝜌xx (kΩ)
+(c)
+1.5 K
+-5.0
+-2.5
+0.0
+2.5
+10
+15
+20
+0 GPa
+0.3 GPa
+0.7 GPa
+1.2 GPa
+1.6 GPa
+(d)
+𝜌yx (kΩ)
+Vg
+15
+20
+25
+(e)
+𝜌xx (kΩ)
+1.5 K
+0 V
+Increasing P
+-0.5
+0.0
+0.5
+1.0
+-20
+-10
+0
+10
+20
+(f)
+𝜌yx (kΩ)
+μ0H (T)
+Increasing P
+FIG. 2. Pressure dependent ρxx and ρyx in a QAHI device. (a) Schematic of pressure
+dependent experiment, depicting the geometry of the pressure cell used and the orientation of the
+sample plane and the external magnetic field. (b) Cartooned depiction of hydrostatic pressure effect
+on CBST unit cell, emphasizing the roughly isotropic compression anticipated in these experiments.
+(c,d) Gate dependent measurements of ρxx (c) and ρyx (d) collected at 1.5 K and 2 T. (e,f) Field
+dependent ρxx (e) and ρyx (f) collected at 1.5 K and a Vg of 0 V. For data presented in panels
+(c-f), 0 GPa data were collected on device P3 while all data at non-zero pressure were collected
+from sample P2.
+17
+
+ 0 GPa
+ 0.33 GPa
+ 0.7 GPa
+ 1.2 GPa
+ 1.6 GPa
+0.5
+1.0
+1.5
+(a)
+(c)
+(b)
+T (K)
+P = 0 GPa
+0.7 GPa
+1.6 GPa
+� yx (h/e
+2)
+0.00
+0.25
+0.50
+0.75
+-5
+0
+3
+0.5
+1.0
+1.5
+(d)
+ 
+ 
+Vg
+T (K)
+-5
+0
+3
+(f)
+Vg
+0.50
+0.75
+1.0
+� xx (h/e
+2)
+-5
+0
+3
+(e)
+Vg
+1
+2
+3
+0.2
+0.4
+0.6
+(g)
+ 
+ 
+� xx (h/e
+2)
+1/T (1/K)
+-0.5
+0.0
+0.5
+-1
+0
+1
+(h)
+� xx
+� xy
+1.6 GPa
+ 
+ 
+� ��(e
+2/h)
+� 0H (T)
+20 mK
+0
+2
+4
+0.0
+0.5
+1.0
+(i)
+ 
+∆ (K)
+P (GPa)
+PC
+FIG. 3. Evolution of topological gap with increasing pressure. (a-f) Temperature and gate
+dependent ρxx and ρyx are presented at a constant field of µ0H = 2 T, and pressures of 0 GPa
+(a,d), 0.7 GPa (b,e), and 1.6 GPa (c,f). Contour lines are included at values of 0.125, 0.25, 0.5,
+and 0.558 h/e2 in ρxx color plots, and at values of 0.5, 0.75, 0.9, and 0.95 in ρyx. (g) Logarithm of
+longitudinal resistance values presented versus 1/T. The linearity of the curves when presented in
+this fashion confirm thermally activated transport behavior of the form ρxx(T) ∝ exp(−∆/2kBT).
+(h) Magnetic hysteresis loop of sample P1 collected at a pressure of 1.6 GPa and temperature of
+20 mK confirming the persistence of high quality quantization at dilution temperatures and high
+pressures. (i) Pressure dependence of topological gap determined by temperature dependencies
+presented in (g). Linear extrapolation to zero predicts a critical pressure Pc of approximately 3.3
+GPa.
+18
+
+-500
+0
+500
+0
+1
+ 
+50
+100
+150
+200
+250
+0.0
+0.5
+1.0
+1.5
+1.6 GPa
+0.9 GPa
+0.1 GPa
+ � xx (h/e
+2)
+� 0H (mT)
+20 mK
+FIG. 4. Evolution of magnetism under hydrostatic pressure. Pressure dependent evolution
+of ρxx hysteresis loops are shown for sample P1 at 20 mK and pressures of 0.1, 0.9, and 1.6 GPa.
+The inset displays the full magnetic hysteresis loops, while the main figure is zoomed to the region
+near µ0Hc shaded in blue in the inset.
+19
+
+Insulator
+QAHI
+Metal
+E   = E
+vb
+F
+ex
+𝛥  = m
+-0.5
+0.0
+0.5
+E-ECNP (eV)
+(a)
+𝜀 = 0 % (P = 0 GPa)
+K
+𝛤
+M
+-100
+-50
+0
+50
+100
+E-ECNP (meV)
+(c)
+𝜀 = -4 % (P = 6.8 GPa)
+(b)
+K
+𝛤
+M
+m
+(d)
+Evb
+0
+2
+4
+6
+-50
+0
+50
+100
+Evb (meV)
+P (GPa)
+(g)
+(h)
+10
+20
+30 0
+1
+2
+3
+4
+𝜀 (%)
+(e)
+m (meV)
+3.5
+4.0
+(f)
+𝜒VV (10
+-10m
+3/mol)
+t
+P
+FIG. 5. Electronic evolution with hydrostatic pressure. (a-d) Calculated electronic band
+structure of a 6 QL thick Sb2Te3 slab with isotropic lattice compressions of 0% (a) and -4%. To
+simplify their comparison, the calculated band structures are shifted vertically so that the center
+of the surface band gap (ECNP ) rather than the calculated EF is positioned at zero energy. (b).
+Zoomed band structures near EF and Γ are shown in (c) (0%) and (d) (4%), highlighting the
+pressure dependencies of the hybridization gap m and the energy of the valence band valley Evb.
+The locations of the zoomed regions are marked in (a) and (b) by boxes. (e-g) Pressure dependence
+of m (e), χV V (f), and Evb (g) are presented with dashed lines as a guide for the eye. (h) Proposed
+zero-temperature topological phase diagram for the CBST system as a function of pressure P and
+film thickness t. Cartooned band structures representative of the electronic ground state are shown
+in the four distinct regions in this topological phase space. In these simplified cartoons, the red
+and blue lines represent the spin split surface bands, while the bulk valence band is presented in
+black.
+20
+

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+Prepared for submission to JCAP
+Constraining primordial
+non-Gaussianity by combining
+next-generation galaxy and 21 cm
+intensity mapping surveys
+Sheean Jolicoeur1, Roy Maartens1,2,3, Simthembile Dlamini1
+1Department of Physics & Astronomy, University of the Western Cape, Cape Town 7535,
+South Africa
+2Institute of Cosmology & Gravitation, University of Portsmouth, Portsmouth PO1 3FX,
+United Kingdom
+3National Institute for Theoretical & Computational Sciences (NITheCS), Cape Town 7535,
+South Africa
+Abstract. Surveys of the matter distribution contain ‘fossil’ information on possible non-
+Gaussianity that is generated in the primordial Universe. This primordial signal survives
+only on the largest scales where cosmic variance is strongest. By combining different surveys
+in a multi-tracer approach, we can suppress the cosmic variance and significantly improve
+the precision on the level of primordial non-Gaussianity. We consider a combination of an
+optical galaxy survey, like the recently initiated DESI survey, together with a new and very
+different type of survey, a 21 cm intensity mapping survey, like the upcoming SKAO survey. A
+Fisher forecast of the precision on the local primordial non-Gaussianity parameter fNL, shows
+that this multi-tracer combination, together with non-overlap single-tracer information, can
+deliver precision comparable to that from the CMB. Taking account of the largest systematic,
+i.e. foreground contamination in intensity mapping, we find that σ(fNL) ∼ 3.
+arXiv:2301.02406v1  [astro-ph.CO]  6 Jan 2023
+
+Contents
+1
+Introduction
+1
+2
+Multi-tracer power spectra
+2
+3
+Modelling the surveys
+3
+3.1
+Noise
+5
+3.2
+Intensity mapping beam and foregrounds
+6
+4
+Fisher forecast
+7
+5
+Conclusion
+9
+1
+Introduction
+Constraining the local primordial non-Gaussianity (PNG) parameter fNL presents an impor-
+tant challenge for next-generation large-scale structure surveys. Local-type PNG affects the
+power spectrum of dark matter tracers by inducing a scale-dependent correction to the tracer
+bias [1, 2]. This correction is strongly suppressed on small to medium scales, but on very
+large scales, it is ∝ fNL(H2
+0/k2). This means that ultra-large survey volumes are required for
+high-precision constraints.
+The Planck survey of the cosmic microwave background (CMB) gives the current state-
+of-the-art constraint σ(fNL) = 5.1 from the bispectrum [3]. Future CMB surveys will lead
+to an improvement on the Planck precision, but not by much and not sufficient to access
+σ(fNL) ≲ 1. This level of precision would enable us to rule out many single-field and multi-
+field inflationary scenarios [4–6]. The hope is that this goal can be achieved by using the
+extra modes in 3-dimensional surveys of the large-scale structure (see e.g. [7–12]). However,
+extracting fNL from these surveys faces formidable difficulties:
+• Observational systematics on very large scales [13–19]. For our simplified Fisher anal-
+ysis, we neglect these systematics, except for the foreground contamination of 21 cm
+intensity mapping (see below).
+• Astrophysical complications entailed in the modelling of tracer clustering, in particular,
+the uncertainties in modelling the scale-dependent bias [20, 21]. We use a simplified
+model since a full treatment requires simulations, beyond the scope of this paper.
+• A theoretical systematic arising from the neglect of lensing magnification and other
+relativistic lightcone effects on the power spectrum, which can bias the best-fit fNL
+[22–35]. Although the bias on best-fit can be large, the error σ(fNL) is not significantly
+affected and we therefore omit these lightcone effects.
+• Cosmic (or sample) variance that dominates ultra-large scale measurements. We deal
+with this problem via a multi-tracer approach – which also helps to mitigate uncorrelated
+systematics.
+– 1 –
+
+Recent constraints on fNL have used the BOSS galaxy [36] and eBOSS quasar [37]
+samples, with the tightest constraint to date of σ(fNL) = 21 from eBOSS [38]. The much
+larger volumes of upcoming surveys should facilitate significant improvement on this precision.
+But if we rely on individual surveys using the power spectrum of single tracers, it turns out
+that σ(fNL) ≲ 1 is not achievable – even if we neglect systematics and complexities in scale-
+dependent bias [11]. The fundamental problem is cosmic variance.
+A way to evade cosmic variance is the multi-tracer technique [39–48]. Forecasts indicate
+that multi-tracing next-generation surveys can surpass the CMB precision, and in some cases
+can achieve σ(fNL) ≲ 1 [47–60]. The performance of the multi-tracer improves when the
+difference in tracer properties, especially the Gaussian clustering biases, is significant. It also
+helps to suppress uncorrelated systematics.
+In this paper, our aim is not to identify survey combinations that can achieve σ(fNL) ≲ 1
+– for this purpose, one typically requires the very high number densities of photometric sur-
+veys [47, 51]. Instead, our aim is to investigate the improvements over single-tracer constraints
+when using a new type of spectroscopic large-scale structure survey – neutral hydrogen (HI)
+intensity mapping of the 21 cm emission line – in combination with spectroscopic galaxy sur-
+veys. Such a pair of spectroscopic samples has very different clustering biases and systematics,
+which accentuates the advantages of a multi-tracer analysis.
+We use a simple Fisher forecast on mock surveys, which are based on next-generation
+surveys that are starting to observe or close to starting. The two surveys we have in mind
+are: the Dark Energy Spectroscopic Instrument (DESI), with the Bright Galaxy Sample
+(BGS) and Emission Line Galaxy (ELG) samples [61, 62], and the Square Kilometre Array
+Observatory (SKAO), with intensity mapping surveys in Bands 1 and 2 [56, 63]. Galaxy
+number count surveys are well established, but a cosmological HI intensity mapping survey
+has not yet been implemented.
+However, pilot surveys on the SKAO precursor telescope
+MeerKAT have already been used to:
+(a) measure the cross-power between MeerKAT and WiggleZ galaxy surveys [64], and
+(b) start developing a pipeline for the planned SKAO surveys [65, 66].
+We combine pairs of these surveys at low and high redshifts, using the Fisher single-
+tracer constraints in the non-overlap volume and the multi-tracer constraints in the overlap
+volume. The results show a reasonable improvement for the low-redshift pair, and a significant
+improvement for the high-redshift pair, compared to the standard single-tracer forecasts for
+σ(fNL).
+The combination of all low- and high-redshift surveys improve on Planck, with
+σ(fNL) ∼ 3. This constraint is based on avoiding foreground contamination of HI intensity
+mapping on very large scales. Foreground contamination of HI intensity mapping has a strong
+effect on the precision: if we neglect this foreground contamination, we find that σ(fNL) ∼ 1.
+2
+Multi-tracer power spectra
+At first order in perturbations, the density (or temperature) contrast of a tracer A is related
+to the matter density contrast δ by
+∆A(z, k) =
+�
+bA(z) + f(z)µ2�
+δ(z, k) ,
+(2.1)
+where bA is the (Gaussian) clustering bias, µ = ˆk · n is the projection along the line-of-sight
+direction n, and f is the linear growth rate. We assume bA has a known redshift evolution
+βA, following [67], so that
+bA(z) = bA0 βA(z) ,
+(2.2)
+– 2 –
+
+where the amplitude bA0 is a free parameter.
+We highlight the point that a multi-tracer
+approach reduces the impact of this assumption on σ(fNL) [35].
+The Fourier power spectra at tree-level are given by
+�
+∆A(z, k)∆B(z, k′)
+�
+= (2π)3PAB(z, k)δD(k + k′) ,
+(2.3)
+where the linear matter power spectrum (computed using CLASS [68]) is
+PAB =
+�
+bA + fµ2��
+bB + fµ2�
+P.
+(2.4)
+In the presence of local PNG, the bias acquires a scale-dependent correction:
+ˆbA(z, k) = bA(z) + bAφ(z)
+fNL
+M(z, k) ,
+M(z, k) =
+2
+3Ωm0H2
+0
+D(z)
+gin
+T(k) k2 .
+(2.5)
+Here T(k) is the matter transfer function (normalized to 1 on very large scales), D is the
+growth factor (normalized to 1 today) and gin is the initial growth suppression function,
+defined deep in the matter era. For ΛCDM
+gin = 3
+5
+�
+1 + z
+�
+D
+�
+1 + 2f
+3Ωm
+�
+.
+(2.6)
+The non-Gaussian bias factor bAφ is halo-dependent and should be determined with the
+aid of simulations [20, 69, 70]. A simplified model reduces it to
+bAφ(z) = 2δc
+�
+bA(z) − pA
+�
+,
+(2.7)
+where the critical collapse density is given by δc = 1.686 and pA are halo-dependent constants.
+In the simplest (universal) halo model, pA = 1 for all tracers A.
+But this model is not
+consistent with simulations [20, 69, 70]. An improved (but still over-simplified) model allows
+the constant pA to vary with tracer. For galaxies chosen by stellar mass, simulations indicate
+that the rough approximation [69]
+pg ≈ 0.55 ,
+(2.8)
+is an improvement, and we use this for the DESI-like samples. For HI intensity mapping, we
+use the approximation [70]
+pH ≈ 1.25 .
+(2.9)
+The fNL terms in the power spectra are of order H2
+0/k2 on ultra-large scales, k ≲ keq,
+where T ≈ 1. Local PNG therefore dominates the power on ultra-large scales – but these
+are also the scales where cosmic variance is largest. We deal with the cosmic variance via a
+multi-tracer analysis, which includes the information from all auto- and cross-power spectra
+of the two (or more) tracers (see section 4).
+3
+Modelling the surveys
+We consider two mock spectroscopic surveys:
+A = g: galaxy survey, similar to DESI surveys [61, 62];
+A = H: 21 cm HI intensity mapping (IM) survey, similar to SKAO surveys [56, 63, 71].
+In both cases, we have a low- and a high-redshift survey. The sky and redshift coverage
+of the individual and overlapping surveys are given in Table 1.
+– 3 –
+
+Table 1. Sky area and redshift range of surveys.
+Survey
+Sample
+Ωsky
+ttot
+redshift
+�
+103 deg2�
+�
+103 hr
+�
+range
+g (DESI-like)
+BGS
+14
+-
+0.00–0.50
+ELG
+14
+-
+0.60–1.70
+H (SKAO-like)
+Band 2
+20
+10
+0.10–0.58
+Band 1
+20
+10
+0.35–3.05
+g × H (low z)
+BGS × Band 2
+10
+5
+0.10–0.50
+g × H (high z)
+ELG × Band 1
+10
+5
+0.60–1.70
+0
+1
+2
+3
+z
+10−4
+10−3
+10−2
+ng [ h3 Mpc
+3]
+BGS
+ELG
+HI
+0.1
+0.2
+0.3
+0.4
+0.5
+¯TH [ mK]
+Figure 1. Background comoving galaxy number densities (red, left-hand y-axis) and HI IM temper-
+ature (blue, right-hand y-axis).
+The one-parameter model (2.2) for the Gaussian clustering bias of the galaxy surveys
+follows [29]
+bg(z) =
+bg0
+D(z) .
+(3.1)
+Fiducial values are bg0 = 1.34 for the Bright Galaxy Sample (BGS) and bg0 = 0.84 for the
+Emission Line Galaxy (ELG) sample. For the HI IM surveys we use a fit based on [72, 73]:
+bH(z) = bH0
+�
+1 + 0.823 z − 0.0546 z2�
+with fiducial value bH0 = 0.842 .
+(3.2)
+The background comoving number density ¯ng for the BGS and ELG samples is modelled fol-
+lowing [29], which leads to the smoothed curves shown in Figure 1. For HI IM, the background
+brightness temperature is modelled via the fit given in [65, 74]:
+¯TH(z) = 0.0559 + 0.2324 z − 0.0241 z2 mK ,
+(3.3)
+which is also shown in Figure 1.
+– 4 –
+
+3.1
+Noise
+In galaxy surveys, the shot noise (assumed to be Poissonian) is
+P shot
+gg
+(z) =
+1
+¯ng(z) .
+(3.4)
+Then the total signal that we measure for the galaxy auto-power spectrum is
+˜Pgg(z, k, µ) = Pgg(z, k, µ) + P shot
+gg
+(z) ,
+(3.5)
+where Pgg is given by (2.4).
+In IM surveys, there is shot noise, but also thermal (or instrumental) noise. The Poisso-
+nian shot noise (see [75] for non-Poissonian corrections) is derived in a halo-model framework
+and given by [72, 76],
+P shot
+HH (z) =
+1
+¯nH(z) =
+1
+¯ρ 2
+H(z)
+�
+dM Nh(M, z) M2
+H(M, z) ,
+(3.6)
+where ¯nH is the effective average comoving number density of HI, ¯ρH is the average comoving
+HI density, Nh is the halo mass function (average comoving halo number density per mass)
+and MH is the average HI mass in a halo of mass M. However, on the linear scales that we
+consider, the shot noise is much smaller than the thermal noise and can be safely neglected
+[72, 76].
+The thermal noise in HI IM depends on the sky temperature in the radio band, the survey
+specifications and the array configuration (single-dish or interferometer). For the single-dish
+mode of SKAO-like surveys, the thermal noise power spectrum is [77–79]:
+P therm
+HH
+(z) =
+Ωsky
+2ν21ttot
+(1 + z)r(z)2
+H(z)
+�Tsys(z)
+¯TH(z)
+�2
+1
+Nd
+,
+(3.7)
+where ν21 = 1420 MHz is the rest-frame frequency of the 21 cm emission, ttot is the total
+observing time, and the number of dishes is Nd = 197 (with dish diameter Dd = 15 m). The
+system temperature is modelled as [80]:
+Tsys(z) = Td(z) + Tsky(z) = Td(z) + 2.7 + 25
+�400 MHz
+ν21
+(1 + z)
+�2.75
+K,
+(3.8)
+where Td is the dish receiver temperature (see [71]). The total signal is then
+˜PHH(z, k, µ) = PHH(z, k, µ) + P shot
+HH (z) + P therm
+HH
+(z) ≈ PHH(z, k, µ) + P therm
+HH
+(z) .
+(3.9)
+The noise for all surveys is shown in Figure 2.
+In the case of the cross-power spectrum PgH, cross-shot noise arises if the halos that
+host galaxies and HI overlap. Following the arguments given in [71, 81], we assume that the
+cross-shot noise may be neglected. The total cross-power signal is then
+˜PgH = PgH with P shot
+gH
+≈ 0 .
+(3.10)
+– 5 –
+
+0.0
+0.5
+1.0
+1.5
+z
+102
+103
+104
+P shot
+gg
+[h−3 Mpc3]
+BGS
+ELG
+1
+2
+3
+z
+101
+102
+103
+P therm
+HH
+[h−3 Mpc3]
+Band 2
+Band 1
+Figure 2.
+Shot noise for galaxy surveys (left) and thermal noise for IM surveys (right).
+3.2
+Intensity mapping beam and foregrounds
+HI IM surveys like the SKAO surveys have poor angular resolution, which results in power loss
+on small transverse scales, i.e. for large k⊥ = (1 − µ2)1/2k. This effect is typically modelled
+by a Gaussian beam factor [77]:
+Db(z, k, µ) = exp
+�
+−(1 − µ2)k2r(z)2θb(z)2
+16 ln 2
+�
+with
+θb(z) = 1.22 λ21(1 + z)
+Dd
+.
+(3.11)
+HI IM surveys are also contaminated by foregrounds that are much larger than the HI
+signal. Since these foregrounds are spectrally smooth, they can be separated from the non-
+smooth signal on small to medium scales. However, on very large radial scales, i.e. for small
+k∥ = µk, the signal becomes smoother and therefore the separation fails. A comprehensive
+treatment includes simulations of foreground cleaning of the HI signal (e.g. [15, 18]). For a
+simplified Fisher forecast we can instead use a foreground avoidance approach, by excising
+the regions of Fourier space where the foregrounds are significant. This means removing large
+radial scales, which can be modelled via an exponential suppression factor [73, 82]:
+Dfg(k, µ) = 1 − exp
+�
+−
+�µk
+kfg
+�2�
+.
+(3.12)
+We choose a typically used value:
+kfg = 0.01 h Mpc−1 .
+(3.13)
+Figure 3 illustrates the consequences of foreground avoidance for the monopoles of the HI
+power spectra, PAH,0. It is clear that large-scale (small k) power in PAH,0 is lost and that the
+effect of local PNG on these large scales is suppressed by foreground avoidance. Note that
+fNL reduces the HI power on very large scales at z = 0.3, since bH < 1.25, which leads to
+bHφ < 0 in (2.7). Also note that the beam effect on PAH,0, which tends to suppress small scale
+(large k) modes, is negligible at the low redshift, but becomes apparent at higher redshift.
+In summary, the effects of the radio telescope beam and radio foreground avoidance lead
+to the following modifications of the power spectra PAH:
+PHH(z, k, µ) → Dfg(k, µ) Db(z, k, µ)2 PHH(z, k, µ) ,
+(3.14)
+PgH(z, k, µ) → Dfg(k, µ) Db(z, k, µ) PgH(z, k, µ) .
+(3.15)
+– 6 –
+
+10−3
+10−2
+10−1
+k [h Mpc−1]
+101
+102
+103
+104
+PAB, 0 [h−3 Mpc3]
+k = 0.01
+BGS
+Band 2
+BGS × Band 2
+fNL = 0
+fNL = 10
+10−3
+10−2
+10−1
+k [h Mpc−1]
+101
+102
+103
+104
+PAB, 0 [h−3 Mpc3]
+k = 0.01
+ELG
+Band 1
+ELG × Band 1
+fNL = 0
+fNL = 10
+Figure 3. The monopole power spectra at z = 0.3 (left) and z = 1.0 (right).
+4
+Fisher forecast
+For a multi-tracer combination of two dark matter tracers g and H, the data vector of power
+spectra is
+P =
+�
+Pgg , PgH , PHH
+�
+.
+(4.1)
+We exclude the noise from the data vector, since the noise is independent of the cosmolog-
+ical and nuisance parameters. (The noise appears in the covariance, as given below.) The
+standard cosmological parameters are measured on medium to small scales and are effectively
+independent of the ultra-large scale parameter fNL. Fixing their values could bias the best-fit
+value of fNL but is unlikely to have any significant impact on σ(fNL). We include in the Fisher
+analysis the cosmological parameters that directly affect the large-scale amplitude (σ8,0) and
+shape (ns) of the power spectrum.
+We therefore consider the following cosmological and
+nuisance parameters:
+ϑα =
+�
+σ8,0, ns, fNL; bg0, bH0
+�
+.
+(4.2)
+Here we assume that the degeneracies between bA and σ8,0, and between ¯TH and σ8,0, have
+been broken by other surveys that focus on medium to small scales.
+The covariance for the multi-tracer power spectra is given by [83, 84]:
+Cov(P , P ) =
+k3
+f
+4πk2∆k
+2
+∆µ
+�
+�
+�
+�
+�
+�
+�
+˜P 2
+gg
+˜Pgg ˜PgH
+˜P 2
+gH
+˜Pgg ˜PgH
+1
+2
+� ˜Pgg ˜PHH + ˜P 2
+gH
+�
+˜PHH ˜PgH
+˜P 2
+gH
+˜PHH ˜PgH
+˜P 2
+HH
+�
+�
+�
+�
+�
+�
+�
+.
+(4.3)
+Here ∆k and ∆µ are the bin-widths for k and µ and kf is the fundamental mode, corresponding
+to the longest wavelength, which is determined by the comoving survey volume of the redshift
+bin centred at z:
+V (z) = Ωsky
+3
+�
+r
+�
+z + ∆z
+2
+�3
+− r
+�
+z − ∆z
+2
+�3�
+=
+� 2π
+kf(z)
+�3
+.
+(4.4)
+– 7 –
+
+Then the multi-tracer Fisher matrix in a redshift bin is
+F P
+αβ =
++1
+�
+µ=−1
+kmax
+�
+k=kmin
+∂α P · Cov(P , P )−1 · ∂β P T ,
+(4.5)
+where ∂α = ∂ / ∂ϑα. We choose the bin-widths and kmin following [85–87]:
+∆µ = 0.04 ,
+∆k = kf ,
+kmin = kf .
+(4.6)
+The smallest scale (largest k) is chosen to ensure that linear perturbations remain accurate,
+since we do not require information from nonlinear scales:
+kmax = 0.08(1 + z)2/(2+ns) h Mpc−1 .
+(4.7)
+The galaxy and HI IM surveys do not have the same sky area and redshift ranges
+(see Table 1).
+The overlap redshift ranges are obvious; for the sky areas, we assume a
+nominal overlap area of 10,000 deg2. The multi-tracer applies in the overlap redshift range
+and overlap sky area. However, we can add the independent Fisher matrices obtained in the
+non-overlapping regions of the two individual surveys to the multi-tracer Fisher matrix in the
+overlapping region [71]. In detail, the non-overlap region is
+• the non-overlap sky area of each survey, across the full redshift range of each survey;
+• the overlap sky area, across the non-overlap parts of the redshift ranges of each survey.
+Then the full Fisher matrix (denoted by g ⊗ H) is
+F g⊗H
+αβ
+= F P
+αβ(overlap) + F g
+αβ(non-overlap) + F H
+αβ(non-overlap).
+(4.8)
+Figure 4 shows the 1σ error contours computed from this Fisher matrix after marginalising
+over the bias parameters in (4.2). We use the fiducial values σ8,0 = 0.8102, ns = 0.9665 and
+fNL = 0, keeping other cosmological parameters fixed to their Planck 2018 best-fit values [88].
+Table 2 lists the marginalised σ(fNL) constraints, including the improvements (in parentheses)
+that follow when we ignore the HI IM foregrounds, i.e. when kfg = 0.
+– 8 –
+
+0.80
+0.83
+σ8, 0
+−40
+0
+40
+fNL
+0.96
+0.98
+ns
+σ8, 0 = 0.8102
+0.96
+0.98
+ns
+ns = 0.9665
+−20
+20
+fNL
+fNL = 0.0
+BGS
+Band 2
+BGS ⊗ Band 2
+0.806
+0.814
+σ8, 0
+−5
+0
+5
+fNL
+0.962
+0.966
+0.970
+ns
+σ8, 0 = 0.8102
+0.963
+0.970
+ns
+ns = 0.9665
+−5
+0
+5
+fNL
+fNL = 0.0
+ELG
+Band 1
+ELG ⊗ Band 1
+Figure 4. 1σ contours, including foreground avoidance in HI intensity mapping and marginalising
+over bias parameters.
+Table 2. Cumulative marginalised error σ(fNL) for: the single-tracer surveys; the low-z and high-z
+multi-tracer pairs – including single-tracer information from the non-overlap volumes as in (4.8); and
+the sum of the multi-tracer pairs.
+Values in parenthesis correspond to ignoring foregrounds, i.e., with kfg = 0.
+Survey
+σ(fNL)
+BGS
+20.8
+ELG
+3.5
+Band 2
+96.3 (57.7)
+Band 1
+4.3
+(1.3)
+BGS ⊗ Band 2
+9.2
+(2.3)
+ELG ⊗ Band 1
+3.2
+(1.3)
+BGS ⊗ Band 2 + ELG ⊗ Band 1
+2.9
+(1.1)
+5
+Conclusion
+We applied a simplified Fisher forecast analysis in order to gain insights into the improvements
+on σ(fNL) from a new multi-tracer combination of large-scale structure surveys that will
+be made possible by the upcoming HI intensity mapping surveys with the SKAO. For the
+spectroscopic galaxy samples, we used mock surveys based on DESI. This allowed for multi-
+tracer pairs at low and high redshifts, in order to see the level of improved precision from the
+higher volume at high redshifts.
+We included all available Fisher information from these mock surveys, using the multi-
+tracer Fisher matrix on the overlap volume (sky area and redshift range) and the single-tracer
+Fisher matrices on the non-overlap volumes, as in (4.8). Our results for σ(fNL) are summarised
+– 9 –
+
+in Table 2, which gives the single-tracer results for the 4 samples, the full multi-tracer results
+from (4.8) for the low- and high-redshift pairs, and finally the results from the Fisher sum of
+these pairs. The impact of foreground avoidance can be seen from the results (in parenthesis)
+when foregrounds are ignored, corresponding to kfg = 0 in (3.12).
+Figure 4 shows the 1σ contours for the 3 cosmological parameters (σ8, 0, ns, fNL),
+marginalising over the 2 bias nuisance parameters (bg0, bH0).
+As expected, the foreground avoidance severely degrades the constraining power of the
+HI intensity surveys. Without foregrounds, the high-redshift Band 1 survey would perform
+as well as the multi-tracer combination with the ELG survey: σ(fNL) = 1.3. The reduced
+precision on fNL for single-tracer HI intensity constraints is consistent with the findings of [15],
+where foreground cleaning is simulated. However, using the multi-tracer with a spectroscopic
+galaxy survey helps to reduce the effect of foregrounds on σ(fNL), as shown by Table 2.
+The multi-tracer analysis is also expected to mitigate other systematics in the galaxy and
+HI intensity samples. For example, HI intensity mapping is also affected by radio frequency
+interference, receiver noise, calibration errors and polarisation leakage. Incorporating these
+and the galaxy systematics requires extensive simulations, beyond the scope of our paper.
+– 10 –
+
+Acknowledgements
+We thank Dionysis Karagiannis for very helpful comments. We are supported by the South
+African Radio Astronomy Observatory (SARAO) and the National Research Foundation
+(Grant No. 75415).
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+– 15 –
+

CNE4T4oBgHgl3EQfeQ0p/content/tmp_files/2301.05097v1.pdf.txt

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+Study of JavaScript Static Analysis Tools for
+Vulnerability Detection in Node.js Packages
+Tiago Brito∗, Mafalda Ferreira, Miguel Monteiro, Pedro Lopes, Miguel Barros, José Fragoso Santos, Nuno Santos
+{∗tiago.de.oliveira.brito, mafalda.baptista, miguel.figueiredo.monteiro, pedro.daniel.l, miguel.v.barros, jose.fragoso, nuno.m.santos}@tecnico.ulisboa.pt
+INESC-ID / IST, Universidade de Lisboa, Portugal
+Abstract—With the emergence of the Node.js ecosystem,
+JavaScript has become a widely-used programming language
+for implementing server-side web applications. In this paper, we
+present the first empirical study of static code analysis tools for
+detecting vulnerabilities in Node.js code. To conduct a comprehen-
+sive tool evaluation, we created the largest known curated dataset
+of Node.js code vulnerabilities. We characterized and annotated
+a set of 957 vulnerabilities by analyzing information contained in
+npm advisory reports. We tested nine different tools and found
+that many important vulnerabilities appearing in the OWASP
+Top-10 are not detected by any tool. The three best performing
+tools combined only detect up to 57.6% of all vulnerabilities in
+the dataset, but at a very low precision of 0.11%. Our curated
+dataset offers a new benchmark to help characterize existing
+Node.js code vulnerabilities and foster the development of better
+vulnerability detection tools for Node.js code.
+I. INTRODUCTION
+JavaScript has become one of the most popular programming
+languages for implementing server-side web applications.
+A driving factor in this trend has been the emergence of
+Node.js [1]. Node.js is a cross-platform, back-end runtime
+environment that executes JavaScript code. Essentially, it can
+be used as a web container, housing JavaScript code that
+handles HTTP requests. Pivoted around Node.js, there is also
+an ecosystem of third-party packages managed by the Node
+Package Manager (npm). Currently, npm stores thousands of
+packages that web developers can readily import into their
+code, either for writing web applications or other packages.
+The widespread adoption of Node.js makes the development
+of effective JavaScript vulnerability scanners a pressing matter.
+For one, the JavaScript [2] language features various constructs
+that display subtle behaviors. When employed by inexperienced
+code developers, these constructs may all too easily lead to
+the introduction of vulnerabilities. In addition, the manual
+detection of code vulnerabilities is complicated by the intricate
+npm inter-package dependency system. In some cases, correct
+packages may become the source of security bugs as a result
+of ill-use by other packages. In others, buggy packages may
+end up propagating vulnerabilities up in the dependency chain
+to correct packages [3]. This combination of factors opens
+up the path for serious security breaches in web applications.
+By exploiting security bugs, an attacker may be able to take
+over the entire server and/or affect many users through SQL
+injection, remote code execution, and other attacks [4, 5].
+An effective technique to prevent security vulnerabilities
+from creeping into production code is to integrate security
+analysis tools as part of Continuous Integration/Continuous
+Deployment (CI/CD) pipelines. Using automatic vulnerability
+detection tools, developers can seamlessly receive prompt
+feedback about potentially existing security flaws in their code.
+This enables them to apply the necessary fixes at an early code
+development stage, thus helping them to improve the reliability
+of their software. In the same vein, JavaScript developers can
+benefit from code analysis tools that allow them to detect and
+fix security flaws inside npm packages. Ideally, such tools
+should have high detection quality (i.e., low false-positive rate),
+and high coverage (i.e., low false-negative rate).
+Motivated by this need, we set out to evaluate the effec-
+tiveness of existing JavaScript vulnerability detection tools at
+analyzing Node.js packages. We found a large body of work
+on client-side JavaScript security [6, 7, 8], and some recent
+work in the study of vulnerabilities in npm packages [4, 5].
+However, no prior work has focused on evaluating tools that
+analyze server-side JavaScript code vulnerabilities, let alone
+on studying their effectiveness at finding security flaws in npm
+packages. As it turns out, performing this task is rather involved,
+given the absence of a gold standard for classifying such tools,
+and the lack of a comprehensive vulnerability dataset that can
+be used for benchmarking purposes.
+In this paper, we present the first empirical study aimed at
+evaluating existing JavaScript vulnerability detection tools on
+Node.js packages. We focus exclusively on fully automatic,
+static code analysis tools that can be used in CI/CD pipelines.
+This excludes tools [4, 9, 10] that expect additional inputs, such
+as test suites, or tools that perform simple checks on known
+vulnerable dependencies [11, 12]. In total, we screened 40
+analysis tools for JavaScript and selected nine that can detect
+vulnerabilities at continuous integration time: NodeJsScan [13],
+CodeQL [14], ODGen [15], Graudit [16], InsiderSec [17],
+ESLint SSC [18], Microsoft’s DevSkim [19], Mosca [20] and
+Drek [21]. We executed them against a curated dataset created
+by us containing npm packages with annotated vulnerabilities,
+mainly: path traversal, cross-site scripting, insecure transfer
+using HTTP, resource exhaustion/denial-of-service, prototype
+pollution, OS command injection, code injection, and improper
+input validation. Then, we checked whether these tools can
+correctly identify these vulnerabilities.
+Given that there is no curated dataset of Node.js vulnerabili-
+ties, our first step was to develop our own. Building this dataset
+was in itself a challenging endeavor because we needed to
+1
+arXiv:2301.05097v1  [cs.CR]  12 Jan 2023
+
+identify real vulnerabilities in a large dataset of npm packages.
+Our starting point was the npm system itself. The npm system
+runs a vulnerability report service that results in the generation
+of the so-called advisory reports. These consist of textual
+descriptions of security vulnerabilities identified inside specific
+packages. These are real security vulnerabilities collectively
+identified by the Node.js developer community. Reports may
+also include an advice to upgrade the package to a fixed version.
+As such, advisory reports provide a reliable source for building
+our dataset. Unfortunately, these reports are not represented
+in a format that allows for automatic processing. Moreover,
+some of them may contain errors and therefore cannot be used
+unless a thorough analysis and verification are performed.
+To overcome these difficulties, we manually analyzed 1359
+advisory reports covering an equal number of vulnerable npm
+package versions. These advisories represent 74% of all the
+vulnerabilities officially reported inside benign npm package
+versions until June 2021. In this process, we identified several
+anomalies in the advisory reports. We have then generated a
+curated dataset covering 957 of these advisories extended with
+annotations that specify the precise location of the reported
+code vulnerabilities. We found that the location of a large
+fraction of existing vulnerabilities can be fully expressed
+through source-sink pair annotations. Our dataset can help
+the research community to i) characterize the vulnerabilities
+already detected within the npm ecosystem, and ii) benchmark
+vulnerability detection tools. Our dataset is publicly available1.
+We tested the pre-selected tools against our dataset and found
+they perform rather poorly, missing many vulnerabilities (low
+true positive rate/recall) and showing a high false positive rate
+(low precision). On average, they were able to correctly identify
+only 15.1% of the total number of vulnerabilities in our dataset.
+The combination of the three best-performing tools detects
+57.6% of all vulnerabilities, albeit with only 0.11% precision.
+The best performing tools, ESLint SSC and CodeQL, manage
+to detect 41.5% and 31.3% across all types of vulnerabilities
+and reach their peaks when it comes to identifying prototype
+pollution (79.2% for CWE-471 and 86.1% for CWE-1321) and
+path traversal (71.2%) vulnerabilities, respectively. Of the 957
+known vulnerabilities in the dataset, 324 (33.8%) were not
+detected by any of the selected tools. Some of the causes are
+tied to fundamental limitations of state-of-the-art code analysis
+techniques when it comes to analyzing server-side, JavaScript
+code vulnerabilities in the npm ecosystem. Addressing these
+limitations is an interesting research direction for future work.
+In summary, our paper makes four contributions: (i) a
+curated dataset with 957 real-world vulnerabilities in npm
+package versions, which will be fundamental to evaluate
+future advancements in static analysis tools for detection of
+vulnerabilities in Node.js applications, (ii) a survey of existing
+vulnerability detection tools for JavaScript / Node.js code, (iii)
+a quantitative assessment of the vulnerability detection toolset
+against our curated dataset, and (iv) a study of the main causes
+1https://github.com/VulcaN-Study/Supplementary-Material
+2016
+2017
+2018
+2019
+2020
+2021
+2022
+0
+1000
+2000
+Severity
+Low
+Moderate
+High
+Critical
+Figure 1: Evolution of published advisories over time.
+of missing important vulnerabilities in npm packages, which
+opens up several research avenues in this field.
+II. STUDY DESIGN
+A. Background
+Node.js features a package manager system named Node
+Package Manager (npm), which currently stores thousands
+of third-party packages. Presently, it is difficult to guarantee
+the absence of security vulnerabilities in packages uploaded
+to npm because there is no systematic code triage in place.
+Consequently, the npm community mainly relies on third-
+party vulnerability reports to identify the potentially vulnerable
+packages that have already been included in the ecosystem.
+The npm system alerts JavaScript developers whenever
+they use a package version reported as vulnerable. These
+vulnerability alerts are called npm advisories, as their purpose
+is to advise developers to update the vulnerable dependencies
+to either a fixed package version or to select another package
+entirely. When someone identifies a potential vulnerability
+in an npm package, they produce a vulnerability report and
+submit it to the npm security team, which checks the report,
+notifies package maintainers, and publishes the advisory, either
+when the package maintainers release a fix or if they remain
+unresponsive for longer than 45 days [22]. Typically, an
+advisory report includes: package name, affected versions,
+description of the vulnerability, effects and references, commits,
+and/or code examples that help trigger the vulnerability. This
+information is then made available to developers in the advisory
+page (see example page for advisory 315 in [23]). Figure 1
+displays the evolution of the number of published advisories
+since npm’s inception until October 11th 2022, broken down
+according to their severity level. This number has steadily
+grown at nearly 40 advisories per month; about 70% cover
+vulnerabilities considered to pose risks of high/critical severity.
+B. Research Questions and Scope
+In this work, we investigate four main research questions:
+RQ1. How to obtain an annotated dataset of vulnerabilities
+in server-side JavaScript code? To evaluate JavaScript vul-
+nerability detection tools, we require an annotated vulnerability
+dataset to compare the output of a given tool against ground
+truth data. The npm repository provides an excellent source for
+retrieving both (i) an extensive collection of vulnerable Node.js
+applications (i.e., vulnerable npm package versions), and (ii)
+information about real-world vulnerabilities (documented by
+the advisory database). Unfortunately, this information cannot
+2
+
+be used as-is from existing advisories. First, advisories often
+lack relevant information about the reported vulnerability (e.g.,
+the exact code location of the vulnerability within the package).
+Second, in many cases, most of the explanations regarding
+the reported vulnerability are given in external references,
+where information tends to be inconsistent and unstructured.
+Third, some advisories may be incorrect in places (e.g., the
+classification of the vulnerability type), which may lead to the
+mischaracterization of existing JavaScript vulnerabilities. These
+obstacles preclude an automated advisory analysis approach.
+RQ2. What is the state-of-the-art of existing security-orien-
+ted static analysis tools for Node.js code? We are interested
+in assessing which static vulnerability detection tools are
+available. We need to distinguish between a broader set of
+code analysis tools which can serve many different purposes
+(e.g., detecting programming malpractices) from those that
+are specifically oriented toward the detection of vulnerabilities.
+Furthermore, we aim to analyze which code analysis techniques
+are employed by these tools. This will help us to better assess
+the strengths and weaknesses of each technique when evaluating
+each tool.
+RQ3. How effective are available detection tools in uncov-
+ering vulnerabilities in JavaScript code? We are interested
+in empirically determining and characterizing how precise the
+publicly available vulnerability detection tools are at identifying
+vulnerabilities in known vulnerable JavaScript code.
+RQ4. What are the main reasons for missing the detection
+of vulnerabilities? We aim to understand the key limitations
+of existing static vulnerability detection tools that explain their
+failure to detect known vulnerabilities in JavaScript code.
+The research questions above have a twofold goal: (i)
+characterize vulnerabilities in npm packages in the wild, and
+(ii) evaluate the effectiveness of existing static JavaScript vul-
+nerability detection tools. To better set the reader’s expectations
+about our study, we further clarify these subgoals and discuss
+other relevant directions we left outside the scope of this work.
+Firstly, our study is narrowed toward the characterization
+of vulnerabilities reported inside npm packages. As such,
+our results cannot be extrapolated to characterize the typical
+programming flaws introduced by JavaScript developers during
+the code development stage; nor is this our goal. Developers
+may use vulnerability detection tools in their continuous
+integration frameworks that may capture some vulnerabilities
+before the code is deployed to npm. This means that some
+security flaws may have been fixed prior to deploying the code
+into production. Also note that, given that we only analyze
+formerly identified vulnerabilities, our curated dataset may
+not be representative of all existing JavaScript vulnerabilities
+lingering inside npm packages; this is also not our purpose. In
+contrast, we intend that our curated dataset contains a large
+set of confirmed real-world vulnerabilities that can be used for
+assessing existing (and future) vulnerability detection tools.
+Secondly, we focus exclusively on fully automatic vulnerabil-
+ity analysis tools that can be easily integrated into existing code
+Internet
+npm 
+website & registry
+Advisory 
+Collection 
+Engine
+Dockerfiles
+Tool
+Configs
+1)
+Tool 
+Execution 
+Engine
+2)
+3)
+Analyst
+4)
+5)
+6)
+Tool 
+Outputs
+Package
+Code
+Advisory 
+Metadata
+VulcaN 
+Reviews
+API
+Analysis 
+Engine
+VulcaN
+Figure 2: VulcaN architecture.
+review pipelines. This excludes the only three existing dynamic
+analysis tools for detecting vulnerabilities in Node.js code:
+SyNode [4], NodeSec[9] and Affogato [10]. These tools require
+a set of unit tests covering all security sensitive behaviors of
+the package to be analysed. Such test suites must be manually
+written as the automatic generation of high-coverage test suits
+for Node.js applications is still an open problem. Nevertheless,
+our curated dataset can still be used as a benchmark for
+evaluating the effectiveness of these excluded tools.
+C. Study Methodology
+Our approach to answering the questions above is to perform
+an empirical study consisting of the following three tasks:
+Task 1. Manual analysis of advisory reports and building
+the annotated dataset: We manually analyzed all advisories,
+filling in the missing information, namely by identifying the
+code location that triggers the vulnerability. This information
+is part of our curated dataset used in the following tasks.
+Task 2. Selection and execution of code analysis tools for
+vulnerability detection in JavaScript code: We searched both
+the literature and the web for code analysis tools that can be
+used for vulnerability detection. We executed all of the selected
+tools on all the vulnerable packages in our curated dataset.
+Task 3. Evaluation of tool output according to the ground
+truth extracted from the annotated dataset: We automati-
+cally compare a tool’s result with the corresponding dataset
+annotations, considering the verbosity levels of each tool.
+To support our methodology, we developed VulcaN, a
+testbed for analyzing vulnerability detection tools for Node.js
+packages. VulcaN is an execution and analysis framework to
+collect vulnerable versions of npm packages, run vulnerability
+detection tools over all collected packages, and help security
+analysts perform the vulnerability analysis. Currently, VulcaN
+supports nine tools (see Section IV) and has these main features:
+• an interface to download the latest published advisories;
+• a Docker-based extension for adding more analysis tools;
+• an interface for accessing both the metadata and the output
+of the analysis tools for each advisory;
+3
+
+• an interface for annotating each advisory with a review
+file, which contains the code location of the vulnerability
+and the classification of the results of the analysis tools;
+• an interface for automatically comparing the output of
+the tools with the corresponding review in our curated
+dataset, this includes a parser for the ouput of each tool;
+• a web API exposing the curated dataset and the classifi-
+cation of the analysis tools.
+Figure 2 shows the architecture of VulcaN and how each
+component is used in the context of our empirical study.
+First, the Advisory Collection Engine crawls the npm advisory
+website [24] and collects the available metadata about all
+published advisories saving it in a MongoDB database (1).
+The latest vulnerable version for each advisory, referenced
+in the collected metadata, is then downloaded via the npm
+registry. Second, the Tool Execution Engine builds a docker
+image for each tool, according to the specified Dockerfile, and
+runs the respective container for all downloaded packages,
+storing the output of each tool locally (2). Third, the advisory
+metadata, code, and tool outputs are fed to the Analysis
+Engine, which generates an environment for advisory report
+analysis (3). The analyst accesses the environment (4) and
+produces a review file comprising an accurate description
+of the vulnerable code location, which is then submitted to
+the framework (5). Finally, the Analysis Engine processes
+the submitted information, automatically compares the tools’
+outputs with the analyst reviews (dataset) using a dedicated
+parser for each tool, and exposes it through a web API (6).
+III. DATASET OF VULNERABILITIES (RQ1)
+In this section, we address RQ1, explaining how we created
+our curated dataset of JavaScript code vulnerabilities.
+A. Selection and Validation of Reports
+To create our dataset, we collected a snapshot of the existing
+npm advisories until the end of June 2021. Then, through
+manual analysis, we excluded some advisories and fixed
+inconsistencies in the remaining ones (see Table I). Out of
+the 1828 advisories from the original snapshot, we excluded
+469, keeping 1359 for further analysis. Next, we present our
+exclusion criteria and discuss the detected inconsistencies.
+Excluded advisories. As of June 30th 2021, there were 1828
+advisories published in npm. Of these 1828 advisories, 416 are
+categorized by npm as Embedded Malicious Code (CWE-506):
+these are packages designed with malicious intent, named very
+similarly to real legitimate packages so as to deceive developers
+into installing them. These packages are not relevant for our
+study, which focuses only on unintentional vulnerabilities.
+From the remaining 1412 advisories, we excluded 31 for
+lacking available code. Lastly, out of the resulting 1381
+vulnerable package versions, 22 were excluded for not including
+JavaScript code; instead, they had pre-transpiled variants such
+as CoffeeScript [25] and TypeScript [26], which prevented us
+from analyzing their source code directly. Consequently, in the
+end, VulcaN successfully collected 1359 advisories and their
+corresponding package versions for further manual analysis.
+Exclusions & Inconsistencies
+# of Advisories
+Malware Packages
+416
+Missing Package Code
+31
+Missing JavaScript Code
+22
+Incorrect Vulnerable Version
+42
+Missing External References
+291
+Imprecise CWE
+101
+Lack of Analysis Information
+402
+Table I: Number of advisories excluded or inconsistent.
+Detected inconsistencies. During the manual analysis, we
+noticed several inconsistencies in the collected advisories. Most
+notably, only a small minority of advisories comes with the ex-
+act code locations that trigger their corresponding vulnerability.
+Furthermore, some advisories provide an incorrect vulnerable
+package version, i.e., the advisory metadata points to a package
+version that does not contain the described vulnerability. When
+the advisory does not come with additional external references,
+which is the case for 21% of the analyzed advisories, correcting
+the incorrect vulnerable package version anomaly can be
+quite challenging, as the advisory metadata alone is generally
+insufficient for pinpointing the correct package version. Another
+detected anomaly is the imprecise classification of vulnerability
+type/category. Most of the times this imprecision is subjective,
+as a Common Weakness Enumeration (CWE) [27] class can
+be a subcategory (child) of another more general CWE. This is
+particularly common for Path Traversals (CWE-22) and Code
+Injections (CWE-94), to which more precise classes can be
+attributed; in particular, CWE-23 (Relative Path Traversal) and
+CWE-24 (another specific Path Traversal variant) to CWE-22
+and CWE-95 (Eval Injection) to CWE-94. Some vulnerabilities
+are simply miscategorised; for instance, sometimes Code
+Injection (CWE-94) vulnerabilities are categorized as Cross-
+Site Scripting (CWE-79). During the analysis, we detected 63
+cases of vulnerability miscategorization (different CWE) and 21
+cases of incorrect vulnerable package version referenced in the
+advisory. The remaining cases lack the CWE categorization.
+B. Analysis of Reported Vulnerabilities
+We analyzed the vulnerabilities in the selected 1359 npm
+advisories. Our goal was to characterize the vulnerability
+landscape of the npm package ecosystem by studying the dis-
+tribution of existing vulnerabilities according to their category
+and assess the potential security risks posed by the affected
+packages. From the 1359 advisories manually analyzed, we
+managed to verify the vulnerability for 957 advisories: these
+are the ones included in our dataset and characterized in this
+study. The remaining advisories (402) did not include sufficient
+information to successfully verify the vulnerability.
+Figure 3 displays the cumulative distribution function (CDF)
+of the number of vulnerabilities of our dataset ranked by their
+CWE category. This distribution is heavily skewed toward a
+relatively small number of CWE categories, i.e., a large fraction
+of vulnerabilities pertains to a restricted set of categories. In
+particular, the top-10 CWEs cover 665 advisories, i.e., 69% of
+the total number of verified vulnerabilities.
+4
+
+0
+25
+50
+75
+100
+Number of CWE categories (ordered by occurrence)
+200
+400
+600
+800
+Number of advisories
+665 (69%)
+957 (100%)
+Top 10 CWE
+All Reviewed Advisories
+Figure 3: CDF of # of reviewed advisories ranked by CWE.
+To estimate the potential security risks of such vulnerabilities,
+we mapped each of the top-10 CWE categories to the latest
+OWASP ranking (from 2021). OWASP is an organization that
+works to raise awareness about web security and ultimately
+improve it. The OWASP Top 10 [28] list is a popular document
+representing a broad consensus about the most critical security
+risks to web applications. Updated every few years, this
+document describes risks, such as injection attacks, broken
+authentication, and known vulnerable dependencies. As shown
+in Table II, most vulnerability types can be mapped to a top-10
+web security risk. This means that npm packages have well-
+known risks that security professionals are familiar with. Most
+notably, 9 out of 10 CWE categories (i.e., 576 advisories) ap-
+pear in the top-3 OWASP list. This translates to approximately
+60% of the total number of analyzed vulnerabilities in our
+dataset (957). In other words, many reported vulnerabilities
+can introduce serious security flaws in web applications.
+C. Our Curated Dataset
+Based on the selected advisories, we created a curated dataset
+aimed at providing a baseline for assessing the effectiveness
+of vulnerability detection tools. It comprises: (i) the code for
+the vulnerable version of the npm package indicated in the
+advisory, and (ii) a corresponding review file. This file contains
+ground truth information that allows us to validate the output
+of a given tool when analyzing that specific package version.
+Review example: Listing 1 shows the created review file for
+advisory 315 [23]. This advisory reports the presence of a code
+injection vulnerability in package summit-0.1.22. A review is
+a JSON object that contains several fields that describe: i)
+the advisory identifier (id), ii) the vulnerability type as per
+the CWE taxonomy (cwe), iii) the affected package version
+(package_link), and iv) the vulnerability location expressed
+as a source/sink pair. A source/sink is specified as a JSON
+object with fields denoting: the file name (file); the line
+number (lineno); and the corresponding line of code (code).
+Vulnerability location: The location fields are used to deter-
+mine if the output of a given vulnerability detection tool is
+correct. Besides using (one or multiple) source/sink pairs to
+locate a vulnerability, we can also employ (one or multiple)
+block patterns, which indicate contiguous code regions in
+which the flaw exists. This form of representation is necessary
+for vulnerability types that cannot be expressed as source/sink
+#
+CWE
+Security Risk
+# Occurrences
+1
+CWE-22
+1. Broken Access Control
+146
+2
+CWE-79
+3. Injection
+99
+3
+CWE-400
+-
+89
+4
+CWE-78
+3. Injection
+75
+5
+CWE-818
+2. Cryptographic Failures
+75
+6
+CWE-471
+3. Injection
+48
+7
+CWE-20
+3. Injection
+41
+8
+CWE-1321
+3. Injection
+36
+9
+CWE-94
+3. Injection
+33
+10
+CWE-77
+3. Injection
+23
+Table II: Possible mapping of most occurring vulnerabilities in dataset
+with OWASP Top 10 Web Security Risks (2021) [29]: Path Traversal
+(CWE-22), Cross-site Scripting (CWE-79), Resource Exhaustion
+(CWE-400), Insufficient Transport Layer Protection (CWE-818), OS
+Command Injection (CWE-78), Modification of Assumed-Immutable
+Data (CWE-471), Improper Input Validation (CWE-20), Improperly
+Controlled Modification of Object Prototype Attributes (CWE-1321),
+Code Injection (CWE-94), and Improper Neutralization of Special
+Elements used in a Command (CWE-77).
+{
+"advisory": { "id": 315, "cwe": "CWE-94" },
+"package_link": "registry.npmjs.org/summit/-/summit-0.1.22.tgz",
+"vulnerability": [
+{
+"source": {
+"file": "lib/drivers/search/pouch.js",
+"lineno": 4,
+"code": "return function search (opts) {"
+},
+"sink": {
+"file": "lib/drivers/search/pouch.js",
+"lineno": 20,
+"code": "eval(opts.filter);"
+}
+}
+]
+}
+Listing 1: Example of VulcaN review file for advisory 315.
+pairs, e.g., usage of HTTP instead of HTTPS which allows
+for MITM attacks (CWE-818). Interestingly, we found that,
+in most cases (78%), the exact location of a vulnerability is
+very clear, e.g., a call to eval in a code injection. These cases
+can be represented by source/sink pairs, while the remaining
+cases (22%) can be represented using code blocks that cover all
+vulnerability-relevant code. Although the block size depends
+on the vulnerability, in our dataset the average block size is six
+lines-of-code. Moreover, since the review files can be processed
+automatically, we believe that our curated dataset will be useful
+for benchmarking purposes beyond the scope of our work.
+Methodology: Vulnerability locations were identified manually.
+To account for their possible mislabeling, each advisory was
+analyzed by two authors at separate times and their results cross-
+checked. Two authors performing cross-validation disagreed in
+84 reviews (8.8% of 957 reviews). Most inconsistencies were
+differences in source/sink pairs. These cases were resolved by
+selecting the correct source/sink pair or specifying a superset of
+source/sink pairs. In rare cases, one author failed to locate the
+vulnerability. These cases were handled by jointly reviewing
+the location identified by the other author.
+5
+
+Included Tools
+Only Package
+Source Code (C0)
+Not Available
+or Proprietary (C1)
+Not Scriptable
+Interface (C2)
+Not Security
+Oriented (C3)
+Other Exclusionary
+Reasons
+NodeJsScan [13] 3
+CodeQL [14] 3
+ODGen [15] 3
+Graudit [16] 3
+InsiderSec [17] 3
+ESLint SSC [18] 3
+MS DevSkim [19] 3
+Mosca [20] 3
+Drek [21] 3
+SyNode [4]
+NodeSec [9]
+Affogato [10]
+Beyond Security [30]
+Checkmarx [31]
+Fortify [32]
+Veracode [33]
+Kiuwan [34]
+CodeSonar [35]
+Thunderscan [36]
+WhiteHat [37]
+JsPrime [38] 3
+Codeburner [39] 3
+SonarQube [40] 3
+CodeWarrior [41] 3
+WALA [42]
+PMD [43]
+Aether [44]
+Coala [45]
+JsHint [46] 3
+EsComplex [47]
+Coverty Scan [48]
+DeepScan [49]
+AppInspector [50] 3
+TAJS [51]
+SAFE [52]
+ESLint SP [53] 3
+Mozilla ScanJs [54] 3
+SemGrep [55]
+JAW [56] 3
+Joern [57, 58] 3
+Table III: Tools included in / excluded from this study. Excluded for reasons beyond C0, C1, C2, & C3: ESLint Security Plugin and Mozilla
+ScanJs use ESLint rules subsumed by ESLint SSC’s; SemGrep requires specially-crafted rules for security purposes and is the backbone of
+NodeJsScan; JAW only implements rules for detecting client-side CSRF; Joern supports JavaScript inspection, but does not support default
+rules for detection. Some tools excluded due to C1 were tested using free trials, but failed to comply with additional criteria.
+Dataset size: From the 1359 analyzed advisories, we were
+able to manually verify 957 review files (70%) at the time of
+this paper submission. For each review file we confirmed the
+exact location of the reported vulnerability. For this reason, we
+are confident to include these reviews in our curated dataset.
+IV. VULNERABILITY DETECTION TOOLS (RQ2)
+We now focus on RQ2, explaining how we selected the
+tools considered in our study. We specify our eligibility criteria,
+survey the existing tools that satisfy them, and classify these
+tools according to the detection technique that they employ.
+A. Tool Selection Criteria and Selection Process
+We focus specifically on fully automatic tools for analysis
+of npm packages. In particular, to select a given tool, it must:
+C0. Depend only on the package source code: The tool
+requires only the source code of the package to analyze. This
+excludes tools that require a test suite to guide the analysis.
+C1. Be available and transparent: The tool is publicly
+available and implements a technique that is non-proprietary.
+Its source code does not need to be open as long as the tool’s
+code analysis techniques can be clearly characterized, e.g.,
+through available documentation, rule set, and usage examples.
+C2. Have a scriptable interface: The tool must support a
+command-line interface (CLI), or similar interaction, allowing
+it to be executed and its output analyzed via a script. This
+facilitates the scalability and automation of the analysis.
+C3. Be security-oriented: The tool must identify vulnerabil-
+ities or security bad practices in JavaScript. This excludes
+tools that only construct artifacts, such as control-flow graphs,
+produce warnings about coding styles and conventions, or
+produce statistical information about the code, such as code
+metrics, that might be irrelevant from a security standpoint.
+Based on the above criteria, we ended up selecting nine tools
+for our testing purposes. We started by examining the academic
+literature [4, 9, 10, 15, 56, 57] and searching the Internet,
+including OWASP lists [59, 60], repository collections [61,
+62, 63] and other websites [64, 65, 66], for suitable tools
+for vulnerability detection in server-side JavaScript code. Most
+tools we screened were developed by the industry and the open-
+source community. In total, we first collected 40 JavaScript
+analysis tools. This full list is presented in Table III.
+After inspecting all 40 analysis tools, we found that we
+needed to manually test 19 tools, which are annotated with
+the symbol 3 in Table III. These 19 tools were tested against
+the Damn Vulnerable Node Application (DVNA) [67], a web
+application written in JavaScript that was purposely built with
+a range of vulnerabilities matching the OWASP Top 10 Web
+Security Risks. After executing each of the 19 tools against
+the vulnerable application, we excluded those that do not allow
+the analysis to be automated via a script and also those that
+fail to show security-oriented results.
+Out of the remaining 19 tools, we selected 14 tools that are
+available, transparent, can be automated, and show security-
+oriented results. Out of these 14 tools, we also excluded ESLint
+Security Plugin, Mozilla ScanJs, SemGrep, JAW, and Joern
+as explained in the caption of Table III. Consequently, we
+ended up with 9 distinct candidates that represent proper fully
+automatic vulnerability detection tools for Node.js applications.
+B. Detection Techniques
+By manually analyzing the source code of the nine selected
+tools, we characterized them according to the employed tech-
+nique for finding vulnerabilities. It is important to understand
+how these techniques work as they can have a significant
+impact on the effectiveness of the tools. We categorize these
+techniques using three classes: graph-based analysis, syntax-
+based analysis and keyword-based analysis. Table IV maps
+every selected tool to its corresponding detection technique.
+Graph-based analysis tools work by first constructing a graph-
+based model of the program to be analyzed. Such models
+usually coalesce into a single graph-like data structure various
+types of statically computed program artefacts, including:
+abstract syntax trees, control-flow graphs, and dependency
+graphs. The obtained data structure can be then inspected
+using queries written in domain-specific languages (DSL)
+especially designed for specifying vulnerable code patterns.
+For instance, typical queries aim at identifying code-flow paths
+through which user-controllable inputs can reach dangerous
+6
+
+Technique
+Tool
+Version
+Graph-based
+Analysis
+CodeQL
+2.2.6
+ODGen
+–
+Syntax-based
+Analysis
+NodeJsScan
+0.2.8
+ESLint SSC
+**
+Keyword-based
+Analysis
+Graudit
+2.8
+InsiderSec
+2.0.5
+MS DevSkim
+0.4.109
+Mosca
+0.8
+Drek
+1.0.3
+Table IV: Vulnerability detection technique employed by each tool.
+**ESLint SSC was used with eslint@7.32.0.
+sinks. CodeQL is one of such tools that models source code as
+database records. These records can be queried using SQL-like
+statements that are specified in the form of rules/queries.
+Syntax-based analysis tools employ a technique that searches
+the code to be analyzed for insecure syntax-aware patterns.
+Patterns can express simple control-flow conditions, e.g., calling
+a function with a particular variable. In contrast to graph-based
+analysis, this technique operates directly on source code and
+does not typically cater for more intricate dependency analysis
+or for matching patterns across multiple files. NodeJsScan is
+an example of such tools. It is used in GitLab’s CI/CD [68].
+Keyword-based analysis tools employ a code analysis tech-
+nique that searches the code to be analyzed for strings
+associated with potentially insecure code. This search is
+typically performed through the use of regular expressions.
+Note that keyword-based analysis does not model the AST
+of the program to analyze. Consequently, it is considerably
+less expressive than graph- and syntax-based analysis, as it
+cannot reason about fine-grained control-flow interactions, often
+operating on a single line of code at a time.
+C. How Different Detection Techniques Work
+To better understand how the aforementioned vulnerability
+detection techniques work, we examine, as an example, the
+advisory 315 of our dataset. According to the information
+available in the advisory page [23], this example consists
+of a code injection vulnerability (CWE-94), which allows a
+malicious user to run arbitrary code on the targeted execution
+platform. In this type of attack the adversary is only limited by
+the expressiveness of the injected language. JavaScript code
+injections at the server can have more significant impact than
+those at the client-side, given that Node.js has fewer security
+barriers (e.g., no sandbox), and a larger and privileged API
+facilitating access to critical system resources (e.g., file system).
+The vulnerable code snippet of the advisory 315 is given
+in Listing 2. In this example, the variable opts is bound to a
+user-controlled object with the properties filter and collection,
+which can be trivially tainted with a maliciously crafted
+input to produce valid JavaScript code that reaches the eval
+function. One can leverage this vulnerability to execute arbitrary
+JavaScript code, including OS-level commands by using a
+payload like the one given in Listing 3.
+Using graph-based analysis: Both CodeQL and ODGen adopt
+graph-based analysis. Listing 4 shows an example of a CodeQL
+1
+function search (opts) {
+2
+if (!opts.filter && opts.collection) {
+3
+if (typeof opts.collection === 'string') {
+4
+opts.filter =
+"function filter (doc) { return doc.type === '" +
+opts.collection + "'}";
+�→
+�→
+5
+} else { ... }
+6
+eval(opts.filter);
+7
+opts.filter = filter;
+8
+}
+9
+}
+Listing 2: Code injection vulnerability (npm advisory 315).
+1
+opts.collection = `'};
+2
+const exec = require("child_process").exec;
+3
+exec("cat /etc/passwd", (err, stdout, stderr) => {
+4
+console.log(stdout); }); var a={ hello: 'world`;
+5
+search(opts)
+Listing 3: Exploit for code injection (npm advisory 315).
+rule designed to detect calls to the eval function using user-
+controlled inputs. In order to create this CodeQL rule, one
+starts by specifying the appropriate configuration, that is, a
+code description of the targeted sources and sinks. In this
+case, we are interested in code flows from remote flow sources,
+described by the predicate isSource (lines 3 to 5), to the eval
+sink, described by the predicate isSink (lines 6 to 8). Then
+the main query (lines 11 to 13) states that, using the specified
+configuration (EvalTaint cfg), CodeQL should find code paths
+from the specified source to the specified sink. The output of
+this query is a string with a description of the source-sink pairs
+that match the query. This particular rule is a simplified version
+of one of the CodeQL rules [69] executed inside VulcaN.
+In general, graph-based analysis works well for taint-tracking,
+but it requires every source and sink to be explicitly encoded
+into rules. These sources and sinks change over time as
+languages evolve and new popular third-party packages are
+created. This is why the community has started to work on
+automatically generating such taint-tracking specifications [71].
+Using syntax-based analysis: NodeJsScan helps us showcase
+syntax-based analysis. Listing 5 lists an excerpt of the Node-
+JsScan rule that detects potentially vulnerable uses of eval. To
+be applied, this rule must match two related patterns. First, eval
+must occur inside a function receiving two or more arguments
+(line 2). Then, eval must be called with a parameter computed
+using one of the given arguments (line 4). NodeJsScan includes
+analogous rules for other vulnerability types [73].
+Similarly to the previous technique (graph-based), syntax-
+based analysis also suffers from the source-sink specification
+limitation. Additionally, there are some other limitations
+specific to syntax-based analysis. For example, it may lead
+to a high number of false positives, as it is not expressive
+enough to capture the dependencies of the variables occurring
+in the patterns; e.g., it will detect all calls to eval regardless
+of whether or not their given input can be controlled by
+the user. Furthermore, it commonly leads to rule overfitting,
+resulting in over-specific rules that match known examples
+of vulnerabilities, but are not general enough to capture other
+instances of the same vulnerability. In Listing 5, the eval call is
+7
+
+1
+class EvalTaint extends TaintTracking::Configuration {
+2
+EvalTaint() { this = "EvalTaint" }
+3
+override predicate isSource(Node node) {
+4
+node instanceof RemoteFlowSource
+5
+}
+6
+override predicate isSink(Node node) {
+7
+node = globalVarRef("eval").getACall().getArgument(0)
+8
+}
+9
+}
+10
+11
+from EvalTaint cfg, Node source, Node sink
+12
+where cfg.hasFlow(source, sink)
+13
+select sink, "Eval with user input from \$@.", source
+Listing 4: CodeQL rule for eval taint-tracking [70].
+1
+patterns:
+2
+- pattern-inside: function $FUNC($REQ, $RES, ...) {...}
+3
+- pattern-either:
+4
+- pattern: eval(..., <... $REQ.$QUERY ...>, ...)
+Listing 5: NodeJsScan rule for eval detection [72].
+only detected when it occurs inside the body of a specific type
+of function declaration. Besides ignoring eval calls at the top
+level, this pattern also ignores calls to eval which occur inside
+the body of JavaScript functions declared using alternative
+syntactic constructs (e.g. function constructor and lambdas).
+Using keyword-based analysis: The tool we use to illustrate
+keyword-based analysis is Graudit. The following is an excerpt
+of a Graudit rule that detects the use of eval:
+eval[[:space:]]*\(
+Here, we see a regular expression pattern that simply detects
+any call to the eval function. This technique suffers from several
+limitations such as the source-sink specification problem and
+a high number of false positives. Graudit includes many other
+rules for dangerous sinks in Node.js applications [74].
+V. EFFECTIVENESS OF THE TOOLS (RQ3)
+In this section, we focus on our third research question
+(RQ3). To perform a quantitative and qualitative assessment
+of the selected tools, we begin by specifying the evaluation
+metrics and methodology we used to rank the tools. Then
+we present our findings, relying on the result of running the
+selected tools across all 957 advisories of our curated dataset.
+A. Evaluation Methodology
+Tool evaluation metrics: To evaluate the selected tools, we use
+two main metrics: true positive rate (TPR) and precision (P).
+The TPR represents the proportion of the total vulnerabilities
+that are correctly detected by a given tool, i.e. the true positives
+(TP): TPR=TP/|vulnerabilities|. The TPR is useful to assess the
+raw detection rate of a tool without considering the influence
+of false positives (FP), i.e., its results that do not match
+the reported advisory. Precision represents the proportion of
+correctly classified positive cases: P=TP/(TP+FP). This metric
+is useful to assess if a tool produces too many false positives
+that can unnecessarily consume analysts’ resources.
+Tool classification score: To compute the evaluation metrics
+for a given tool, we need to analyze the output that it generates
+<report_mosca>
+<Path>/src/lib/drivers/search/pouch.js</Path>
+<Title>Possible code injection</Title>
+<Description>
+Command injection is an attack in which the goal is
+execution of arbitrary commands on the host
+operating system via a vulnerable application.
+�→
+�→
+</Description>
+<Level>High</Level>
+<Match>eval\s?\(|setTimeout|setInterval</Match>
+<Result>Line: 20 - eval(opts.filter);</Result>
+</report_mosca>
+Listing 6: Snippet of Mosca output classified with Score A.
+/src/lib/drivers/search/pouch.js-19- }
+/src/lib/drivers/search/pouch.js:20: eval(opts.filter);
+/src/lib/drivers/search/pouch.js-21- opts.filter = filter;
+Listing 7: Snippet of Graudit output classified with Score B.
+when applied to analyzing a specific vulnerability. Given that
+each tool outputs the vulnerability analysis results in its own
+specific, unstandardized format, we characterize a tool’s output
+according to a common discrete classification score:
+• Score A: The tool correctly detects and classifies the
+vulnerability reported in the advisory (true positive).
+• Score B: The tool shows a warning for the vulnerable
+code, but does not explicitly classify the finding as a
+vulnerability (true positive).
+• Score C: The tool only shows results that do not match
+the vulnerability in the advisory report (false positives).
+• Score D: The tool produces no output (false negative).
+We split the TP results according to two distinct classes: A
+means an explicit vulnerability notification, and B a security
+warning notification. The tools ranked with score A provide a
+richer output to the user and, thus, more information about the
+detected vulnerability. As an example, consider the output of
+two tested tools, Mosca and Graudit, with regards to advisory
+315 shown in Listings 6 and 7, respectively. Although both
+outputs flag the vulnerable eval call reported by the review file
+of Listing 1, Mosca’s output clearly identifies a possible code
+injection, provides a description, a severity level, and the line
+of code containing the vulnerability. On the other hand, Graudit
+only shows the vulnerable line of code without explaining how
+or why it flags that particular snippet. For this reason, the
+output of Mosca is classified with Score A while the output of
+Graudit is classified with Score B.
+This discrete classification is also important to account for
+tools that might flag for the vulnerability at a place that is only
+close to it (textually, or on the AST). Considering this, we
+require that tools must clearly identify the vulnerable statement
+for some vulnerabilities, e.g., code injections and others that
+can typically be pinpointed to a single statement, while for
+other vulnerability types multiple lines-of-code are acceptable.
+Two authors performed a cross-check of all tools’ outputs to
+guarantee fairness of the tool classification in these cases.
+B. Analysis Performance
+We gauge analysis performance by measuring tools’ execu-
+tion time for all 957 advisories on a machine with an Intel
+8
+
+Tool
+Min
+Max
+Mean
+St. Dev.
+Q-90
+Total
+ODGen
+1.236
+3653.043
+148.757
+385.629
+370.969
+110823.8
+CodeQL
+1.712
+736.546
+119.570
+98.755
+177.550
+28696.8
+NodeJsScan
+20.023
+984.453
+99.562
+121.246
+230.216
+23795.4
+ESLint SSC
+0.592
+3556.665
+29.871
+237.799
+25.465
+7139.1
+Graudit
+0.042
+14.632
+0.550
+1.624
+0.704
+131.3
+InsiderSec
+0.000
+243.000
+5.749
+25.186
+7.000
+1374.0
+MS DevSkim
+0.276
+186.338
+6.393
+23.262
+8.044
+1527.9
+Drek
+0.300
+6.649
+1.022
+0.865
+1.949
+244.3
+Mosca
+0.005
+245.498
+7.408
+25.166
+12.216
+1770.5
+Table V: Summary statistics of the analysis times (in seconds) taken
+by the tested tools across all 957 reviewed advisories.
+ODGen
+CodeQL
+NodeJsScan
+ESLint SSC
+Graudit
+InsiderSec
+MS DevSkim
+Drek
+Mosca
+0
+20
+40
+60
+80
+100
+Percentage of Advisories
+146
+286
+98
+103
+75
+294
+212
+288
+231
+324
+414
+513
+158
+376
+210
+456
+515 426
+530
+146 225
+791
+500
+732
+475
+A
+B
+C
+D
+Figure 4: Score distribution for each tool.
+Xeon E3-1220 v3 @ 3.10GHz processor and 32GB of memory.
+Table V shows several statistics of the execution times taken
+by each tool to analyze our dataset. When compared to all other
+tools, ODGen, CodeQL, NodeJsScan and ESLint SSC require
+considerably more time. To analyze all 957 packages, ODGen
+took over 30 hours (110k seconds), CodeQL took nearly 8
+hours (27k seconds), NodeJsScan took nearly 7 hours (24k
+seconds), while ESLint SSC took nearly 2 hours (7k seconds).
+All other tools are considerably more efficient, taking at most
+30 minutes to analyze all packages.
+The mean execution time of ODGen, CodeQL and Node-
+JsScan is 148.8, 119.6 and 99.6 seconds, respectively. ODGen,
+CodeQL and NodeJsScan tools are slower because their detec-
+tion techniques involve modeling statically computed structures.
+These operations are more complex than performing keyword-
+based matching searches (see Section IV-B). Depending on
+the size of the package and on the CI/CD pipeline restrictions,
+these tools may end up being exceedingly slow.
+C. Results Across the Entire Dataset
+Figure 4 displays the score distribution for each tool across
+our entire dataset, and Table VI shows the evaluation metrics
+for each tool. Globally, the tested tools perform rather poorly.
+We can draw the following main observations:
+1. Some tools have very low TPR: Counting A and B
+scores as successful detections, we see that InsiderSec, Drek
+and Mosca only detect 7 (0.7%), 15 (1.6%) and 25 (2.6%)
+vulnerabilities, respectively. Hence, these tools fail to detect
+most vulnerabilities of the dataset.
+2. The tools with best TPR have very low precision: The
+tools that have higher TPR are: ODGen, Graudit, ESLint SSC,
+Graph
+Syntax
+Keyword
+0
+25
+50
+75
+100
+Percentage of Advisories
+320
+196
+104
+226
+197
+356
+443
+516
+262
+92
+140
+A
+B
+C
+D
+Figure 5: Score distribution for each detection technique.
+and CodeQL. Unfortunately, Graudit and ESLint SSC also
+have a considerable number of false positives, which tends to
+erode the confidence of application developers in vulnerability
+detection tools. Graudit detects 219 vulnerabilities (22.9%),
+but it also reports over 109k FPs, giving it an overall precision
+of just 0.2%. A higher number of FPs is expected from a
+keyword-based tool like Graudit, as many of its string signatures
+often match non-vulnerable code snippets. ESLint SSC has
+the highest TPR (41.5%). However, it is also the tool with the
+highest number of reported FPs (over 389k) and, consequently,
+the lowest precision (0.1%). This is because ESLint SSC
+includes many rules from different ESLint plugins, some of
+which are simple matches (akin to keyword-based analysis)
+with greedy behaviour, leading to a higher number of FPs.
+3. Graph-based analysis has the best detection capability:
+Figure 5 shows the scores according to a particular detection
+technique. These results show that graph-based analysis reports
+a significantly larger number of results with score A (explicit
+vulnerability notifications). Syntax and keyword-based analysis
+look fairly similar, with reasonable detection rates, but also a
+high number of reports containing only false positive results.
+When considering the results of both tools in this category,
+i.e., ODGen and CodeQL, they strike a better balance between
+true positives and precision. CodeQL detects 300 vulnerabilities
+(31.3%) and has a significantly higher precision (7.8%), when
+compared to most other tools, while ODGen detects 154
+vulnerabilities (16.1%). This number is significantly lower
+than CodeQL’s, but it represents a much higher precision
+(23.8%) than any other tool tested. Although both these
+tools do not have the highest TPR, most of their detected
+vulnerabilities were classified with the A score, meaning that
+the reported information is richer and more meaningful to the
+user. Consequently, CodeQL and ODGen are the most balanced
+tools, achieving a reasonable detection rate (TPR) and less
+FPs, when compared to other tools with similar TPR. We also
+note that both these tools have the potential for being further
+improved by extending them with additional rules.
+4. Combining multiple tools increases TPR, but also lowers
+the overall precision: The combination of the two best tools
+(CodeQL and ESLint SSC) detects 508 vulnerabilities (53.1%),
+albeit with only 0.12% precision. If we add the third best
+tool (Graudit), we detect more vulnerabilities (551/57.6%), but
+the precision further decreases to 0.11%. Finally, combining
+9
+
+Scope
+ODGen
+CodeQL
+NodeJsScan
+ESLint SSC
+Graudit
+InsiderSec
+MS DevSkim
+Drek
+Mosca
+TP (%) FP (P%) TP (%) FP (P%) TP (%) FP (P%) TP (%) FP (P%) TP (%) FP (P%) TP (%) FP (P%) TP (%) FP (P%) TP (%) FP (P%) TP (%) FP (P%)
+CWE-22
+70
+136
+104
+416
+56
+257
+110
+25467
+122
+3101
+2
+401
+0
+368
+0
+1057
+0
+241
+(47.9)
+(34.0)
+(71.2)
+(20.0)
+(38.4)
+(17.9)
+(75.3)
+(0.4)
+(83.6)
+(3.8)
+(1.4)
+(0.5)
+(0.0)
+(0.0)
+(0.0)
+(0.0)
+(0.0)
+(0.0)
+CWE-79
+1
+33
+26
+843
+6
+924
+29
+123477
+11
+23634
+0
+28
+0
+3990
+0
+6353
+1
+1664
+(1.0)
+(2.9)
+(26.3)
+(3.0)
+(6.1)
+(0.6)
+(29.3)
+(0.0)
+(11.1)
+(0.0)
+(0.0)
+(0.0)
+(0.0)
+(0.0)
+(0.0)
+(0.0)
+(1.0)
+(0.1)
+CWE-400
+4
+40
+13
+212
+2
+374
+39
+21936
+4
+2795
+0
+22
+0
+1026
+0
+74
+1
+271
+(4.5)
+(9.1)
+(14.6)
+(5.8)
+(2.2)
+(0.5)
+(43.8)
+(0.2)
+(4.5)
+(0.1)
+(0.0)
+(0.0)
+(0.0)
+(0.0)
+(0.0)
+(0.0)
+(1.1)
+(0.4)
+CWE-78
+22
+40
+43
+416
+2
+121
+29
+7405
+4
+1567
+0
+15
+0
+269
+3
+380
+3
+190
+(29.3)
+(35.5)
+(57.3)
+(9.4)
+(2.7)
+(1.6)
+(38.7)
+(0.4)
+(5.3)
+(0.3)
+(0.0)
+(0.0)
+(0.0)
+(0.0)
+(4.0)
+(0.8)
+(4.0)
+(1.6)
+CWE-818
+0
+11
+16
+57
+0
+97
+1
+6990
+3
+1431
+1
+8
+64
+1093
+0
+393
+0
+150
+(0.0)
+(0.0)
+(21.3)
+(21.9)
+(0.0)
+(0.0)
+(1.3)
+(0.0)
+(4.0)
+(0.2)
+(1.3)
+(11.1)
+(85.3)
+(5.5)
+(0.0)
+(0.0)
+(0.0)
+(0.0)
+CWE-471
+11
+19
+13
+54
+0
+295
+38
+7466
+0
+2632
+0
+0
+0
+249
+0
+220
+0
+438
+(22.9)
+(36.7)
+(27.1)
+(19.4)
+(0.0)
+(0.0)
+(79.2)
+(0.5)
+(0.0)
+(0.0)
+(0.0)
+(0.0)
+(0.0)
+(0.0)
+(0.0)
+(0.0)
+(0.0)
+(0.0)
+CWE-20
+5
+19
+4
+25
+0
+116
+19
+7947
+4
+911
+0
+13
+0
+181
+1
+26
+2
+294
+(12.2)
+(20.8)
+(9.8)
+(13.8)
+(0.0)
+(0.0)
+(46.3)
+(0.2)
+(9.8)
+(0.4)
+(0.0)
+(0.0)
+(0.0)
+(0.0)
+(2.4)
+(3.7)
+(4.9)
+(0.7)
+CWE-1321
+3
+12
+5
+78
+0
+92
+31
+18130
+0
+4468
+0
+9
+0
+0
+0
+301
+0
+465
+(8.3)
+(20.0)
+(13.9)
+(6.0)
+(0.0)
+(0.0)
+(86.1)
+(0.2)
+(0.0)
+(0.0)
+(0.0)
+(0.0)
+(0.0)
+(0.0)
+(0.0)
+(0.0)
+(0.0)
+(0.0)
+CWE-94
+4
+18
+5
+79
+2
+159
+16
+17930
+10
+7159
+1
+23
+0
+358
+8
+858
+8
+199
+(12.1)
+(18.2)
+(15.2)
+(6.0)
+(6.1)
+(1.2)
+(48.5)
+(0.1)
+(30.3)
+(0.1)
+(3.0)
+(4.2)
+(0.0)
+(0.0)
+(24.2)
+(0.9)
+(24.2)
+(3.9)
+CWE-77
+8
+11
+11
+90
+1
+32
+9
+5390
+0
+892
+0
+1
+0
+1
+0
+52
+0
+97
+(34.8)
+(42.1)
+(47.8)
+(10.9)
+(4.3)
+(3.0)
+(39.1)
+(0.2)
+(0.0)
+(0.0)
+(0.0)
+(0.0)
+(0.0)
+(0.0)
+(0.0)
+(0.0)
+(0.0)
+(0.0)
+Other CWE
+26
+154
+60
+1283
+34
+2548
+76
+147552
+61
+60895
+3
+104
+17
+7930
+3
+9159
+10
+2868
+(8.9)
+(14.4)
+(20.5)
+(4.5)
+(11.6)
+(1.3)
+(26.0)
+(0.1)
+(20.9)
+(0.1)
+(1.0)
+(2.8)
+(5.8)
+(0.2)
+(1.0)
+(0.0)
+(3.4)
+(0.3)
+Dataset
+154
+493
+300
+3553
+103
+5015
+397
+389690
+219
+109485
+7
+624
+81
+15465
+15
+18873
+25
+6877
+(16.1)
+(23.8)
+(31.3)
+(7.8)
+(10.8)
+(2.0)
+(41.5)
+(0.1)
+(22.9)
+(0.2)
+(0.7)
+(1.1)
+(8.5)
+(0.5)
+(1.6)
+(0.1)
+(2.6)
+(0.4)
+Table VI: TP, TPR (%), FP and Precision (P%) for each tool by CWE. TPR highlights: green (TPR ≥ 50%) or yellow (50% > TPR ≥
+15%). When TPR is highlighted we also highlight the FP and P columns: yellow (50% > P ≥ 15%), light red (15% > P ≥ 2.5%) and dark
+red (P < 2.5%). The CWEs are: Path Traversal (CWE-22), Cross-site Scripting (CWE-79), Resource Exhaustion (CWE-400), Insufficient
+Transport Layer Protection (CWE-818), OS Command Injection (CWE-78), Modification of Assumed-Immutable Data (CWE-471), Improper
+Input Validation (CWE-20), Code Injection (CWE-94), Improper Neutralization of Special Elements used in a Command (CWE-77), and
+Improperly Controlled Modification of Object Prototype Attributes (CWE-1321).
+both graph-based tools, CodeQL and ODGen, allows for the
+detection of 339 vulnerabilities (35.4%) with a precision of
+7.7%. This shows that combining the best tools can increase
+the TPR, but at the cost of also increasing the number FPs,
+which limits the advantage of such an approach.
+D. Results Across Specific Vulnerability Types
+We now assess the performance of the tools when focusing
+on particular types of vulnerabilities. We concentrate on two
+main aspects: i) studying the types of vulnerabilities that the
+tools detect more frequently, and ii) analyzing which types of
+vulnerabilities can be detected simultaneously by several tools.
+1. Most frequently detected vulnerability types: From the
+analysis of Table VI, we highlight seven CWEs that are detected
+most often regardless of the used tool: CWE-22, CWE-471,
+CWE-78, CWE-79, CWE-94, CWE-77, and CWE-1321. These
+are colored in yellow and green in Table VI.
+CWE-22 (path traversal) is the only type clearly detected by
+all four best performing tools (ODGen, ESLint SSC, Graudit,
+and CodeQL). This is because path traversal can be found
+statically by searching for well-known dangerous sinks in
+the Node.js API, e.g., the functions readFile, writeFile and
+createReadStream. The difference in precision between these
+four tools lies in that ESLint SSC and Graudit simply match
+these function calls, while ODGen and CodeQL report only
+cases where the path is tainted by user input.
+CWE-78 and CWE-77 (OS command injection), CWE-79
+(cross-site scripting) and CWE-94 (code injection) correspond
+to classic injection vulnerabilities. Detecting these vulnerabil-
+ities depends on the sets of sinks considered by each tool.
+CodeQL detects more OS command injections, while ESLint
+SSC detects more code injections because each have more
+extensive rulesets for those particular vulnerabilities. Both
+tools detect about the same number of XSS vulnerabilities.
+Both CWE-471 (Modification of Assumed-Immutable Data)
+and CWE-1321 (Improperly Controlled Modification of Object
+Prototype Attributes) are umbrella CWEs for several prototype
+tampering and prototype pollution vulnerabilities, for which
+both CodeQL and ESLint SSC have various rules. We expected
+ODGen to perform better at detecting prototype pollution vul-
+nerabilities (CWE-471), as this is one of its central goals [15].
+Although ODGen’s results for this vulnerability (22.9%) fall
+short of those by CodeQL (27.1%) and ESLint SSC (79.2%),
+ODGen does achieve a much higher precision (36.7%) than
+CodeQL (19.4%) and, especially, ESLint SSC (0.5%).
+2. Vulnerability types detected by the three best performing
+tools: Figure 6 shows the intersections of TPs for the top-10
+CWEs. We can see a substantial intersection for CWE-22,
+where all three tools detect the same 85 vulnerabilities. This
+happens because path traversals are easy to find statically
+using a limited set of known dangerous sinks from the Node.js
+API, which all tools share. For CWE-79, both CodeQL and
+ESLint SSC detect about the same number of vulnerabilities,
+but only about half intersect with each other. This is due to
+differences in the rules regarding XSS sources and sinks. CWE-
+471 shows a significant intersection, but ESLint SSC detects
+several vulnerabilities that CodeQL misses. This is because
+ESLint SSC’s rules have a wider range of sinks. Other CWEs
+have less intersections because their rulesets differ. For example,
+CodeQL is the only tool with specific rules to detect resource
+downloads over HTTP, hence the results for CWE-818.
+10
+
+2
+4
+4
+7
+13
+17
+85
+CWE-22
+12
+13
+11
+6
+0
+2
+3
+CWE-79
+2
+26
+9
+2
+0 1
+1
+CWE-400
+24
+7
+18
+3
+1
+CWE-78
+14
+1
+1 2
+CWE-818
+2
+26
+11
+CWE-471
+2
+12
+1
+1 3
+CWE-20
+26
+5
+CWE-1321
+3
+7
+0
+2
+0
+7
+1
+CWE-94
+5
+4
+2
+CWE-77
+CodeQL
+ESLint SSC
+Graudit
+Figure 6: Intersections of TPs of the 3 best tools for top-10 CWEs.
+5
+10
+15
+20
+Precision (%)
+0
+200
+400
+600
+# Lines of Code (LOC)
+CWE-22
+CWE-78
+CWE-77
+CWE-471
+CWE-79
+CWE-818
+CWE-94
+CWE-400
+CWE-1321
+CWE-20
+Figure 7: Correlation between Query LOC and Precision.
+E. Results as a Function of Queries and Ruleset
+To study the relationship between ruleset/queries and vulner-
+ability detection results, we take CodeQL as an example and
+show, in Figure 7, how the precision of this tool compares with
+the number of lines of code of the specific queries CodeQL uses
+to detect the vulnerabilities in the Top-10 CWE categories in
+the dataset. Although this cannot be taken as a general rule, we
+can see that, in most cases, the precision is higher for smaller
+queries. Taking into account each CWE, simpler vulnerabilities,
+like CWE-22, can be detected using smaller queries with higher
+precision, while more complex vulnerabilities to detect, like
+CWE-1312, are harder to detect even when using larger (more
+complex) queries.
+It is then clear that vulnerability detection results can be
+influenced by the ruleset/queries executed by the tool. Using a
+small (specific) ruleset may improve the precision in detecting
+a specific vulnerability. However, in the context of a CI/CD
+pipeline, application developers do not know beforehand which
+specific ruleset to select to detect the (unknown) vulnerability.
+Consequently, it is reasonable to apply the most comprehensive
+ruleset available for the tool. This is the approach we used in
+this paper. For every tool, we selected the most comprehensive
+and complete ruleset, by either combining rules into a single
+tool execution or executing the tool multiple times using a
+different rule for every execution and combining the results. We
+also used the rules available off-the-shelf instead of developing
+or customizing rules. This allows us to reflect how developers
+will use these tools as most of them are not technically versed
+to improve the ruleset specified by the tool developers.
+VI. REASONS FOR MISSED DETECTION (RQ4)
+In RQ4, we study why existing tools fail to detect certain
+vulnerabilities. Table VII shows the number of undetected
+CWE
+CWE Description
+OWASP
+Undetected
+%
+CWE-79
+Cross-site Scripting
+50 / 99
+50.5%
+CWE-400
+Resource Exhaustion
+-
+44 / 89
+49.4%
+CWE-78
+OS Command Injection
+20 / 75
+26.7%
+CWE-20
+Improper Input Validation
+16 / 41
+39.0%
+CWE-22
+Path Traversal
+12 / 146
+8.2%
+CWE-94
+Code Injection
+10 / 33
+30.3%
+CWE-818
+Insecure Transport Layer
+10 / 75
+13.3%
+CWE-287
+Improper Authentication
+9 / 9
+100.0%
+CWE-471
+MAID
+8 / 48
+16.7%
+CWE-200
+Information Exposure
+8 / 14
+57.1%
+Others
+-
+-
+137 / 286
+47.9%
+Total
+-
+-
+324 / 957
+33.9%
+Table VII: Number of vulnerabilities undetected by any tool.
+Limitation
+Advisory
+CWE
+Vulnerability
+L1
+63
+CWE-730
+CVE-2015-9241
+567
+CWE-287
+CVE-2017-11429
+L2
+165
+CWE-818
+CVE-2016-10583
+305
+CWE-22
+CVE-2016-1000249
+L3
+26
+CWE-287
+CVE-2014-10067
+92
+CWE-200
+CVE-2016-10533
+L4
+113
+CWE-89
+CVE-2016-10554
+43
+CWE-79
+CVE-2014-9772
+L5
+1469
+CWE-471
+CVE-2017-1000048
+313
+CWE-502
+CVE-2017-5954
+Table VIII: Examples of undetected vulnerabilities by cause (Lx).
+vulnerabilities grouped by CWE and their mapping to OWASP
+Top 10 Web Security Risks (2021) [29]. Of the 957 known
+vulnerabilities in the dataset, 324 vulnerabilities (33.9%) were
+not detected by any of the selected tools. To understand the
+underlying reasons, we have manually analyzed a sample
+of undetected vulnerabilities. So far, we have identified the
+following five main tool limitations. In Table VIII, we map
+these limitations with some vulnerability categories (CWE) and
+provide specific examples of undetected vulnerabilities (CVE).
+L1. Cross-package vulnerabilities: The selected tools come
+with pre-defined sets of manually written rules, typically
+focusing solely on popular APIs. We noticed that some
+undetected vulnerabilities exist in code that invokes functions of
+third-party packages that map directly to known dangerous code,
+e.g., wrappers to OS-level commands. These vulnerabilities
+could have been found by testing all package dependencies
+(can be thousands of other packages [3]), or by using a more
+complete set of rules and queries (covering additional sources
+and sinks). However, the manual maintenance of such lists
+of sources and sinks is impractical as the Node.js ecosystem
+expands. Existing work [71] tries to automatically extract taint
+specifications (sources and sinks) from JavaScript libraries,
+which partially solves the issue of incomplete rules, but requires
+the constant dynamic testing of every new npm package.
+For instance, Listing 8 shows a code snippet of a command
+injection vulnerability for advisory 1440. This problem exists
+because user-controlled data reaches an exec sink inside the
+third-party package comandante, a package meant to ease the
+execution of OS-level commands. The tools fail to recognize
+this vulnerability because the usual command injection sinks
+are not directly present in the analyzed code, but are instead
+inside a third-party dependency that is not modelled by the
+11
+
+1
+// Snippet of ./gnuplot.js:
+2
+var run = require('comandante');
+3
+4
+module.exports = function () {
+5
+var plot = run('gnuplot', []);
+6
+plot.print = function (data, options) {
+7
+plot.write(data);
+8
+// (...)
+9
+};
+10
+// (...)
+11
+}
+Listing 8: Command Injection (advisory 1440) - NPM and Github
+Advisories [75, 76].
+{
+"scripts": {
+"preinstall":
+"""wget http://s.qdcdn.com/17mon/17monipdb.zip &&
+unzip -p 17monipdb.zip 17monipdb.dat > 17monipdb.dat"""
+}
+}
+Listing 9: Insecure Transport Layer in package.json of ipip-coffee
+package (advisory 279) - CVE-2016-10673.
+vulnerability detection rules of each tool, i.e., they failed to
+include the write function as a potential dangerous sink.
+L2. Limited analysis scope: In addition to JavaScript code
+files, Node.js projects depend on several other components,
+such as configuration files, front-end template code, testing
+frameworks, etc. However, by analyzing only the JavaScript
+code in isolation, certain vulnerabilities can be missed. As an
+example, npm packages contain a package.json file which may
+include bootstrap scripts. In several analyzed packages, these
+scripts are used to download resources over HTTP. As it turns
+out, using HTTP allows for man-in-the-middle attacks, where
+resources are replaced by malicious payloads. While some
+tools can detect insecure downloads if they are performed by
+the main JavaScript code (e.g., by searching for HTTP URLs),
+they cannot detect downloads issued from package.json.
+Listing 9 shows a snippet of the package.json file for the ipip-
+coffee package, in which an external resource is downloaded
+over HTTP. This allows for man-in-the-middle attacks that
+might compromise the server. In this particular example, this
+vulnerability can only be detected if the package.json file is
+also considered when performing the vulnerability analysis.
+L3. Lack of contextual knowledge: Packages may expose
+sensitive information, e.g., by logging plaintext passwords to a
+file. These vulnerabilities are application-specific and require
+contextual knowledge of which data is sensitive. The analyzed
+tools, however, are not designed to gain contextual knowl-
+edge and thus miss vulnerabilities that depend upon it, e.g.,
+application-specific leaks. To help detect such vulnerabilities,
+a possible approach is to annotate application inputs, objects,
+or data flows with sensitivity levels, and check which system
+resources handle the annotated features during the execution.
+Listing 10 shows an example of a Credential Exposure
+vulnerability, in which plaintext passwords are logged to the
+console. The code snippet itself seems benign until one becomes
+aware that the key variable holds security-critical information.
+1
+// Snippet of ./lib/odbc.js:
+2
+if(exports.debug) {
+3
+console.log("""%s odbc.js : pool[%s] :
+4
+pool.close() - processing pools %s - connections: %s""",
+5
+getElapsedTime(), self.index, key, connections.length);
+6
+}
+Listing 10: Credential Exposure (advisory 1185) - SNYK-JS-IBMDB-
+459762 [77].
+1
+// Snippet of ./protect/lib/rules/xss.js
+2
+const xssSimple = new
+RegExp('((%3C)|<)((%2F)|/)*[a-z0-9%]+((%3E)|>)', 'i')
+�→
+3
+const xssImgSrc = new RegExp('((%3C)|<)((%69)|i|(%49))((%6D)
+4
+|m|(%4D))((%67)|g|(%47))[^\n]+((%3E)|>)', 'i')
+5
+6
+function isXss(value) {
+7
+return xssSimple.test(value) || xssImgSrc.test(value)
+8
+}
+9
+// Example attack payload:
+10
+// <input type="image" src onerror="alert('XSS')">
+Listing 11: XSS (advisory 1116) - CVE-2018-1000160.
+This contextual knowledge is needed to detect the vulnerability
+but is difficult to extract using automated tools.
+L4. Incorrect sanitization: Application developers often use
+regular expressions to detect malicious inputs. However, regular
+expressions are complex, and developers usually do not test
+them thoroughly, allowing sanitization bypasses to occur.
+Sanitization errors are often hard to detect statically, as they
+require dynamically testing each regular expression ensuring
+that they generate semantically valid inputs that can both bypass
+the validation and effectively trigger the vulnerability.
+For instance, Listing 11 shows a code fragment containing
+two regular expressions that aim to prevent potential XSS
+vulnerabilities. However, these regular expressions are not
+entirely correct as there still exist some specially crafted inputs,
+such as the one shown in the comment of Listing 11, that can
+bypass this validation and launch an XSS attack.
+L5. Inability to cope with JavaScript dynamicity: Specific
+features of JavaScript can lead to vulnerabilities that are hard
+to detect by static analysis tools. For example, object-based
+inheritance, extensible objects, and dynamic typing are key
+features of JavaScript, which can lead to prototype pollution,
+authentication bypass, and business logic vulnerabilities.
+Listing 12 shows a type of Prototype Pollution vulnerability
+present in the qs package, which is a querystring parsing library
+that allows developers to create objects within query strings.
+For example, the string ’foo[bar]=baz’ is converted to the
+object {foo:{bar:’baz’}}. Usually, this package protects
+against attacks that try to overwrite the existing prototype
+properties of an object. However, in this vulnerable version,
+the protection can be circumvented by prefixing the name of
+the parameter with character [ or ], as shown in the proof-
+of-concept exploit code shown in Listing 12. Consequently,
+calling toString() on the object will throw an exception. This
+can subvert the application logic, potentially allowing attackers
+to work around security controls, modify data, and make the
+application unstable. The selected tools miss this example
+because they fail to model how objects change depending on
+the instructions applied to them, specially the object prototype.
+12
+
+1
+// Snippet of ./lib/parse.js:
+2
+module.exports = function (str, opts) {
+3
+var options = opts || {};
+4
+var tempObj = typeof str === 'string' ? parseValues(str,
+options) : str;
+�→
+5
+var obj = options.plainObjects ? Object.create(null) : {};
+6
+7
+var keys = Object.keys(tempObj);
+8
+for (var i = 0; i < keys.length; ++i) {
+9
+var key = keys[i];
+10
+var newObj = parseKeys(key, tempObj[key], options);
+11
+obj = Utils.merge(obj, newObj, options);
+12
+}
+13
+return Utils.compact(obj);
+14
+};
+15
+// Proof-of-Concept exploit code:
+16
+qs.parse("]=toString", { allowPrototypes: false })
+// {toString = true} <== prototype overwritten
+�→
+Listing 12: Prototype Override (advisory 1469) - CVE-2017-1000048.
+From these limitations, we can extract actionable insights on
+the applicability of static code analysis tools for vulnerability
+detection in Node.js code. On one hand, these tools can
+potentially overcome limitations L1 and L2 by both employing
+improved strategies for maintaining taint specifications, and by
+considering all the appropriate analysis scopes for Node.js
+code. On the other hand, every static analysis tool will
+struggle to overcome limitations L3, L4, and L5, because
+they fail to capture behavioral and contextual information that
+is only available at runtime when the package is executed
+with appropriate, and application-specific, test inputs. To this
+end, it seems that the approaches employed by current static
+vulnerability detection tools can mainly be used successfully to
+detect classic injection-style vulnerabilities even if all the tools
+tested in this paper cannot do so with reasonable precision.
+VII. THREATS TO VALIDITY
+1. Even though our dataset is composed of real known-
+vulnerable npm packages, there may be an implicit bias towards
+vulnerabilities that are easier to analyze and more common
+across different programming languages (i.e., not specific to
+JavaScript code). Thus, since our curated dataset may not be
+fully representative of all vulnerabilities in Node.js applications,
+a tool that can detect all the vulnerabilities of our dataset may
+still miss other unreported ones.
+2. We may have missed some relevant tool, failed to evaluate
+an analyzer that excels above all tested tools in our study,
+or overlooked third-party detection rules that produce better
+results. To reduce this risk, we will promote the reproducibility
+of our evaluation by providing both the source code of VulcaN
+and our curated dataset.
+3. Both the labeling of vulnerable packages and identification of
+their vulnerable code snippets were performed manually. Given
+the challenges of manual code inspection, these annotations
+could be mislabeled. To mitigate this risk, all vulnerabilities
+were analysed by at least two authors at separate times and
+we will make our dataset available for public scrutiny.
+4. A potential concern is whether our study is susceptible
+to survivor bias. For instance, assuming hypothetically that
+all the packages that we analyze had already been analyzed
+using CodeQL during the code development phase, and that
+the vulnerabilities reported by CodeQL had been accordingly
+fixed by the developer prior to package release on npm, then
+the number of vulnerabilities effectively detected by CodeQL
+could be higher than those reported in our study. This would
+misleadingly suggest that the quality of CodeQL is worse than
+what it is in reality. Note, however, that such a comprehensive
+characterization of each tool is beyond the scope of this work.
+In our study, we concentrate on evaluating tools’ ability to
+detect, not all possible vulnerabilities, but only those that have
+been officially reported in npm packages already in production.
+VIII. RELATED WORK
+The literature covers many tools for detecting vulnerabilities
+in Web applications, including static [78, 79, 80], dynamic [81,
+82], and hybrid analysis tools [83, 84, 85, 86], often combining
+different types of program analysis techniques, such as fuzzing
+(e.g. [81, 82]), control-flow and data-flow analysis (e.g. [57, 79,
+83, 85]), and symbolic execution (e.g. [80, 84, 86]). The great
+majority of these tools is, however, aimed at PHP-based Web
+applications, with considerably fewer tools targeting JavaScript
+applications. Most of the existing tools for JavaScript are aimed
+at client-side JavaScript code and its specific vulnerabilities: for
+instance, DOM-based XSS [87, 88], unrestricted inclusion of
+third-party cross-origin scripts [89], and potentially malicious
+flows via client-side persistent storage [8].
+Graph-based vulnerability scanners: State-of-the-art static
+vulnerability analysis techniques often work by first computing
+a static model describing the dynamic behaviour of the
+application to be analyzed. Most notably, code property graphs
+(CPGs) [57] were proposed as a compact representation
+of an application’s behaviour. With CPGs, one can encode
+specific vulnerability types as simple graph traversals, which
+can, in turn, be expressed using graph query languages and
+then executed on top of off-the-shelf graph databases (e.g.
+Neo4J [90]). Code property graphs have successfully been
+applied to find SQL injection, XSS, and CSRF vulnerabilities
+in PHP applications [79, 85]. Furthermore, they are at the
+core of CodeQL [14]. For JavaScript, code property graphs
+were employed by JAW [56] and ODGen [15], for client-side
+and server-side JavaScript respectively. In our work, we have
+extensively evaluated CodeQL and ODGen as representative
+state-of-the-art, graph-based vulnerability scanners.
+Vulnerability studies & analyzers for Node.js applications:
+Unlike client-side JavaScript applications, which run in the
+browser, Node.js application code is not sandboxed. Recent
+empirical studies [3, 91] have shown that, contrary to popular
+belief, npm applications are often poorly maintained and
+tested, with a significant percentage (up to 40%) of all
+packages depending on code with at least one publicly known
+vulnerability. Furthermore, after reviewing more than 200K
+npm applications, Staicu et al. [4] concludes that 20% of the
+analyzed applications either directly or indirectly make use
+of an injection API. Despite this security-critical situation,
+there is only a small number of research tools for detecting
+vulnerabilities in Node.js applications and their underlying
+13
+
+infrastructure, most of which based on dynamic code analysis.
+For instance, Synode [4] aims to prevent injection attacks
+in Node.js applications, and NodeSec [9] aims to detect
+vulnerabilities in Node.js applications. The authors of [5]
+and [92] design specific dynamic analysis for finding regular
+expression denial of service (ReDoS) vulnerabilities. The
+authors of [93] also apply dynamic analysis and symbolic
+execution to detect attacks that leverage hidden properties in
+client- and server-side JavaScript. There are also academic
+works that employ static analysis techniques for detecting
+vulnerabilities in Node.js, but most focus on detecting prototype
+pollution vulnerabilities [94, 95]. ODGen [15] is the only purely
+static code analysis tool developed by the academia that aims
+to detect several types of vulnerabilities in Node.js.
+Empirical studies of vulnerability analyzers: Several em-
+pirical studies aim at characterizing the efficacy of existing
+white-box vulnerability detection tools (e.g. [88, 96, 97]).
+Durieux et al. [96] evaluated 9 automated analysis tools for
+Ethereum Smart Contracts. The authors created a curated
+dataset consisting of 69 annotated vulnerable smart contracts, as
+well as a raw dataset consisting of 47,518 smart contracts. They
+report that only 42% of the vulnerabilities on the annotated
+dataset were detected, with the highest ranking tool having an
+accuracy of 21%. Melicher et al. [88] evaluated 3 automated
+static analysis tools for detecting DOM-based XSS in client-
+side JavaScript code (Esflow [98], ScanJS [54], and Burp
+Suite Pro [99]). They created a dataset with 3219 confirmed
+vulnerabilities. However, many security flaws in server-side
+code for Node.js do not exist on the client-side (e.g., SQL
+injections), and vice-versa. As such, the dataset from [88] is
+not representative enough of server-side vulnerabilities. Finally,
+Nunes et al. [97] evaluate five free static analysis tools for
+detecting SQL injection and XSS vulnerabilities in PHP web
+applications using a dataset comprising 134 WordPress plugins.
+In contrast to the studies referenced above, our paper presents
+the first empirical study targeting fully automated vulnerability
+detection tools for npm packages. Our study comes with a
+comprehensive manually-annotated dataset based on confirmed
+real-world vulnerabilities.
+IX. CONCLUSIONS
+This paper presented an empirical study of static analysis
+tools for detecting vulnerabilities in Node.js packages. To
+conduct this study, we built VulcaN, an automated analysis
+framework, using which we created the largest known curated
+dataset of Node.js packages with well-characterized security
+vulnerabilities. Currently, our curated dataset includes 745
+reviews that accurately identify the exact location of known
+vulnerabilities inside affected npm packages. We found that
+the nine evaluated tools fail to detect many vulnerabilities and
+exhibit high false positive rates. Additionally, we show that
+many important vulnerabilities appearing in the OWASP Top-
+10 are not detected by any evaluated tool or even when using
+the combination of all tools.
+We believe that our curated dataset will substantially con-
+tribute to enabling future research on automatic vulnerability
+detection tools for server-side JavaScript applications. To this
+end, we have made this dataset publicly available.
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+arXiv:2301.02883v1  [hep-ph]  7 Jan 2023
+One-Loop Electron Mass and QED Trace Anomaly
+Michael I. Eides∗
+Department of Physics and Astronomy,
+University of Kentucky, Lexington, KY 40506, USA
+Abstract
+Electron mass is considered as a matrix element of the energy-momentum trace in the rest frame.
+The one-loop diagrams for this matrix element are different from the textbook diagrams for the
+electron mass renormalization. We clarify connection between the two sets of diagrams and explain
+analytically and diagrammatically why the results of both calculations coincide.
+∗ Email address:meides@g.uky.edu
+Typeset by REVTEX
+1
+
+I.
+INTRODUCTION
+Hadron energy-momentum tensor (EMT), its matrix elements, anomalous trace, form
+factors and multipole expansion is now a vibrant field of research. EMT form factors describe
+interaction of particles with weak external gravitational field [1, 2]. For a long time there
+was no way to measure form factors of hadron EMT, but the situation changed when it was
+realized that they are connected to the generalized parton distribution functions which can
+be measured in deeply virtual Compton scattering and other hard exclusive reactions, see,
+e.g., [3–8].
+Due to nonperturbative nature of the low-energy QCD theoretical studies of hadron grav-
+itational form factors and other hadron EMT properties use general gauge theory principles,
+lattice QCD, QCD inspired low-energy models and model theories which admit quantitative
+analysis, see e.g., [9–16] and numerous other papers.
+A new insight into the EMT properties could arise from consideration of EMT in theories
+which allow perturbative treatment. While perturbative approach is clearly impossible in
+QCD, one can consider a simpler gauge theory, namely QED, and hope to acquire some
+experience which would be useful for the hadronic world. One-loop QED contributions to
+the EMT form factors, matrix elements and trace were calculated for a free electron and
+electron in the Coulomb field in a number of old and recent papers [17–28].
+One-loop electron mass renormalization in the mass-shell renormalization scheme is a
+textbook problem discussed in every introductory quantum field theory textbook, see, e.g.,
+[29]. Consideration of EMT suggests another perspective on this classical problem. One can
+calculate electron mass as a matrix element of the EMT trace. The diagrams describing this
+matrix element do not coincide with the well known diagrams for the one-loop corrections
+to the electron mass. We will calculate electron mass with the help of both sets of different
+contributions and explain why they produce coinciding results.
+II.
+MATRIX ELEMENTS OF EMT AND MASS OF PARTICLES
+General formulae for EMT T µν(x) follow from its definition as a conserved two-index
+symmetric tensor. Due to translational invariance
+2
+
+⟨p′|T µν(x)|p⟩ = ei(p′−p)·x⟨p′|T µν(0)|p⟩,
+(1)
+where |p⟩ is a particle eigenstate with momentum p and states here are normalized rela-
+tivistically, ⟨p′|p⟩ = 2Ep(2π)3δ(3)(p − p′).
+The Hamiltonian is a three-dimensional integral, H =
+�
+d3xT 00(x), and on the one hand
+�
+d3x⟨p′|T 00(x)|p⟩ = 2E2
+p(2π)3δ(3)(p − p′),
+(2)
+and on the other hand (see Eq. (1))
+�
+d3x⟨p′|T 00(x)|p⟩ = (2π)3δ(3)(p − p′)⟨p′|T 00(0)|p⟩.
+(3)
+Hence,
+⟨p|T 00(0)|p⟩ = 2E2
+p,
+(4)
+and due to Lorentz invariance
+⟨p|T µν(0)|p⟩ = 2pµpν,
+⟨p|T µ
+µ(0)|p⟩ = 2m2.
+(5)
+In the rest frame and with the nonrelativistic normalization of states ⟨p′|p⟩ = (2π)3δ(3)(p −
+p′)
+⟨0|T µ
+µ(0)|0⟩ = m,
+⟨0|T 00(0)|0⟩ = m.
+(6)
+These relations hold both elementary particles and for bound states, and are obviously valid
+in any relativistic field theory. Below we will consider the first of these equations for an
+electron in QED.
+Symmetric EMT tensor is conserved in a translationally invariant relativistic field theory
+and it is not renormalized as any conserved operator T µν
+0
+= [T µν]R. It is well known that
+EMT trace in gauge theories acquires an anomalous contribution [30–32] and has the form
+T µ
+0 µ = (1 + γm(e0)) ¯ψ0m0ψ0 + β(e0)
+2e0
+F 2
+0 = (1 + γm(e))[ ¯ψmψ]R + β(e)
+2e [F 2]R,
+(7)
+where β(e)/2e = α/6π, γm(e) = 3α/2π.
+The left hand side in Eq. (7) is renorminvariant and then the sum of the operators on
+the right hand side (RHS) is also renorminvariant. There are subtleties with separation of
+3
+
+the terms on the right hand side in a sum of renorminvariant operators beyond one loop,
+see [15, 33–38].
+We are going to consider matrix element of the anomalous trace in Eq. (7) for an electron
+at rest in the one-loop approximation. We will be working in the renormalized perturbation
+theory and use the mass-shell renormalization scheme. Then, according to Eq. (6), this
+matrix element should be equal to the physical electron mass m and describe one-loop mass
+renormalization. At the same time the diagrams which contribute to this matrix element do
+not coincide with the well known mass renormalization diagrams. Our goal is to clarify from
+the diagrammatic and analytic perspectives, why two different diagrammatic descriptions
+lead to the identical results1.
+III.
+ONE-LOOP MASS RENORMALIZATION AND EMT ANOMALOUS TRACE
+FOR A FREE ELECTRON
+Let us recall one-loop electron mass renormalization in the mass-shell scheme with dimen-
+sional regularization. We collected the well known relevant formulae in the Appendix. In
+the mass-shell renormalization scheme the counterterm δm(2) kills the ultraviolet divergence
+in the regularized but not renormalized self-energy diagram Σ(p) and preserves the physical
+mass at m
+=
+m
++
+−
+m
+δm
+i
+Σ(m)
+FIG. 1. One-loop mass renormalization.
+m = m + Σ(m) − δm(2),
+(8)
+where δm(2) = Σ(m). This expression is illustrated in Fig. 1. The factor i before the self-
+energy diagram in Fig. 1 is included in the standard diagrammatic definition of Σ(p), see
+e.g., [29].
+1 One-loop matrix element of the EMT trace in the electron state was calculated in the mass-shell renor-
+malization scheme with the momentum cutoff in [23], but the relationship between two sets of diagrams
+was not addressed there.
+4
+
+Next we turn to the matrix element of the EMT trace in Eq. (6) for the electron. The
+leading contribution to the matrix element of the term (β(e0)/2e0)F 2
+0 in Eq. (7) is of order
+α2 and we ignore it. It is sufficient to calculate the one-loop matrix element
+T = ⟨0|m0(1 + γm) ¯ψ0ψ0|0⟩.
+(9)
+In the one-loop approximation
+T = ⟨0|(1 + γm)m0 ¯ψ0ψ0|0⟩ ≈ ⟨0|(1 + γm)(m − δm(2)) ¯ψ0ψ0|0⟩
+(10)
+= m + mδZ2 − δm(2) + mγm + Γm(m)
+where Γm(m) is the one-loop diagram for the scalar vertex m ¯ψψ, see Fig. 2.
+mδZ2
+Γm(m)
+δm(2)
+−
+T =
++
++
+m
++
+γmm
+FIG. 2. One-loop matrix element of the EMT trace.
+All terms on the RHS in Eq. (10) and in Fig. 2, except Γm(m), are known from the one-loop
+mass-shell renormalization scheme (see Eq. (A7) and Eq. (A8)) and only Γm(m) requires
+calculation. After an easy one-loop calculation we obtain
+Γm(m) = α
+4πm
+�8
+ǫ − 4γ + 2 ln λ2
+m2 + 4 ln µ2
+m2 + 2 + 4 ln(4π)
+�
+.
+(11)
+Next we use Γm(m), the renormalization constants in Eq. (A7) and Eq. (A8), and γm =
+3α/2π to calculate the sum in Eq. (10)
+T = m + mδZ2 − δm + mγm + Γm(m) = m.
+(12)
+Thus we confirmed that the matrix element T of the anomalous EMT trace in the one-loop
+approximation is equal the physical electron mass, as it should be. Comparing Eq. (8) and
+Eq. (12) (and the respective Figs. 1 and 2) we see that
+Σ(m) = Γm(m) + mδZ2 + γmm.
+(13)
+5
+
+At this stage it is unclear why the different sets of diagrams in Fig. 1 and Fig. 2 produce
+coinciding results. To figure out a deeper reason why this happens we expand the unrenor-
+malized electron self-energy Σ(/p) in the Taylor series near the physical mass
+Σ(/p) = Σ(/p = m) + (/p − m)Σ′(/p = m) + O((/p − m)2)
+(14)
+= δm(2) + (/p − m)δZ2 + O((/p − m)2).
+Differentiating with respect to m we obtain at /p = m
+mdΣ(/p)
+dm
+|/p=m = mdΣ(/p = m)
+dm
+− mΣ′(/p = m).
+(15)
+Notice that (see Fig. 3)
+mdΣ(p)
+dm
+= Γm(p),
+(16)
+i
+Γm(p)
+Σ(p)
+m d
+dm
+�
+�
+=
+FIG. 3. Logarithmic mass derivative of Σ(p).
+which holds due to the identity
+m d
+dm
+�
+1
+/p − m
+�
+=
+1
+/p − mm
+1
+/p − m.
+(17)
+Then Eq. (15) can be written in the form
+Γm(m) + mδZ2 = mdΣ(/p = m)
+dm
+,
+(18)
+and Eq. (13) turns into (δm(2) = Σ(/p = m))
+Σ(m) = mdΣ(/p = m)
+dm
++ γmm ≡ md(δm(2))
+dm
++ γmm.
+(19)
+We calculate the derivative on the RHS using the explicit expression for δm(2) in Eq. (A7)
+and obtain (see Fig. 4)
+6
+
+md(δm(2))
+dm
+= δm(2) − µdδm(2)
+dµ
+= δm(2) − γmm,
+(20)
+where at the last step we used the definition of the electron mass anomalous dimension.
+i
+Σ(m)
+Σ(m)
+m d
+dm
+�
+�
+=
+−
+γmm
+i
+FIG. 4. Logarithmic mass derivative of Σ(m).
+Thus we proved by direct calculation that Eq. (19) holds and the expressions in Eq. (8)
+and Eq. (12) (and the respective sets of diagrams in Fig. 1 and Fig. 2) coincide.
+IV.
+CONCLUSIONS
+We have shown that the standard mass renormalization in Fig. 1 and the sum of the
+diagrams for the matrix element of the EMT trace in Fig. 2 coincide. This happens due
+to two important relationships. First, the one-loop diagram for a scalar vertex is equal to
+the logarithmic derivative of the self-energy diagram, see Eq. (16) and Fig. 3. Second, the
+mass renormalization counterterm (self-energy at /p = m) is equal to its own logarithmic
+derivative plus the product of mass and its anomalous dimension, see Eq. (19) and Fig. 4.
+The calculations above are made in the one-loop approximation, but we expect that they
+can be generalized to any number of loops. Really, connection between an arbitrary diagram
+and its logarithmic derivative with respect to the fermion mass, and the connection between
+such derivative and the fermion mass anomalous dimension do not depend on the number
+of loops, and these are the only essentail steps in the derivation above.
+Appendix A: Standard one-loop electron mass renormalization
+Some well known results are collected below.
+We use dimensional regularization and
+mass-shell renormalization. The QED Lagrangian in this scheme is
+L0 = −1
+4F 2
+0 + ¯ψ0(i/∂ − m0)ψ0 − e0 ¯ψ0 /A0ψ0 = L + δL,
+(A1)
+7
+
+where
+L = −1
+4F 2 + ¯ψ(i/∂ − m)ψ − µ
+ǫ
+2e ¯ψ /Aψ,
+(A2)
+δL = −1
+4δZ3F 2 + ¯ψ(iδZ2/∂ − δm)ψ − µ
+ǫ
+2eδZ1 ¯ψ /Aψ,
+(A3)
+L + δL = −1
+4Z3F 2 + iZ2 ¯ψ/∂ψ − mZm ¯ψψ − eZ1µ
+ǫ
+2 ¯ψ /Aψ,
+(A4)
+and
+Z1 = 1 + δZ1,
+Z2 = 1 + δZ2,
+Z3 = 1 + δZ3,
+mZm = m(1 + δZm) = m + δm.
+(A5)
+We define δm = m−m0 = m−mZmZ−1
+2 . In the one-loop approximation δm = m−m0 =
+m − mZmZ−1
+2
+≈ −mδZm + mδZ2 = −δm + mδZ2.
+The renormalized one-loop self-energy Σr(p) is
+Σr(p) = α
+2π
+� 1
+0
+dx
+�
+(2m − x/p)
+�2
+ǫ − γ + ln(4π) + ln
+µ2
+−x(1 − x)p2 + xλ2 + (1 − x)m2
+�
+− (m − x/p)
+�
+− (/pδZ2 − δm)|/p→m
+≈ Σ(m) + (/p − m)Σ′(/p = m) − (/p − m)δZ2 + (δm − mδZ2)
+= 3α
+4π m
+�2
+ǫ − γ + ln(4π) + ln µ2
+m2 + 4
+3
+�
+− (/p − m) α
+4π
+�2
+ǫ − γ + ln(4π) + ln µ2
+m2 + 2 ln λ2
+m2 + 4
+�
+− (/p − m)δZ2 + (δm − mδZ2),
+(A6)
+where Σ(p) is the dimensionally regularized self-energy, µ is the auxiliary dimensional reg-
+ularization mass, λ is the IR photon mass and ǫ = 4 − d.
+The one-loop counterterms are
+δm(2) ≡ mδZ2 − δm = Σ(m) = 3α
+4πm
+�2
+ǫ − γ + ln(4π) + ln µ2
+m2 + 4
+3
+�
+,
+(A7)
+δZ2 = Σ′(/p = m) = − α
+4π
+�2
+ǫ − γ + ln(4π) + ln µ2
+m2 + 2 ln λ2
+m2 + 4
+�
+.
+(A8)
+8
+
+ACKNOWLEDGMENTS
+This work was supported by the NSF grant PHY- 2011161.
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+11
+

CtE1T4oBgHgl3EQfDwP-/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf,len=443
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='02883v1  [hep-ph]  7 Jan 2023  One-Loop Electron Mass and QED Trace Anomaly  Michael I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' Eides∗  Department of Physics and Astronomy,  University of Kentucky, Lexington, KY 40506, USA  Abstract  Electron mass is considered as a matrix element of the energy-momentum trace in the rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  The one-loop diagrams for this matrix element are different from the textbook diagrams for the  electron mass renormalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' We clarify connection between the two sets of diagrams and explain  analytically and diagrammatically why the results of both calculations coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  ∗ Email address:meides@g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='uky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='edu  Typeset by REVTEX  1  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  INTRODUCTION  Hadron energy-momentum tensor (EMT), its matrix elements, anomalous trace, form  factors and multipole expansion is now a vibrant field of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' EMT form factors describe  interaction of particles with weak external gravitational field [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' For a long time there  was no way to measure form factors of hadron EMT, but the situation changed when it was  realized that they are connected to the generalized parton distribution functions which can  be measured in deeply virtual Compton scattering and other hard exclusive reactions, see,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=', [3–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  Due to nonperturbative nature of the low-energy QCD theoretical studies of hadron grav-  itational form factors and other hadron EMT properties use general gauge theory principles,  lattice QCD, QCD inspired low-energy models and model theories which admit quantitative  analysis, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=', [9–16] and numerous other papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  A new insight into the EMT properties could arise from consideration of EMT in theories  which allow perturbative treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' While perturbative approach is clearly impossible in  QCD, one can consider a simpler gauge theory, namely QED, and hope to acquire some  experience which would be useful for the hadronic world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' One-loop QED contributions to  the EMT form factors, matrix elements and trace were calculated for a free electron and  electron in the Coulomb field in a number of old and recent papers [17–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  One-loop electron mass renormalization in the mass-shell renormalization scheme is a  textbook problem discussed in every introductory quantum field theory textbook, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=',  [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' Consideration of EMT suggests another perspective on this classical problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' One can  calculate electron mass as a matrix element of the EMT trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' The diagrams describing this  matrix element do not coincide with the well known diagrams for the one-loop corrections  to the electron mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' We will calculate electron mass with the help of both sets of different  contributions and explain why they produce coinciding results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  MATRIX ELEMENTS OF EMT AND MASS OF PARTICLES  General formulae for EMT T µν(x) follow from its definition as a conserved two-index  symmetric tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' Due to translational invariance  2  ⟨p′|T µν(x)|p⟩ = ei(p′−p)·x⟨p′|T µν(0)|p⟩,  (1)  where |p⟩ is a particle eigenstate with momentum p and states here are normalized rela-  tivistically, ⟨p′|p⟩ = 2Ep(2π)3δ(3)(p − p′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  The Hamiltonian is a three-dimensional integral, H =  �  d3xT 00(x), and on the one hand  �  d3x⟨p′|T 00(x)|p⟩ = 2E2  p(2π)3δ(3)(p − p′),  (2)  and on the other hand (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' (1))  �  d3x⟨p′|T 00(x)|p⟩ = (2π)3δ(3)(p − p′)⟨p′|T 00(0)|p⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  (3)  Hence,  ⟨p|T 00(0)|p⟩ = 2E2  p,  (4)  and due to Lorentz invariance  ⟨p|T µν(0)|p⟩ = 2pµpν,  ⟨p|T µ  µ(0)|p⟩ = 2m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  (5)  In the rest frame and with the nonrelativistic normalization of states ⟨p′|p⟩ = (2π)3δ(3)(p −  p′)  ⟨0|T µ  µ(0)|0⟩ = m,  ⟨0|T 00(0)|0⟩ = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  (6)  These relations hold both elementary particles and for bound states, and are obviously valid  in any relativistic field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' Below we will consider the first of these equations for an  electron in QED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  Symmetric EMT tensor is conserved in a translationally invariant relativistic field theory  and it is not renormalized as any conserved operator T µν  0  = [T µν]R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' It is well known that  EMT trace in gauge theories acquires an anomalous contribution [30–32] and has the form  T µ  0 µ = (1 + γm(e0)) ¯ψ0m0ψ0 + β(e0)  2e0  F 2  0 = (1 + γm(e))[ ¯ψmψ]R + β(e)  2e [F 2]R,  (7)  where β(e)/2e = α/6π, γm(e) = 3α/2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  The left hand side in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' (7) is renorminvariant and then the sum of the operators on  the right hand side (RHS) is also renorminvariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' There are subtleties with separation of  3  the terms on the right hand side in a sum of renorminvariant operators beyond one loop,  see [15, 33–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  We are going to consider matrix element of the anomalous trace in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' (7) for an electron  at rest in the one-loop approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' We will be working in the renormalized perturbation  theory and use the mass-shell renormalization scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' Then, according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' (6), this  matrix element should be equal to the physical electron mass m and describe one-loop mass  renormalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' At the same time the diagrams which contribute to this matrix element do  not coincide with the well known mass renormalization diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' Our goal is to clarify from  the diagrammatic and analytic perspectives, why two different diagrammatic descriptions  lead to the identical results1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  ONE-LOOP MASS RENORMALIZATION AND EMT ANOMALOUS TRACE  FOR A FREE ELECTRON  Let us recall one-loop electron mass renormalization in the mass-shell scheme with dimen-  sional regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' We collected the well known relevant formulae in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' In  the mass-shell renormalization scheme the counterterm δm(2) kills the ultraviolet divergence  in the regularized but not renormalized self-energy diagram Σ(p) and preserves the physical  mass at m  =  m  +  −  m  δm  i  Σ(m)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' One-loop mass renormalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  m = m + Σ(m) − δm(2),  (8)  where δm(2) = Σ(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' This expression is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' The factor i before the self-  energy diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' 1 is included in the standard diagrammatic definition of Σ(p), see  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=', [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  1 One-loop matrix element of the EMT trace in the electron state was calculated in the mass-shell renor-  malization scheme with the momentum cutoff in [23], but the relationship between two sets of diagrams  was not addressed there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  4  Next we turn to the matrix element of the EMT trace in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' (6) for the electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' The  leading contribution to the matrix element of the term (β(e0)/2e0)F 2  0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' (7) is of order  α2 and we ignore it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' It is sufficient to calculate the one-loop matrix element  T = ⟨0|m0(1 + γm) ¯ψ0ψ0|0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  (9)  In the one-loop approximation  T = ⟨0|(1 + γm)m0 ¯ψ0ψ0|0⟩ ≈ ⟨0|(1 + γm)(m − δm(2)) ¯ψ0ψ0|0⟩  (10)  = m + mδZ2 − δm(2) + mγm + Γm(m)  where Γm(m) is the one-loop diagram for the scalar vertex m ¯ψψ, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  mδZ2  Γm(m)  δm(2)  −  T =  +  +  m  +  γmm  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' One-loop matrix element of the EMT trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  All terms on the RHS in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' (10) and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' 2, except Γm(m), are known from the one-loop  mass-shell renormalization scheme (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' (A7) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' (A8)) and only Γm(m) requires  calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' After an easy one-loop calculation we obtain  Γm(m) = α  4πm  �8  ǫ − 4γ + 2 ln λ2  m2 + 4 ln µ2  m2 + 2 + 4 ln(4π)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  (11)  Next we use Γm(m), the renormalization constants in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' (A7) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' (A8), and γm =  3α/2π to calculate the sum in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' (10)  T = m + mδZ2 − δm + mγm + Γm(m) = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  (12)  Thus we confirmed that the matrix element T of the anomalous EMT trace in the one-loop  approximation is equal the physical electron mass, as it should be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' Comparing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' (8) and  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' (12) (and the respective Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' 1 and 2) we see that  Σ(m) = Γm(m) + mδZ2 + γmm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  (13)  5  At this stage it is unclear why the different sets of diagrams in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' 2 produce  coinciding results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' To figure out a deeper reason why this happens we expand the unrenor-  malized electron self-energy Σ(/p) in the Taylor series near the physical mass  Σ(/p) = Σ(/p = m) + (/p − m)Σ′(/p = m) + O((/p − m)2)  (14)  = δm(2) + (/p − m)δZ2 + O((/p − m)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  Differentiating with respect to m we obtain at /p = m  mdΣ(/p)  dm  |/p=m = mdΣ(/p = m)  dm  − mΣ′(/p = m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  (15)  Notice that (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' 3)  mdΣ(p)  dm  = Γm(p),  (16)  i  Γm(p)  Σ(p)  m d  dm  �  �  =  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' Logarithmic mass derivative of Σ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  which holds due to the identity  m d  dm  �  1  /p − m  �  =  1  /p − mm  1  /p − m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  (17)  Then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' (15) can be written in the form  Γm(m) + mδZ2 = mdΣ(/p = m)  dm  ,  (18)  and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' (13) turns into (δm(2) = Σ(/p = m))  Σ(m) = mdΣ(/p = m)  dm  + γmm ≡ md(δm(2))  dm  + γmm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  (19)  We calculate the derivative on the RHS using the explicit expression for δm(2) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' (A7)  and obtain (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' 4)  6  md(δm(2))  dm  = δm(2) − µdδm(2)  dµ  = δm(2) − γmm,  (20)  where at the last step we used the definition of the electron mass anomalous dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  i  Σ(m)  Σ(m)  m d  dm  �  �  =  −  γmm  i  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' Logarithmic mass derivative of Σ(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  Thus we proved by direct calculation that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' (19) holds and the expressions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' (8)  and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' (12) (and the respective sets of diagrams in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' 2) coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  CONCLUSIONS  We have shown that the standard mass renormalization in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' 1 and the sum of the  diagrams for the matrix element of the EMT trace in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' 2 coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' This happens due  to two important relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' First, the one-loop diagram for a scalar vertex is equal to  the logarithmic derivative of the self-energy diagram, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' (16) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' Second, the  mass renormalization counterterm (self-energy at /p = m) is equal to its own logarithmic  derivative plus the product of mass and its anomalous dimension, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' (19) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  The calculations above are made in the one-loop approximation, but we expect that they  can be generalized to any number of loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' Really, connection between an arbitrary diagram  and its logarithmic derivative with respect to the fermion mass, and the connection between  such derivative and the fermion mass anomalous dimension do not depend on the number  of loops, and these are the only essentail steps in the derivation above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  Appendix A: Standard one-loop electron mass renormalization  Some well known results are collected below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  We use dimensional regularization and  mass-shell renormalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' The QED Lagrangian in this scheme is  L0 = −1  4F 2  0 + ¯ψ0(i/∂ − m0)ψ0 − e0 ¯ψ0 /A0ψ0 = L + δL,  (A1)  7  where  L = −1  4F 2 + ¯ψ(i/∂ − m)ψ − µ  ǫ  2e ¯ψ /Aψ,  (A2)  δL = −1  4δZ3F 2 + ¯ψ(iδZ2/∂ − δm)ψ − µ  ǫ  2eδZ1 ¯ψ /Aψ,  (A3)  L + δL = −1  4Z3F 2 + iZ2 ¯ψ/∂ψ − mZm ¯ψψ − eZ1µ  ǫ  2 ¯ψ /Aψ,  (A4)  and  Z1 = 1 + δZ1,  Z2 = 1 + δZ2,  Z3 = 1 + δZ3,  mZm = m(1 + δZm) = m + δm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  (A5)  We define δm = m−m0 = m−mZmZ−1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' In the one-loop approximation δm = m−m0 =  m − mZmZ−1  2  ≈ −mδZm + mδZ2 = −δm + mδZ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  The renormalized one-loop self-energy Σr(p) is  Σr(p) = α  2π  � 1  0  dx  �  (2m − x/p)  �2  ǫ − γ + ln(4π) + ln  µ2  −x(1 − x)p2 + xλ2 + (1 − x)m2  �  − (m − x/p)  �  − (/pδZ2 − δm)|/p→m  ≈ Σ(m) + (/p − m)Σ′(/p = m) − (/p − m)δZ2 + (δm − mδZ2)  = 3α  4π m  �2  ǫ − γ + ln(4π) + ln µ2  m2 + 4  3  �  − (/p − m) α  4π  �2  ǫ − γ + ln(4π) + ln µ2  m2 + 2 ln λ2  m2 + 4  �  − (/p − m)δZ2 + (δm − mδZ2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  (A6)  where Σ(p) is the dimensionally regularized self-energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' µ is the auxiliary dimensional reg-  ularization mass,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' λ is the IR photon mass and ǫ = 4 − d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  The one-loop counterterms are  δm(2) ≡ mδZ2 − δm = Σ(m) = 3α  4πm  �2  ǫ − γ + ln(4π) + ln µ2  m2 + 4  3  �  ,  (A7)  δZ2 = Σ′(/p = m) = − α  4π  �2  ǫ − γ + ln(4π) + ln µ2  m2 + 2 ln λ2  m2 + 4  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  (A8)  8  ACKNOWLEDGMENTS  This work was supported by the NSF grant PHY- 2011161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  [1] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' Kobzarev and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' Okun, “Gravitational interaction of fermions,” Zh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' Eksp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' Teor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  43, 1904-1909 (1962).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  [2] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' Pagels, Energy-Momentum Structure Form Factors of Particles, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' 144, 1250-1260  (1966) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='1103/PhysRev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='1250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content='  [3] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' Ji, Gauge-Invariant Decomposition of Nucleon Spin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'}
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@@ -0,0 +1,1198 @@
+arXiv:2301.00556v1  [q-bio.PE]  2 Jan 2023
+Competition of alliances in a cyclically dominant eight-species population
+Junpyo Parka, Xiaojie Chenb, and Attila Szolnokic,∗∗
+aDepartment of Applied Mathematics, Kyung Hee University, Yongin 17104, Republic of Korea
+bSchool of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China
+cInstitute of Technical Physics and Materials Science, Centre for Energy Research, P.O. Box 49, H-1525 Budapest, Hungary
+Abstract
+In a diverse population, where many species are present, competitors can fight for surviving at individual
+and collective levels. In particular, species, which would beat each other individually, may form a specific
+alliance that ensures them stable coexistence against the invasion of an external species. Our principal goal
+is to identify those general features of a formation which determine its vitality. Therefore, we here study a
+traditional Lotka-Volterra model of eight-species where two four-species cycles can fight for space. Beside
+these formations, there are other solutions which may emerge when invasion rates are varied. The complete
+range of parameters is explored and we find that in most of the cases those alliances prevail which are formed
+by equally strong members. Interestingly, there are regions where the symmetry is broken and the system
+is dominated by a solution formed by seven species. Our work also highlights that serious finite-size effects
+may emerge which prevent observing the valid solution in a small system.
+Keywords:
+cyclic dominance, alliances, competition
+1. Introduction
+If a species is stronger and beats the other one
+then how can the “weaker” species survive? This
+is a fundamental problem of ecology to explain the
+amazing biodiversity of life [1, 2]. A possible expla-
+nation could be the so-called intransitive or cycli-
+cally dominated relation among competing species
+where a third (or more) species beat the preda-
+tor species, hence establishing a delicate balance
+among all competitors. In the simplest case, these
+relations remind the well-known rock-scissors-paper
+game and this type of interaction can really be
+observed among animals, plants, or even among
+bacterias [3, 4, 5, 6, 7, 8, 9]. Evidently, a three-
+member loop can be easily enlarged for more par-
+ticipants, as it was made in the extended Lotka-
+Volterra model [10, 11, 12, 13]. Motivated by the
+complexity of such interactions, researchers have
+studied related models actively over the last decade
+and several interesting observations have been made
+[14, 15, 16, 17, 18].
+∗Email
+addresses:
+junpyopark@khu.ac.kr;
+xiao-
+jiechen@uestc.edu.cn; szolnoki.attila@ek-cer.hu
+∗∗Corresponding author
+It would be almost impossible to sum all impor-
+tant observations, such as reported in Refs. [19, 20,
+21, 22, 23, 24, 25]. Instead, the reader is referred
+to specific review papers about the milestones along
+this research path [26, 27, 28, 29]. Nevertheless, it is
+also worth noting here that the mentioned cyclic or
+intransitive relation among participants should not
+necessarily be defined by a Lotka-Volterra-type mi-
+croscopic rule, but could be the result of collective
+behavior among competing strategies in evolution-
+ary game models [30, 31, 32, 33, 34, 35, 36, 37, 38]
+The most exciting question is whether we can
+predict the direction of evolution based on the food-
+web that defines the microscopic dynamics? Can
+we identify general principles which may inform us
+about the vitality of an alliance? An early observa-
+tion was, for instance, that when two three-member
+cycles fight then the one, where the inner invasion
+is faster, is more viable [39].
+The picture, how-
+ever, is more subtle, because a faster general rota-
+tion does not necessarily provide an advantage if
+the members of the trio are unequal. In the lat-
+ter case, the triplet becomes vulnerable no matter
+the average of inner invasion rates is higher than in
+the rival triplet which is formed by less aggressive,
+Preprint submitted to Chaos Solitons and Fractals
+January 3, 2023
+
+but equally strong partners [40]. Notably, hetero-
+geneous invasion rates may emerge temporarily or
+locally [41, 42]. Or, the extension of the original
+food-web by adding a reverse link can also change
+a stable solution [43]. Furthermore, the number of
+members could also be a decisive factor because a
+smaller loop is generally stronger than a large one
+[44].
+In this work we introduce a minimal eight-species
+model where two four-member loops fight for space.
+The interaction between these quartets practically
+establishes a traditional Lotka-Volterra circle of
+eight species, which can be described by an inva-
+sion rate. Not to break the balance between the
+quartets, we assume that the inner invasion is uni-
+form and equally strong in both formations. The
+only difference between them is we introduce two
+short-cuts in one of the loops which generates extra
+interactions within the related alliance. Our major
+question is how the stability of alliances change due
+to this extension and can we identify new solutions
+which would be hidden otherwise?
+It could also
+be interesting whether the previously established
+principles, observed for simpler systems formed by
+trials and duets, remain valid when we apply them
+to alliances formed by larger groups.
+2. Alliances formed at different levels
+To make our observations comparable with pre-
+vious works [45, 46, 47, 48, 49, 50], we assume
+that species are distributed on an L × L square
+lattice where periodic boundary conditions are ap-
+plied. Each node is occupied by one of the species
+which are denoted by Xi where indexes are from
+i = 0 to 7. During an elementary step, we select
+a neighboring pair nodes. If they are occupied by
+the same species or represent species which are neu-
+tral then nothing happens. Otherwise, an invasion
+happens with a specific probability and predator
+species invades prey species. The microscopic rules
+are defined in the following way:
+XiXi+1
+γ−→ XiXi
+(1)
+XiXi+2
+α−→ XiXi
+(2)
+X2X6
+β−→ X2X2, X4X0
+β−→ X4X4,
+(3)
+where i + 1 and i + 2 are calculated in cyclic man-
+ner. The parameters α, β and γ define the proba-
+bilities of successful invasions between the involved
+predator-prey neighbors.
+γ
+α
+β
+1
+2
+4
+0
+3
+5
+6
+7
+Figure 1: Food-web of competing species. In the basic model
+eight species are invade cyclically each other with probabil-
+ity γ, similar to the extended Lotka-Volterra model.
+Ad-
+ditionally, we introduce a cyclic inner invasion among odd
+and separately among even species with probability α. To
+break the symmetry among the quartets, we also introduce
+an inner invasion between species “2” and “6” and between
+species “4” and “0” with probability β. Importantly, we use
+the color-code of species in later figures where spatial distri-
+butions are presented.
+For a deeper insight about the model, in Fig. 1 we
+show the food-web of competing species. Our first
+note is about the biggest loop where all species are
+involved, according to the extended Lotka-Volterra
+type system. Here every species is simultaneously
+a predator and a prey of another one, hence es-
+tablishing a possible solution.
+The invasion rate
+is γ for all interactions here. Another solution is
+formed by “1”, “3”, “5”, and “7” species who in-
+vade each other cyclically with probability α. For
+later reference, we denote this alliance as A4
+1,3,5,7.
+A similar A4
+0,2,4,6 quartet is formed among species
+“0”, “2”, “4” and “6”. Additionally, we introduce
+an extra chance of invasions among group members
+here. In particular, species “2” invades species “6”
+and species “4” invades species “0” with probabil-
+ity β.
+In this way, there are no neutral pairs in
+the mentioned quartet anymore. As a consequence,
+the average inner invasion is augmented among the
+four species, which could be a support to their via-
+bility. Interestingly, however, there are many other
+possible solutions in this system. For example, the
+new invasions among even-numbered species offer
+2
+
+the chance for trials to emerge: A3
+0,2,4 or A3
+0,2,6 can
+work as a rock-scissors-paper-type solution. Beside
+the mentioned formations, there are A5 (five-), A6
+(six-), or A7 seven-member solutions, as well. Per-
+haps, it is not necessary to specify them because
+the reader can easily construct examples, based on
+the food-web, shown in Fig. 1.
+As we noted, there are three parameters in our
+model and in the following we systematically scan
+the whole parameter field to identify the dominant
+solution for each combination of parameters. For
+this reason, we executed Monte Carlo simulations
+where a Monte Carlo step (MCS) means that on
+average every node has a chance to update its state.
+The linear system size varied between L = 100 to
+L = 3200 and the necessary relaxation steps were
+between 103 to 3 × 105 MCSs. According to the
+standard protocol, simulations were launched from
+an initial state where species are distributed ran-
+domly on the lattice and monitor the ρi concentra-
+tion of Xi for all species. We, however, would like
+to stress that this initial state does not always could
+be a good choice to find the solution which is valid
+in the large size limit. It is because a small sys-
+tem size does not necessarily “offer” equal chance
+for all possible solutions to emerge. Some solution,
+which involves large typical lengths, would emerge
+just later, but to that stage of the evolutionary pro-
+cess other solutions may invade the whole available
+space. Therefore, a more complex solution practi-
+cally has no chance to emerge at a small system size.
+As a consequence, other solutions may win and an
+independent run can easily terminates onto a differ-
+ent solution, no matter we use the same set of model
+parameters. To overcome this uncertainty, we not
+simply enlarged the system size, but also used alter-
+native initial states where larger and homogeneous
+domains of competing species were distributed ran-
+domly on the lattice. Similar approach was used
+previously in Refs.[51, 52], for instance. In this way,
+we can create all types of interfaces which could
+be the foundation to build more sophisticated solu-
+tions. To make our analysis more complete, at the
+critical values of control parameters we also gener-
+ated sub-solutions independently in restricted areas
+of space and after we let them fight directly along
+the separating interface [53, 54, 55]. This technique
+can identify the dominant solution unambiguously.
+The details of the above mentioned protocols will
+be specified in the next section.
+3. Results
+3.1. Phase diagrams
+We first summarize our main findings and later
+we give further details about the microscopic mech-
+anisms which are responsible how dominant solu-
+tions emerge.
+According to our simulations, we
+can distinguish three main cases which determine
+the system behavior. The decisive condition is the
+intensity of interaction between the quartets men-
+tioned in Sec. 2. If this interaction is weak, means
+the value of γ is small, then the A4
+{1,3,5,7} solution
+dominates the whole β − α parameter plane.
+In
+other words, only the quartet of “1+3+5+7” sur-
+vive independently of the α and β values.
+For intermediate values of γ, when the invasion
+flow in the large cycle becomes substantial, a con-
+ceptually new behavior emerges. We represent this
+phenomenon by showing the phase diagram ob-
+tained at γ = 0.5.
+Figure 2 shows that beside
+the mentioned A4
+{1,3,5,7} quartet, other solutions be-
+come dominant at certain values of β − α pairs.
+When both α and β are small then the “large cir-
+cle” is the winner, hence all eight species coexist.
+For intermediate β values this solution is replaced
+by A7
+{0,1,2,3,4,6,7} where only species “5” is missing.
+ 0
+ 0.2
+ 0.4
+ 0.6
+ 0.8
+ 1
+ 0
+ 0.2
+ 0.4
+ 0.6
+ 0.8
+ 1
+A4
+{1,3,5,7}
+A7
+{0,1,2,3,4,6,7}
+all
+β
+α
+Figure 2: Phase diagram on β − α parameter plane obtained
+at γ = 0.5. If α is large enough, the quartet of “1+3+5+7”
+species always win. If both β and α are small, all species
+survive. Interestingly, for low α and intermediate β a seven-
+member solution dominates where species “5” is missing.
+Dashed blue line denotes discontinuous, while solid red line
+marks the positions of continuous phase transition points.
+Qualitatively similar system behavior can be ob-
+served when γ is large, hence the flow in the ex-
+3
+
+ternal loop is intensive. In other words, the inter-
+action between the quartets becomes large. This
+is illustrated in Fig. 3 where we present the com-
+plete phase diagram on the β − α plane. The only
+difference between the diagrams is the area, where
+complete eight-species solution dominates, is larger
+and the seven-member solution is shifted toward
+larger β values.
+ 0
+ 0.2
+ 0.4
+ 0.6
+ 0.8
+ 1
+ 0
+ 0.2
+ 0.4
+ 0.6
+ 0.8
+ 1
+A4
+{1,3,5,7}
+A7
+{0,1,2,3,4,6,7}
+all
+β
+α
+Figure 3: Phase diagram on β − α parameter plane obtained
+at γ = 1. The diagram is similar to the one shown in Fig. 2.
+The only difference is the eight-member full solution occu-
+pies larger area in the parameter field, while seven-member
+solution is shifted toward higher β values.
+We must stress that the presented phase dia-
+grams are valid in the large system size limit. If
+the system size is small then we may observe that
+the system can easily terminate onto different so-
+lutions. To illustrate it, in Fig. 4 we present the
+survival probabilities at each (α, β) parameter pairs
+for γ = 1, when the linear system size was L = 100.
+More precisely, we launched the evolution from a
+random initial state and monitored the fractions of
+species until N = L×L MCSs. After, we recorded
+the number of surviving species and repeated the
+whole process 100 times. In this way we can esti-
+mate the probability that S species survive for long
+time.
+The panels of Fig. 4 show these surviving
+probabilities for all possible S values at x different
+(α, β) pairs on the parameter plane.
+These pan-
+els indicate that different solutions may emerge no
+matter we use the same values of (α, β) parameters.
+Therefore, the destination is rarely unambiguous,
+hence the surviving probability equals to 1 only for
+S = 4 when α is high and β is low. Nevertheless,
+0
+0.5
+1
+0
+0.5
+1
+S=1
+0
+0.5
+1
+0
+0.5
+1
+S=2
+0
+0.5
+1
+0
+0.5
+1
+S=3
+0
+0.5
+1
+0
+0.5
+1
+S=4
+0
+0.5
+1
+0
+0.5
+1
+S=5
+0
+0.5
+1
+0
+0.5
+1
+S=6
+0
+0.5
+1
+0
+0.5
+1
+S=7
+0
+0.5
+1
+0
+0.5
+1
+S=8
+0
+0.2
+0.4
+0.6
+0.8
+1
+Figure 4: Heat maps of the survival probability on β − α
+parameter plane obtained for γ = 1 by using 100×100 system
+size. Each panel shows the probability to reach a state which
+contains S species after N = L2 MCSs. The value of S is
+indicated on each panel. The results are averaged over 100
+independent runs.
+the contour of areas around “maximum” values are
+roughly agree with the diagram shown in Fig. 3.
+But, as Fig. 4 illustrated, this system size is insuffi-
+cient to make reliable conclusions about the proper
+system behavior.
+3.2. Detecting phase transitions
+Therefore, to detect the phase transition points
+more precisely, we need to apply systematic finite-
+size analysis. An example is given in Fig. 5 where
+we present the “lack” of species “5” in dependence
+of α at fixed β = 0.75 and γ = 1 values. When the
+value of 1 − ρ5 reaches 1 then the system enters
+onto the mentioned A7
+{0,1,2,3,4,6,7} solution.
+Evi-
+dently, we also checked the portions of other species,
+because ρ5 = 0 is fulfilled in other solutions, too.
+Here the symbols are the average of many indepen-
+dent runs.
+As an example, for L = 100 we exe-
+cuted 2000 times. Importantly, the average hides
+the proper system behavior for small system size,
+because it mixes different destinations. For exam-
+ple, at L = 100, α = 0.15 we never measured
+ρ5 = 0.063.
+Instead, the majority of indepen-
+dent runs terminated onto a state where ρ5 = 0
+or ρ5 = 0.25. This ambiguity disappears for large
+system sizes. The plot also highlights that the tran-
+sition from the eight-member to seven-member so-
+lution is continuous because species “5” vanishes
+gradually as we increase α. The transition, how-
+ever, between A7
+{0,1,2,3,4,6,7} and A4
+{1,3,5,7} phases is
+discontinuous.
+To illustrate another type of phase transition, in
+Fig. 6 we show an alternative order parameter in
+dependence of β at γ = 0.5 and α = 0.01. Here,
+we calculate the difference between the portions of
+4
+
+ 
+ 0.8
+ 
+ 0.9
+ 
+ 1
+ 0
+ 
+ 0.1
+ 
+ 0.2
+ 
+ 0.3
+ 
+ 0.4
+all
+A7
+{0,1,2,3,4,6,7}
+A4
+{1,3,5,7}
+1 - ρ5
+α
+100
+200
+400
+800
+1600
+Figure 5: The absence of species “5”, as an order param-
+eter, in dependence of α at γ = 1, β = 0.75 for different
+system sizes. The linear sizes are indicated in the legend.
+The dominant solutions are marked on the top. The plots
+are the average of 100-2000 runs depending on the system
+size. Lines are just to guide the eye.
+quartets formed by odd- and even-indexed species,
+respectively. When β is small, all available species
+coexist, the system is in the “all” phase, hence the
+mentioned difference is small. When this parame-
+ter becomes 1, then species with even indexes van-
+ish and the quartet of “1+3+5+7” species becomes
+dominant.
+Interestingly, this formation is viable
+even if the inner invasion, due to the tiny α = 0.01,
+is extremely slow. The explanation of this surpris-
+ing behavior is given in the next subsection.
+Similarly to the previously discussed Fig. 5 case,
+the average of the order parameter could be mis-
+leading when the system size is small. This can be
+clearly seen for L = 100 and L = 200, where the av-
+erage is larger than the value for larger system sizes.
+It is because the system can easily be trapped in the
+A4
+{1,3,5,7} state already at small β values. The pos-
+sible destinations, however, are less ambiguous for
+larger sizes, but the jump in the order parameter is
+valid signaling a discontinuous phase transition at
+β = 0.66.
+3.3. Battles of solutions
+In the following, we analyze the possible mech-
+anisms which explain the system behavior summa-
+rized in Fig. 2 and in Fig. 3. To get a deeper insight
+about the dominant processes, which drive the pat-
+ter formation, it is instructive to use a specific ini-
+tial state where we divide the available space into
+two halves and both regions are occupied by one of
+the main quartets formed by odd- or even-indexed
+ 0
+ 0.2
+ 0.4
+ 0.6
+ 0.8
+ 1
+ 0
+ 0.1
+ 0.2
+ 0.3
+ 0.4
+ 0.5
+ 0.6
+ 0.7
+ 0.8
+all
+A4
+{1,3,5,7}
+(ρ1+ρ3+ρ5+ρ7) - (ρ0+ρ2+ρ4+ρ6)
+β
+100
+200
+400
+800
+1600
+Figure 6: The difference between symmetric quartets in de-
+pendence of β at γ = 0.5, α = 0.1. When this order param-
+eter becomes 1, the system enters to a four-species state.
+The dominant solutions are marked on the top. The plots
+are the average over 20-2000 runs, depending on the system
+size. The linear size of squares are indicated.
+species. In this way, we can follow how these solu-
+tions compete, or how new possibilities may emerge
+due to their interaction.
+The β − α parameter plane can be divided into
+four main sectors which fundamentally determine
+the relation of different solutions.
+We first con-
+sider the low α – low β region.
+It is impor-
+tant to note that the A4
+{0,2,4,6} quartet formed
+by even-indexed species is not a proper solution
+here.
+Instead, we can see the battle of A3
+{0,2,4}
+and A3
+{0,2,6} triplets when even-indexed species are
+present. Since species “4” beats species “6”, this
+fight ends up with the victory of the former solu-
+tion.
+This domain is marked by “A” on the left
+panel of Fig. 7. When γ is small then there is just
+a weak interaction between odd- and even-indexed
+species. Therefore, the mentioned domains remain
+compact and they fight along the separating inter-
+face. Finally, A4
+{1,3,5,7} quartet win this battle and
+the whole space will be occupied by the domain,
+marked by “B” on the left panel.
+If γ is high enough then the evolution changes
+drastically. This is illustrated on the right panel
+of Fig. 7. Because of high γ, species “7”, who has
+no predator in the “0+2+4” triplet, can easily en-
+ter into the A3
+{0,2,4} domain. The raid of species
+“7” results in a neutral duo.
+This A2{2, 7} so-
+lution is marked by “C” on this panel. Interest-
+ingly, however, this solution has limited life, be-
+cause the intensive interactions of the original quar-
+tets at the border offer a chance for the complete
+5
+
+A
+B
+C
+D
+Figure 7: Pattern formation in the low α – low β region when
+we launch the evolution from a prepared state where left
+(right) side of the space is occupied by even-indexed (odd-
+indexed) species. In the former case only the A3
+{0,2,4}triplet
+survive, shown by “A” on the left panel. For small γ, when
+the interaction is weak around the main loop, the A4
+{1,3,5,7}
+quartet, marked by “B”, can beat the other solution and
+gradually invades the whole space.
+If γ is high enough,
+species “7” can easily invade the triplet leaving a neutral
+A2
+{2,7} pair behind. But this solution, marked by “C”, can-
+not survive because the interface between the original quar-
+tets is a birthplace of the complete eight-species solution.
+This is marked by “D”. Since γ is high enough, this solution
+is viable and prevails. Color codes agree with those we intro-
+duced in Fig. 1. Parameters are: α = 0.1, β = 0.1, γ = 0.05
+(left panel), and γ = 1 (right panel).
+eight-member solution to emerge.
+This solution
+is marked by “D” on this panel. Because of the
+high γ value, the general invasion flow is intensive
+in the largest loop, which makes it strong against
+A4
+{1,3,5,7}. Notably, the latter quartet is weak due to
+small α. The above described mechanism explains
+the left-down corners of phase diagrams shown in
+Fig. 2 and in Fig. 3.
+We can face a conceptually different situation
+in the large α – small β corner of the parame-
+ter plane, because it provides supporting conditions
+both for A4
+{0,2,4,6} and A4
+{1,3,5,7} quartets. There-
+fore, both solutions would be viable in the absence
+of the other. Importantly, the high α value makes
+the difference from the previously discussed cases,
+which generates a fast rotation withing both quar-
+tets. There is, however, a crucial difference between
+these solutions: while A4
+{1,3,5,7} is formed by equal
+partners, the extra inner invasions described by β
+probability break this delicate balance for the ben-
+efit of species “0” at the expense of species “6”. In
+this way the pattern formed by even-indexed species
+becomes heterogeneous, including larger domains of
+“0” species. This makes the whole alliance vulner-
+able against the attack of the rival well-balanced
+loop.
+This process is illustrated in Fig. 8, where we
+ 0
+ 
+ 0.1
+ 
+ 0.2
+ 
+ 0.3
+ 0
+ 20000
+ 40000
+ 60000
+ 80000  100000  120000
+0
+6
+2
+4
+1 3 5 7
+ρi
+time [MCS]
+Figure 8: The time evolution of species when quartets are
+fighting at high α, low β values.
+Initially both A4
+{0,2,4,6}
+and A4
+{1,3,5,7} solutions evolve independently in separated
+areas of available space. When separating borders opened,
+marked by an arrow, symmetric quartet gradually crowd out
+the rival gang. Color codes are the usual, except white line
+is replaced by grey one. Parameters are: α = 0.5, β = 0.1,
+γ = 0.1, L = 1600.
+monitor the battle of these quartets. Initially, we
+allowed both solutions to emerge peacefully in re-
+stricted areas and the stationary portions of species
+“0” and “6” change, as we described previously. Let
+us stress that in the initial state all species repre-
+sented equally, which is practically invisible because
+the unequal stationary distributions of “0+2+4+6”
+species evolve very fast. After 20000 MCSs, when
+the separating borders opened, the symmetric so-
+lution gradually invades the whole space. Notably,
+the presented simulation was recorded at γ = 0.1,
+where the interaction between the quartets is mod-
+erate. Still, this light communication is capable to
+reveal the advantage of A4
+{1,3,5,7} solution.
+If we
+apply larger γ values then nothing changes concep-
+tually, but the victory of A4
+{1,3,5,7} becomes faster.
+Hence, in agreement with the phase diagrams, the
+symmetric four-member solutions is always the win-
+ner in the mentioned corner of the β − α parameter
+plane.
+The lastly discussed observation also answers one
+of our original questions. Namely, one may argue
+that the introduction of an additional inner inva-
+sion within A4
+{0,2,4,6} solution results in a faster ro-
+tation among these species, which could be useful
+them [39]. But the weakening consequence of sym-
+metry breaking is stronger, as it was also the case
+for three-member loops [40].
+Next, we discuss the small α – large β corner of
+the parameter plane. Here, the situation is partly
+6
+
+similar to the first discussed case. More precisely,
+A4
+{0,2,4,6} solution is unstable and replaced very
+soon by A3
+{0,2,4}. But this triplet is vulnerable, be-
+cause the large heterogeneity of inner flow, (α ≪ β),
+results in huge homogeneous domains, as it was al-
+ready reported in [56] for traditional rock-scissors-
+paper game.
+Such a large homogeneous domain
+is also an easy prey of the rival A4
+{1,3,5,7} quartet.
+This picture is valid for small and medium γ values,
+where the interaction between the even- and odd-
+indexed species is not too strong. For large γ val-
+ues, however, the evolution could be more complex,
+which can be observed more easily from a “random-
+patch” initial state. This starting pattern is illus-
+trated in Fig. 9. As we already mentioned in Sec. 2,
+this prepared initial state can offer all possible in-
+terfaces to be present at the very beginning, which
+is extremely useful when there are large difference
+among the invasion rates. In this way, we can ob-
+serve the valid solution already at smaller system
+size. We stress, however, that the mentioned state
+should be reached from all kind of initial states if
+all available species are present and the system size
+is large enough.
+(a)
+(b)
+(c)
+Figure 9: Alternative initial states from where evolution is
+launched when L = 200.
+Panel (a) shows the traditional
+starting state where every node is uploaded by a randomly
+chosen species. Panel (b) shows a state where two competing
+quartets are generated first. Panel (c) shows a state where
+larger patches of species are distributed randomly. The so-
+lution, which is valid in the large size limit, can be reached
+from all starting points, but the necessary system size could
+be largely different.
+Turning back to the large γ, large β, small α re-
+gion, the above specified prepared initial state can
+help us to identify the valid A7
+{0,1,2,3,4,6,7} solution
+already at relatively small system sizes.
+At first
+sight, this solution may seem weird or exotic, but its
+emergence can be understood if we follow how the
+portion of species change by increasing β. This phe-
+nomenon is illustrated in Fig. 10 where we present
+the stationary fractions of all species when we in-
+crease the intensity of additional invasions at fixed
+γ = 1 and α = 0.2. At β = 0 we have a completely
+symmetric food-web where there are equally strong
+ 0
+ 
+ 0.1
+ 
+ 0.2
+ 0
+ 0.1
+ 0.2
+ 0.3
+ 0.4
+ 0.5
+ 0.6
+ 0.7
+all
+A7
+{0,1,2,3,4,6,7}
+0
+1
+2
+3
+4
+5
+6
+7
+ρi
+β
+Figure 10: Stationary fractions of species in dependence of
+β at γ = 1, α = 0.2. The results are obtained at L = 3200
+system size. Species are denoted to every lines. For clarity
+we only show the lines without symbols here.
+The stable
+phases are shown on the top.
+quartets. Consequently, all species are present at
+the same level here. When we introduce a non-zero
+β then not only the fractions of species “0” and
+“6” start decaying, but simultaneously, the portions
+of their principal preys, species “1” and “7” start
+growing. This increment involves the decay of their
+preys, which are species “2” and “0”. This dou-
+ble stress explains why species “0” suffers the most
+at small β values. Naturally, the decay of species
+“6” affects the population of its principal predator,
+hence the portion of species “5” also a decreasing
+function. One may argue that the decay of species
+“0” should affect its main predator species “7”. But
+the predator of the latter, which is species “6”, can-
+not grow here, hence in sum species “7” should not
+necessarily decrease. This difference between the
+status of species “5” and “7” explains why there are
+diverse consequences when both of their principal
+preys, species “6” and “0”, are attacked directly via
+an inner invasion described by parameter β. Impor-
+tantly, the direct support of species “4” via the in-
+tensive “4”→“2” helps the spreading of species “4”.
+This process is also dangerous for species “5”. On
+the other hand, the intensive “2”→“6” process will
+not simply strengthen species “2”, but also weaken
+its prey species “3”, which consequence is also en-
+joyed by species “4”. When the latter species die
+out, the remaining seven species form a heteroge-
+neous seven-member Lotka-Volterra loop where the
+“weakest” species (species “4”, who beats species
+“6” with probability α) occupies the largest frac-
+tion of space. This effect is an example for the phe-
+7
+
+C
+B
+F
+H
+G
+A
+D
+E
+I
+Figure 11: Intermediate stage of the evolution taken from
+the battlefield when different solutions emerge and fight for
+space. The domains mark the following solutions: A4
+{0,1,3,4}
+(A),
+A5
+{1,3,4,5,7}
+(B),
+A4
+{1,3,5,7}
+(C),
+A5
+{0,2,3,5,7}
+(D),
+A5
+{0,1,3,4,6} (E), A3
+{0,2,4} (F ), A6
+{1,2,3,4,5,7} (G), A5
+{1,2,4,5,7}
+(H), and A5
+{1,2,3,5,7} (I). Finally “C” domain wins. Param-
+eters are α = 0.9, β = 0.9, γ = 1, and L = 400.
+nomenon observed first by Tainaka in the simplest
+three-member loop [57].
+Last, we discuss the remaining large α – large β
+corner of the parameter plane. Here the A4
+{0,2,4,6}
+solution is not stable again, hence the remaining
+A3
+{0,2,4} solution fights against A4
+{1,3,5,7} quartet.
+When γ is low and the interaction is weak between
+these groups then the latter formation wins. This
+evolution is similar to the case we reported in the
+left panel of Fig. 7. Practically, the same happens
+when γ is high, but in this case a large set of solu-
+tions may emerge temporarily. Despite of this di-
+versity, however, the winning alliance remains the
+symmetric quartet.
+To give an impression about
+the variety of different candidates, we present an in-
+termediate snapshot about the “battlefield” taken
+at α = 0.9, β = 0.9, and γ = 1. Figure 11 shows
+that in the intermediate stage of the evolution at
+least nine(!) solutions emerge locally and fight for
+space. But eventually the symmetric A4
+{1,3,5,7} so-
+lution crowds out all other competitors.
+4. Conclusions
+Which are the most important features of a cycli-
+cally dominant alliance that determine its viabil-
+ity against alternative formations?
+Motivated by
+this question, we introduced an eight-species model
+where there are two four-member quartets who in-
+teract each other, hence forming a complete eight-
+member loop, too.
+The original model is com-
+pletely symmetric where the inner invasion within
+the quartets are equally strong. Additionally, we
+break the symmetry and introduced an extra inner
+invasion within one of the loops.
+Based on pre-
+vious observations about three-member loops, one
+may argue that this faster rotation of alliance mem-
+bers could be positive to make them stronger. From
+the other viewpoint, the broken symmetry is always
+detrimental, hence the new opportunity would just
+weaken the involved quartet. It is also worth not-
+ing that the slight alteration of the original “sym-
+metric” food-web offers the chance several potential
+solutions to emerge. Indeed, an armada of candi-
+dates can be observed at intermediate stage of the
+evolution, but it is believed that the final pattern
+is determined by some basic concepts.
+According to our findings, keeping the symme-
+try is vital and the solutions, which maintain a
+balance among their members, are fitter.
+In the
+complete parameter space we studied, it can hap-
+pen that there are more than one solution which
+possess this attractive character. In this case the
+general speed of inner invasion could be a decisive
+factor. If the rotations are equally strong then we
+could give examples when a short or a larger loop is
+the winner. For example, a quartet can beat a trio,
+but a quartet can also beat an octet. Therefore, it
+cannot be made a simple conclusion that a smaller
+or larger alliance is better. Probably, there are two
+competing effects whose relation determines the ac-
+tual fitness of a loop. On one hand, a short loop
+may involve relatively large homogeneous domains,
+which could always be dangerous.
+On the other
+hand, too large loop also means that the reaction of
+the alliance could be delayed, because several inner
+invasions should happen to produce the predator of
+the external intruder. Therefore, the sum of these
+adverse impacts could be case-sensitive.
+Naturally, our present study is a sterile theoreti-
+cal model, but we strongly believe that the observa-
+tions we made could be useful when real systems are
+studied. Our another important message is the im-
+portance of appropriately chosen system size, which
+8
+
+is always a central issue when cyclic dominance is
+present. Otherwise, the observations could be di-
+verse without deeper understanding. This is why
+we should always consider the actual size when we
+want to give predictions about the pattern forma-
+tion in a finite system driven by intransitive inter-
+actions.
+This work was supported by the National Re-
+search Foundation of Korea (NRF) grant funded
+by the Korea government (MSIT) (No.
+NRF-
+2021R1A4A1032924). J.P. was also supported by
+a grant from Kyung Hee University in 2022 (KHU-
+20220901).
+X.C. was supported by the National
+Natural Science Foundation of China (Grants Nos.
+61976048 and 62036002) and the Fundamental Re-
+search Funds of the Central Universities of China.
+A.S. was supported by the National Research, De-
+velopment and Innovation Office (NKFIH) under
+Grant No. K142948.
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+MNRAS 000, 1–?? (2023)
+Preprint 3 January 2023
+Compiled using MNRAS LATEX style file v3.0
+Time-averaging Polarimetric and Spectral Properties of GRBs
+Liang Li1,2,3⋆ Soroush Shakeri4,1,5†
+1ICRANet, Piazza della Repubblica 10, I-65122 Pescara, Italy
+2ICRA and Dipartimento di Fisica, Università di Roma “La Sapienza”, Piazzale Aldo Moro 5, I-00185 Roma, Italy
+3INAF – Osservatorio Astronomico d’Abruzzo, Via M. Maggini snc, I-64100, Teramo, Italy
+4Department of Physics, Isfahan University of Technology, Isfahan 84156-83111
+5ICRANet-Isfahan, Isfahan University of Technology, Isfahan 84156-83111, Iran
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+One of the most fundamental and yet open issues in gamma-ray burst (GRB) physics, is the comprehension of the nature of
+their jet composition. The investigation of joint polarimetric and spectral properties is essential to probe the jet composition and
+radiation mechanism of GRBs. Several distinct categories of jet properties—the “Kinetic-energy-dominated" (KED), “Poynting-
+flux-dominated" (PFD), and “Hybrid-dominated" (HD) jets—have been observed in the observed GRB spectra, and the emission
+dominated by different jet properties is expected to have a different level of polarization (πKED ≲ πHD ≲ πPED). In the present
+paper, we collected a GRB sample in which all the bursts detected by the Gamma-ray Burst Monitor (GBM) on board the
+NASA Fermi Gamma-ray Space Telescope whose polarization measurements are also reported in the literature and the epochs
+of prompt emission are heavily overlapped with their polarization observations, aiming to establish a connection between the
+polarization and jet properties of GRBs, and to confirm the validity of this correlation (πKED ≲ πHD ≲ πPED) from observations.
+With a detailed spectral analysis, we found that all the bursts are classified as the “Hybrid" jet type, implying that one cannot
+rule out that the photosphere emission may also be the possible mechanism powering the high levels of polarization. Moreover,
+we also discovered that the polarization degrees π are tightly correlated with the cosmological rest-frame peak energy (Ep,z) of
+the νFν prompt emission spectrum, the isotropic-bolometric-equivalent emission energy (Eγ,iso), and the blackbody temperature
+(kT). Finally, we present different polarization models in the presence of ordered and random magnetic field configurations
+with the properties of corresponding hybrid jets in order to interpret polarization measurements of the prompt emission in our
+sample.
+Key words: gamma-ray burst: general, radiation mechanisms: non-thermal, radiation mechanisms: thermal
+1 INTRODUCTION
+Gamma-ray bursts (GRBs) are one of the most explosive, and elec-
+tromagnetically the brightest transient phenomena in the Universe,
+occurrence at cosmological distances. After decades of investiga-
+tion, the origin of the jet composition (a hot baryonic-dominated
+fireball or a cold Poynting-flux-dominated outflow), and the radi-
+ation mechanism and energy dissipation mechanism (synchrotron,
+or Comptonization of quasi-thermal emission from the photosphere)
+in gamma-ray burst (GRB) physics are still unclear (e.g., Rees &
+Meszaros 1994; Mészáros & Rees 2000; Rees & Mészáros 2005;
+Pe’er et al. 2006; Dai et al. 2006; Pe’er 2015; Pe’Er & Ryde 2017;
+Zhang 2018; Bégué et al. 2022).
+There are two crucial clues that can in principle be helpful in di-
+agnosing the jet composition of GRBs, as well as their radiation
+mechanism and energy dissipation mechanism. The conventional
+approach is to examine the spectral properties of prompt emission.
+⋆ E-mail: liang.li@icranet.org (LL)
+† E-mail:s.shakeri@iut.ac.ir (SS)
+Theoretically, a thermal component originating from photosphere
+emission, or a non-thermal component originating from synchrotron
+radiation, possibly also from inverse Compton scattering, is often
+expected to be present in GRB spectral analysis. Phenomenologi-
+cally, GRB spectra in the keV-MeV energy range can be typically
+well-delineated by an empirical function, known as the Band func-
+tion (Band et al. 1993), which is generally considered to be a non-
+thermal spectrum. The Band spectrum features a smoothly broken
+power law, with the peak energy Ep ≃ 210 keV (the energy at which
+most of the energy is released) in νFν space and the asymptotic
+power-law photon indices below (α ≃ −0.8) and above (β ≃ −2.5)
+the break energy (e.g., Li et al. 2021). The low-energy spectra dur-
+ing GRB prompt emission phase are closely related to the energy
+distribution of electrons (e.g., Preece et al. 1998; Lloyd & Petrosian
+2000; Geng et al. 2018). This fact can be utilized in order to diag-
+nose GRBs radiation mechanism as well as their jet properties. For
+instance, synchrotron emission predicts two different α values: α=-
+3/2 and α=-2/3 (so-called the line-of-death (LOD) of synchrotron
+emission, Preece et al. 1998) correspond to the fast-cooling and
+slow-cooling synchrotron emission, respectively. It has been shown
+© 2023 The Authors
+arXiv:2301.00576v1  [astro-ph.HE]  2 Jan 2023
+
+2
+Li & Shakeri
+that the synchrotron emission in the presence of a decaying mag-
+netic field can reproduce the Band-like spectrum of the GRB prompt
+phase (Lan et al. 2021). While photosphere models, on the other
+hand, predict much harder values of α; e.g., above α=-2/3. For ex-
+ample, a recent study (Acuner et al. 2020) suggests that the spec-
+tra that prefer the photospheric model all have low-energy power-
+law indices α ∼>-0.5, as long as the data has a high significance.
+Years of observations have revealed, however, that GRBs have di-
+verse spectral properties, making it difficult for a single spectral
+model (such as the Band-alone model) to accurately characterize all
+the spectral shapes. For instance, time-resolved and time-integrated
+spectral analysis inferred from the broadband Fermi observations
+have revealed that GRB prompt emission exhibits remarkably di-
+verse spectral properties (e.g., Abdo et al. 2009; Ryde et al. 2010;
+Axelsson et al. 2012; Acuner et al. 2019; Li 2019a; Li et al. 2019,
+2021, 2022b; Deng et al. 2022). A kinetic-energy-dominated (KED)
+jet characterised by a quasi-thermal Planck-like spectrum has been
+detected in some bursts (e.g. GRBs 090902B, 220426A, Ryde et al.
+2010; Deng et al. 2022; Wang et al. 2022; Song et al. 2022), while
+a cold Poynting-flux-dominated (PFD) outflow characterised by a
+Band (or cutoff power-law1)-only function (Band et al. 1993) has
+been also observed in some other bursts (e.g. GRB 080916C2, GRB
+130606B, and many others, Abdo et al. 2009; Li 2022a), even a
+hybrid-dominated (HD) relativistic outflow with a hot fireball com-
+ponent and a cold Poynting-flux component, characterized by either
+a composited spectral scenario, with a non-thermal component and
+a thermal component, e.g., GRBs 100724B, 110721A, 150314A,
+190114C, and several others, Axelsson et al. 2012; Guiriec et al.
+2011; Wang et al. 2019; Li 2022a; Li et al. 2022b), or a transition
+from a fireball to a Poynting-flux-dominated outflow within a single
+burst (e.g. GRBs 140206B, 160625B, and several others, Li 2019a),
+have also been observed.
+An alternative approach is to investigate their polarization prop-
+erties. Theoretically, photon polarizations play a key role to un-
+derstand the jet composition, angular structure, geometric config-
+uration, magnetic composition and magnetic field configuration of
+GRB jets, and radiation mechanism of GRB jets (Toma et al. 2009a;
+Lundman et al. 2013; Zhang 2014; Zhang et al. 2019). Although
+magnetic field configurations with relatively large coherence lengths
+more than gyroradius of charged particles can generate the same
+energy spectrum via synchrotron mechanism, the level of polariza-
+tion may significantly different for various magnetic field structures.
+Therefore joint spectral and polarization analysis is essential to de-
+termine the magnetic field structure in outflow materials of GRBs
+(Granot 2003; Lyutikov et al. 2003; Granot & Königl 2003a; Kole
+et al. 2020). For instance, the center engine is anticipated to gen-
+erate strong magnetic fields (a highly magnetized jet) and launch
+them concurrently with the relativistic jets. It is unclear, neverthe-
+less, whether the GRB emission is caused by shock dissipation or
+magnetic reconnection, and whether the outflow is dominated by the
+photosphere or synchrotron emission (Toma et al. 2009a).
+In fact, the generation of the polarization signal can be intrinsic
+to the emission process or due to the propagation effects Shakeri
+1 Recent studies (Li 2022b,a) supported by several pieces of additional ev-
+idence (e.g., inconsistent spectral parameter distributions and distinct Amati
+and Yonetoku correlations) have shown that Band-like spectra and CPL-like
+spectra may originate from distinct radiation processes.
+2 It has been demonstrated in recent studies (Guiriec et al. 2015; Vereshcha-
+gin et al. 2022) that a thermal component needs to be added during the initial
+prompt emission of GRB 080916C to obtain an acceptable fit to the spectral
+data.
+& Allahyari (2018). Several emission models (induced synchrotron
+emission, Rybicki & Lightman 1979; photosphere emission, Lund-
+man et al. 2014a; and Compton drag model, Lazzati et al. 2004)
+have been proposed to explain the intrinsic polarization properties
+of relativistic jets during prompt emissions. (i) Synchrotron emis-
+sion model. There are some studies (e.g., Rybicki & Lightman 1979;
+Toma et al. 2009b; Lan & Dai 2020) showing that higher values
+of linear polarized signal (polarization degree π ranging from 20%
+to 70%) is expected to be measured with an ordered magnetic field
+from the synchrotron emission from a relativistic jet. While jets with
+random magnetic fields produce lower levels of polarization, this is
+due to the polarization being canceled out so that the net polariza-
+tion degree being close to zero for an on-axis observer. A polariza-
+tion detection which is less than 15% is believed to be originated
+from a random magnetic field configuration within the jet (Mao &
+Wang 2013). For example, if the emission is dominated by the inter-
+nal shock (IS) model, π is expected to range from 10% (the maxed
+magnetic field configuration) to 70% (the large-scale ordered mag-
+netic field configuration). (ii) Dissipative photosphere model. The
+dissipative photosphere model predicts a relatively low degree of po-
+larization in the γ-ray band. However, a structured jet photosphere
+model might also generate polarized photons by Compton scatter-
+ing, but the degree of polarization would be energy-dependent from
+the synchrotron model in ordered magnetic fields. For instance, it
+is demonstrated that if the jet has considerable structure, the model
+may create polarizations of up to 40% within δΘ ∼ Γ−1. However, in
+the absence of dissipation and below the photosphere the polariza-
+tion is rather limited to values below 15%-20% (Gill et al. 2018). To
+restrict these models, a high-sensitivity gamma-ray polarimeter with
+a broad band-pass to detect energy-dependent polarization signals is
+required (Zhang 2014; Ito et al. 2014; Lundman et al. 2014a; Lund-
+man et al. 2018a). (iii) Internal-collision-induced magnetic recon-
+nection and turbulence (ICMART) model. In the ICMART model
+(Zhang & Yan 2011a), π is expected to range from 60 percent at the
+beginning of the pulse and down to about 10 percent at the end of
+the pulse. A decaying polarization degree is predicted.
+It is highly speculated that the prompt emission is likely expected
+to be strongly polarized owing to its non-thermal origin (a non-
+thermal Band-like spectrum). Observationally, higher levels of lin-
+ear polarization measured from prompt γ-ray emission have been
+reported by several authors (e.g., Coburn & Boggs 2003; Willis
+et al. 2005; McGlynn et al. 2007; Yonetoku et al. 2012a). For in-
+stance, a higher polarization degree π = 80%±20% in GRB 021006
+was claimed by Coburn & Boggs (2003) using the RHESSI data.
+Later, several other cases were also reported, e.g., GRB 930131 (π >
+35%), GRB 960924 (π > 50%), GRB 041219A, GRB 100826A
+(π = 27% ± 11%), GRB 110301A (π = 70% ± 22%), and GRB
+110721A (π = 84%+16%
+−28%). Subsequent observations were also ob-
+served in the optical band during the afterglow emission and were of
+relatively low polarization. Compared with prompt γ-ray emission,
+the levels of linear polarization measured from afterglow emission
+are relatively lower. e.g., GRB 060418 (π < 8%), GRB 090102 (π =
+10.1% ± 1.3%), GRB 091208B (π = 10.4% ± 2.5%), and 120328A
+(π = 28% ± 4%). However, higher degrees of polarization observa-
+tions are still expected to be measured from early reverse shocks, up
+to ∼ 60%.
+Generally speaking, we can study GRB polarization and spec-
+tral properties either in a time-integrated (e.g., Li 2022a) or time-
+resolved (e.g., Li et al. 2021) manner. The former represents average
+polarimetric and spectral properties and is treated as a single-time
+event for the entire emission period. The latter treats the entire emis-
+sion period as divided into multiple-time events, and polarimetric
+MNRAS 000, 1–?? (2023)
+
+Time-averaging Polarimetric and Spectral Properties of GRBs
+3
+and spectral analyses are therefore performed on each event individ-
+ually. The time-integrated method depends more heavily on the sta-
+tistical results of a large sample in order to produce a more trustwor-
+thy result because different bursts have distinct observational prop-
+erties (e.g., angular structure). However, this issue does not arise
+when a time-resolved technique is used in the same burst since it
+ensures that other conditions (e.g., magnetic field configuration) are
+essentially the same and, therefore, makes it easier to obtain reliable
+results.
+Following these lines of argument, the emissions dominated by
+different jet properties (KED, PFD, and HD) may have different lev-
+els of prompt-GRB polarization measurements3 (Li 2019a). As a
+result, a different level of polarization degrees (πKED ≲ πHD ≲ πPED)
+is naturally expected due to different jet properties if other condi-
+tions are basically the same, where πKED, πHD, and πPED are the po-
+larization degree in the KED, HD, and PHD jets, respectively. This
+may provide a method to study the correlations between polarization
+properties and their spectral properties, as well as their jet properties.
+A possible connection between the spectral and polarization proper-
+ties has not yet been firmly established, though recent works provide
+some statistical results (Chattopadhyay et al. 2019; Kole et al. 2020).
+Therefore, we dedicate this work to examining whether any possi-
+ble connections exist between polarization and jet properties, and
+aim to confirm the validity of this correlation (πKED ≲ πHD ≲ πPED)
+from observations. Practically, several important factors need to be
+taken into account in our analysis. (i) In order to potentially evaluate
+all of the frequently-used GRB spectral models and thus diagnose
+the jet properties, our analysis focuses on the bursts detected by the
+Gamma-ray Burst Monitor (GBM, 8 KeV-40 MeV, Meegan et al.
+2009) onboard the NASA Fermi Gamma-ray Space Telescope. (ii)
+To directly compare the spectral and polarization properties and thus
+make our results more trustworthy, we select the bursts during which
+polarization observations and spectral data are available during the
+same epoch. (iii) Realistically, systematic error is frequently at play
+in polarization measurements and different polarization instruments
+have different systematic errors. Therefore, a high-significance sig-
+nal from polarization measurement is required. In this paper, we
+collect a sample of the Fermi-GBM detected bursts along with the
+polarized measurements reported in the literature using the time-
+integrated spectral and polarization analysis approach based on their
+statistical results, aiming to establish a connection between the po-
+larization and jet properties of GRBs.
+The paper is organized as follows. The sample and Methodol-
+ogy are presented in Section 2 and Section 3, respectively. Our re-
+sults and their physical implication are summarized in Section 4 and
+Section 5, respectively. The conclusion is presented in Section 5.
+Throughout the paper, the standard Λ-CDM cosmology with the pa-
+rameters H0 = 67.4 kms−1 Mpc−1, ΩM = 0.315, and ΩΛ = 0.685 are
+adopted (Planck Collaboration et al. 2018).
+3 Polarization measurement is a measurement where π ranges from 0% to
+100% (0% ≤ π ≤ 100%) and includes the non-detection (0%), so measur-
+ing something consistent with zero is a measurement. Polarization detection,
+however, indicates a different meaning, where 0% < π ≤ 100% and excludes
+0%, since we have excluded part of the parameter space equivalent to detec-
+tion measurements due to the fact that a non-detection of flux implies that we
+did not measure something.
+2 THE SAMPLE
+A comprehensive database of GRB polarimetric observations has
+been created in a recent work (Li et al. 2022a) by extensively search-
+ing for those GRBs in the literature whose polarization measure-
+ments have been reported. A total of 73 bursts with polarization
+detections were included in the database, covering a broad wave-
+length range from radio to optical, X-ray, and γ-ray emission (see
+Table 1 in Li et al. 2022a). The prompt emission data of these bursts
+were observed by different satellites (Fermi, Swift, BeppoSAX, and
+BATSE). On the other hand, the diagnosis of jet composition usually
+requires a refined spectral analysis. Among these satellites, Fermi
+covers the broadest energy range in the observation. Consequently,
+in order to fully evaluate all current spectral models we pay spe-
+cial attention to these Fermi-detected bursts. The Gamma-ray Burst
+Monitor (GBM, 8 KeV-40 MeV, Meegan et al. 2009) and the Large
+Area Telescope (LAT, 20 MeV- 300 GeV, Atwood et al. 2009), on-
+board the NASA Fermi Gamma-ray Space Telescope, together pro-
+vide unprecedented spectral coverage for seven orders of magnitude
+in energy (from ∼8 keV to ∼300 GeV). Our statistical analysis in
+the current work includes the prompt γ-ray emission spectral anal-
+ysis and the connection between the spectrum and polarization are
+thus based on the Fermi-detected bursts. On the other hand, we fo-
+cus on the GRBs whose polarization measurements were recorded
+during the prompt emission in the γ-ray band in order to compare
+the polarization and spectral properties during the same time period.
+Following are the specific observational properties of the five fas-
+cinating bursts (see Table 1) that made up the smaller sample as a
+result of these selection criteria.
+• GRB 100826A. On August 26, 2010 at 22:58:22.898 (T0)
+UT, GRB 100826A, was detected by the Fermi/GBM. A t90 of
+(84.993±0.724) s in the 10-1000 keV was measured using the
+GBM data. The GBM lightcurve exhibits a complicated shape with
+multiple-peak pulses, and the fluence (flux integrated over the burst
+duration) in the energy range of 10-1000 keV during the t90 dura-
+tion reported by the GBM team is (1.6388±0.001)×10−4 erg cm−2
+(Figure 2). This burst has no redshift measurement, a value of 2.3
+is estimated using the Yonetoku correlation (Yonetoku et al. 2004).
+Therefore, the isotropic-equivalent energy released from gamma-
+ray emission in the cosmological frame for this burst can be cal-
+culated as, Eγ,iso=(3.39+0.36
+−0.33)×1054 erg. The KED-to-PFD transition
+pattern belonging to the “HD” jet type is diagnosed for this burst
+by using a low-energy spectrum based on a detailed time-resolved
+spectral analysis during prompt emission (see Section 4). Yonetoku
+et al. (2011a) reported that a relatively higher polarization degree
+(πobs=27±11) with a 2.9σ linear polarization signal significance was
+detected during the prompt emission (0-100 seconds after the trig-
+ger) of GRB 100826A using a GAmma-ray Polarimeter (GAP) on
+board a small Japanese solar-power-sail demonstrator, Interplanetary
+Kite-craft Accelerated by Radiation of the Sun (IKAROS).
+• GRB 110301A. On March 01, 2011 at 05:08:43.070 (T0) UT,
+GRB 110301A, was detected by the Fermi-GBM. A t90 of (5.693±
+0.362) s in the 10-1000 keV was measured using the GBM data. The
+lightcurve exhibits a complicated shape with multiple-peak pulses,
+and the fluence in the energy range of 10-1000 keV from T0+0 s
+to T0+5.693 s reported by the GBM team is (3.5891±0.003)×10−5
+erg cm−2 (Figure 3). This burst has no redshift measurement, a
+value of 0.36 is estimated by using the Yonetoku relation (Yone-
+toku et al. 2004). With this estimated value of redshift, the isotropic-
+equivalent energy released from gamma-ray emission in the cosmo-
+logical frame for this burst can be calculated as, Eγ,iso=1.4×1052 erg.
+A KED-to-PFD jet is diagnosed for this burst by using a low-energy
+MNRAS 000, 1–?? (2023)
+
+4
+Li & Shakeri
+spectrum based on a detailed time-resolved spectral analysis during
+prompt emission. A linear polarization signal (3.7σ) with a very high
+polarization degree (π=70±22) for this burst was claimed by Yone-
+toku et al. (2012b) using the IKAROS/GAP data 0-7 seconds after
+the trigger (Figure 3).
+• GRB 110721A. On July 21, 2011 at 04:47:43.761 (T0)
+UT, GRB 110721A, was detected by the Fermi-GBM. A t90 of
+(21.822±0.572) s in the 10-1000 keV was measured using the GBM
+data. The lightcurve exhibits a single-peak pulse, and the fluence in
+the energy range of 10-1000 keV from T0+0 s to T0+21.822 s as re-
+ported by the GBM team is (3.5891±0.003)×10−5 erg cm−2 (Figure
+4). This burst has a value of 0.382 of redshift. With this redshift, we
+calculate the isotropic-equivalent energy released from gamma-ray
+emission in the cosmological frame for this burst, Eγ,iso=3.0×1052
+erg. A peak-KED jet is diagnosed for this burst by using a low-
+energy spectrum based on a detailed time-resolved spectral analysis
+during prompt emission. A linear polarization signal (3.3σ) with a
+very high polarization degree (π=85+16
+−22) for this burst was claimed
+by Yonetoku et al. (2012b) using the IKAROS/GAP data 0-11 sec-
+onds after the trigger (Figure 4).
+• GRB 140206A. On 6 February 2014 at 06:36:12.843 UT (T0),
+a long burst, GRB 140206B, was detected by Fermi-GBM and sev-
+eral other satellites (e.g., INTEGRAL/IBIS). Its GBM lightcurve in
+the 10-900 keV exhibits three clearly separated emission episodes
+(G1, G2, and G3) as shown in the left panel of Figure 5, with T90 of
+146.690±4.419 s was measured by GBM data. GRB 140206B is a
+very bright burst, and the fluence (flux integrated over the burst dura-
+tion) in the energy range of 10-1000 keV from T0+0 s to T0+146.690
+s reported by the GBM team is (1.23±0.003)×10−4 erg cm−2. This
+burst has a value of 2.73 for redshift. With this redshift, we calculate
+the isotropic-equivalent energy released from gamma-ray emission
+in the cosmological frame for this burst, Eγ,iso=2.3×1054 erg. The
+spectral analysis has been performed in great detail in Li (2019a),
+and three different spectral components (KED-to-PFD) have been
+identified, a short-thermalized precursor early on, followed up with
+the main burst with a non-thermal emission later on, and in the last
+with a fainter burst with still a non-thermal emission. Interestingly,
+immediately following up with the short-thermalized precursor, a
+clear polarization signal with 90% confidence was observed 4-26
+seconds after the trigger as reported in Götz et al. (2014) using the
+INTEGRAL/IBIS data. This signal is a linear polarization with an
+up-limit polarization degree of π > 28 observed in the γ-ray energy
+band (Figure 5).
+• GRB 160802A. GRB 160802A was detected by GBM on Au-
+gust 2, 2016 at UT 06:13:29.63. A T90 of (16.384±0.362) s in the
+10-1000 keV was measured using the GBM data. The prompt emis-
+sion light curve shows two clearly separated active periods, with a
+quiescent time interval between the two periods of about 10 sec-
+onds. The earlier period consists of overlapping pulses while the lat-
+ter period shows a clear single-peak pulse. The fluence in the energy
+range of 10-1000 keV from T0+0 s to T0+21.822 s as reported by
+the GBM team is (6.8399±0.0057)×10−5 erg cm−2. This burst has
+no redshift measurement, a value of 0.36 is estimated by using the
+Yonetoku relation (Yonetoku et al. 2004). Using this estimated value
+of redshift, we further calculate the isotropic-equivalent energy re-
+leased from gamma-ray emission in the cosmological frame for this
+burst, Eγ,iso=2.2×1053 erg. A peak-KED jet for each period is di-
+agnosed for this burst by using a low-energy spectrum based on a
+detailed time-resolved spectral analysis during prompt emission. A
+linear polarization signal (∼3σ) with a very high polarization degree
+(π=85±29) for this burst was claimed by Yonetoku et al. (2012b) us-
+ing the AstroSat/GZTI data 0-20.34 seconds after the trigger (Figure
+6).
+3 METHODOLOGY
+3.1 Spectral Analysis Techniques
+In order to diagnose the jet properties for a given burst, a de-
+tailed time-integrated or time-resolved spectral analysis is re-
+quired. The spectral analysis is performed by a pure Python pack-
+age, namely, the Multi-Mission Maximum Likelihood
+Framework (3ML,Vianello et al. 2015). Moreover, a Bayesian
+approach and Markov Chain Monte Carlo (MCMC) iterations to
+explore the best parameter space was used. Our spectral analy-
+sis includes the following main steps. (1). First is to select detec-
+tors, sources, and background intervals; (2). Second, we used the
+Bayesian block method (BBlocks, Scargle et al. 2013) to bin the
+Time-Tagged Events (TTE) lightcurve of the brightest detector (with
+the minimum viewing angle), and the significance (S, Vianello 2018)
+for each BBlocks time bin was also calculated. (3). Third, we use a
+typical GRB spectral model (Band function, Band et al. 1993) to fit
+all the spectra selected by the BBlocks method and the best model
+parameters are obtained by adopting a fully Bayesian approach. For
+a detailed Bayesian spectral analysis and the reduction procedure
+applied to a GRB spectrum, we refer to Li (2019a,b, 2020); Li et al.
+(2021); Li & Zhang (2021).
+3.2 Using the Yonetoku correlation to infer their redshift values
+In order to study the intrinsic properties in the cosmological rest
+frame, a redshift measurement for each burst is needed. Unfortu-
+nately, we currently have 3 bursts (GRB 100826A, GRB 110301A,
+and GRB 160802A) without known redshift. Several works to in-
+fer redshifts using empirical relations of GRB have been reported in
+the literature(Amati et al. 2002; Yonetoku et al. 2004). For example,
+(Yonetoku et al. 2004) discovered a new and much tighter relation-
+ship between the spectral peak energy (Ep) and the peak luminos-
+ity (Lp) using the combined data detected by the BeppoSAX and
+BATSE satellites, and claimed that one can use the Ep-Lp relation to
+estimating redshift without knowing distances in the BSTASE cata-
+log. We, therefore, attempt applying the Yonetoku relation(Yonetoku
+et al. 2004) to estimate redshift values for these bursts.
+Our procedure to estimate the pseudo-redshift using the Yonetoku
+relation includes the following steps.
+First, we perform a spectral fit to the peak spectrum (the brightest
+time bin was selected by using the Bayesian blocks method(Scargle
+et al. 2013) at the highest statistical significance S) of each burst.
+The peak flux Fγ and peak energy Ep on the observer’s frame from
+the spectral fits are thus obtained, where Fγ is the observed peak flux
+integrated between (1-104) keV in units or erg cm−2 s−1.
+Second, in order to use the Ep,z-Lp,iso relation, a bolometric lu-
+minosity in a common cosmological rest-frame energy band (1-104
+keV) is needed. It can be obtained by using the spectral parameters to
+conduct a k-correction extrapolating the observed energy band to 1-
+104 keV. For a given burst, the k-correction factor (kc) can be derived
+using the following procedure. The observed flux Fobs (erg cm−2 s−1),
+in a fixed detector energy bandwidth [e1, e2] (for instance, for the
+Fermi-GBM observation, e1=8 keV, e2=40 MeV), can be written as:
+Fobs
+[e1,e2] =
+� e2
+e1
+EN(E)dE,
+(1)
+MNRAS 000, 1–?? (2023)
+
+Time-averaging Polarimetric and Spectral Properties of GRBs
+5
+where E is in units of keV, and N(E) is a GRB photon number spec-
+trum. The total luminosity emitted in the bandwidth [e1,e2], defined
+in the cosmological rest-frame, is given by:
+L[e1(1+z),e2(1+z)] = 4ˆπD2
+L(z)Fobs
+[e1,e2],
+(2)
+which DL(z) is the luminosity distance. To express the luminosity
+L in the cosmological rest-frame energy band, [E1=1 keV, E2 =104
+keV], common to all sources, the Eq.(2) can be rewritten as:
+L[E1,E2] = 4ˆπD2
+LFobs
+� E1
+1+z , E2
+1+z
+� = 4πD2
+Lk[e1,e2,E1,E2,z]Fobs
+[e1,e2],
+(3)
+where the k-correction factor, kc, is therefore defined as:
+kc = k[e1,e2,E1,E2,z] =
+Fobs
+� E1
+1+z ; E2
+1+z
+�
+Fobs
+[e1,e2]
+=
+� E2/(1+z)
+E1/(1+z) EN(E)dE
+� e2
+e1 EN(E)dE
+,
+(4)
+Last, with the k-correction factors known, the peak luminosity can be
+derived from the observed γ-ray flux Fγ according to L = 4ˆπd2
+LFγkc,
+where dL is the luminosity distance.
+Finally, as long as the prompt emission spectral properties can be
+obtained, the Yonetoku relation can be used to infer redshift. With
+the above Steps, one can use Equation (2) in Yonetoku et al. (2004)
+to estimate their redshift values,
+Lp
+1052ergs−1 = (2.34+2.29
+−1.76)×105
+�Ep(1+z)
+1keV
+�2.0±0.2
+.
+(5)
+Table 2 lists the spectral properties obtained from the peak spec-
+tral analysis and the estimated redshift values inferred from the Yo-
+netoku relation.
+4 RESULTS
+4.1 Spectral Properties and their inferred jet properties
+Five bursts were claimed to have a high-significance polarization
+detection (Table 1) and GBM data (Table 4) taken at their prompt
+emission, thereby providing an “ideal” sample to study the possi-
+ble connection between jet properties and polarization straightfor-
+wardly.
+In practice, either time-integrated or time-resolved spectral analy-
+sis is frequently used to diagnose their jet properties. Several meth-
+ods have been widely used to diagnose the jet properties based on
+their spectral analysis. The simplest method is to use the low-energy
+spectrum (e.g., Preece et al. 1998) to resolve the jet properties for
+a given pulse/burst (Method-I). In this method, a single empirical
+model (such as the Band model) based on a time-resolved tech-
+nique is typically used. The KED jets, therefore, can be defined
+as all α indices, within uncertainties, in a given burst, obtained
+from time-resolved spectral analysis, are systematically above the
+synchrotron limit throughout the entire burst/pulse duration, being
+consistent with a matter-dominated fireball jet. The PFD jets, on
+the other hand, are consistent with the scenario that all α indices,
+within uncertainties, in a given burst are below the synchrotron limit
+throughout the burst/pulse duration, this suggests a Poynting-flux-
+dominated jet. The HD jets represent the moderate scenario, and
+can be presented that some α indices, obtained from time-resolved
+spectral analysis, are above the synchrotron limit (α >2/3) while
+some others are below the synchrotron limit (α <2/3) throughout
+the burst/pulse duration. The HD jets can be further divided into two
+subcategories. (1) the peak-KED pattern since the thermal emission
+component (the spectra that violate the LOD line) is only detected
+around the peak of the pulse (thermal component dominates the peak
+of a pulse/burst); and (2) the KED-to-PFD (thermal to non-thermal
+component) transition pattern (Li 2019a), since the thermal emission
+component is detected at the beginning of the burst, and followed by
+non-thermal emission component. However, it may be difficult to
+classify jets as either KED or PFD jets based on the spectral index
+alone. In contrast to an optically-thin synchrotron emission, a photo-
+spheric quasi-thermal component would, in fact, have a harder low-
+energy spectral index, but this does not guarantee that the jet is KED
+(Gill et al. 2020). Therefore, a more reliable approach is to find the
+best model (Method-II) by comparing various frequently-used spec-
+tral models using certain statistical information criteria, such as the
+Akaike Information Criteria (AIC; Akaike 1974), Bayesian Infor-
+mation Criteria (BIC; Schwarz 1978), and the Deviance Information
+Criterion (DIC; Spiegelhalter et al. 2002; Moreno et al. 2013). Af-
+ter the identification of the best spectral model, one can assess the jet
+properties using the spectral properties inferred from the best model.
+Moreover, we may also directly fit the spectral data with a physical
+model (such as the synchrotron emission model, Burgess et al. 2020;
+Method-III), so as to diagnose the jet properties and any potential
+connection with their polarization properties. Our analysis in this
+task incorporates both Method-I and Method-II (primary method).
+Method III will be used elsewhere.
+We first perform time-integrated spectral analysis (treating the
+entire epoch of polarization observation as one time bin) by using
+various GRB spectral models, including power-law (PL), blackbody
+(BB), cutoff power law (CPL), Band function, PL+BB, CPL+BB,
+and Band+BB, respectively. We adopted both AIC and BIC to eval-
+uate different spectral models and select the preferred one, and the
+preferred model is the one that provides the lowest AIC and BIC
+scores. As such, we define
+• PFD jets: a single non-thermal (Band-like) spectral compo-
+nent was found in the time-integrated spectral analysis, like GRB
+080916C (Abdo et al. 2009).
+• KED jets: a dominate thermal (BB-like) spectral compo-
+nent was found in the time-integrated spectral analysis, like GRB
+090902B (Ryde et al. 2010) and GRB 220426A (Deng et al. 2022).
+• HD jets: a hybrid spectrum of thermal (BB-like) and non-
+thermal (Band-like) components was observed in the time-integrated
+spectral analysis in a single burst, like GRB 110721A (Axelsson
+et al. 2012), and GRB 140206A (e.g., Li 2019a).
+Our refined time-integrated spectral analysis suggests that the
+Band+BB model can best characterize the spectral shape of all the
+five bursts (see Table 3). The global properties of our sample used in
+the spectral analysis are reported in Table 4. These include the GRB
+name (column 1), observed duration T90 of burst4 (column 2), 10-
+1000 KeV fluence (column 3), together with the used detectors (col-
+umn 4), the selected source (column 5) and background (column 6)
+intervals, and the best model (column 7). The best-fit spectral param-
+eters of the sample with the Band+BB model are reported in Table 5,
+including GRB name (column 1), source duration (column 2), and
+corresponding significance (column 3), and Band component nor-
+malization K (column 4), low-energy power-law index α (column
+5), peak energy Ep (column 6) of the νFν spectrum, and high-energy
+power-law index β (column 7); and BB component normalization K
+(column 8), and temperature kT (column 9).
+We next perform time-resolved spectral analysis (treating the
+entire epoch of polarization observation as divided into multiple-
+4 The time interval during which 90% of the total observed counts have been
+detected.
+MNRAS 000, 1–?? (2023)
+
+6
+Li & Shakeri
+time events) for each individual BBlocks time bin using the Band
+model. Temporal evolution of α is presented in Figures 2-6 for GRB
+110826A, GRB 110301A, GRB110721A, GRB 140206A, and GRB
+160802A, respectively. With the BBlock time bins selected and the
+corresponding significance (S) values calculated, the data points are
+shown in different S ranges5: S < 10, 10 ≤ S ≤ 20, and S > 20.
+For a subset of GRBs, a mixture of thermal (blackbody contribution
+from the photosphere emission) and non-thermal (synchrotron emis-
+sion from relativistic electrons) components was observed in a sin-
+gle burst (e.g., Li 2019a). Previous studies (e.g., Burgess et al. 2014)
+showed that the characteristic energy (Ep) of non-thermal emission is
+correlated to the characteristic energy (kT) of thermal emission with
+a power-law relation in form of Ep ∝ T q, where q ranges from ∼1-2.
+Burgess et al. (2014) studied a set of bright Fermi single-pulse GRBs
+and claimed that one can use this correlation to identify whether the
+jet is dominated by kinetic (q ∼ 1) or magnetic energy (q ∼ 2) de-
+pending on the value of the exponent. GRB 110721A was identified
+as the baryonic jet type in Burgess et al. (2014) with q = 1.24±0.11,
+which is also consistent with our finding in the current analysis.
+Our time-integrated spectral analysis indicates that all five bursts
+belong to the HD jet type. This finding is quite interesting since only
+a subset of GRBs (a fairly low percentage) have an observed ther-
+mal component in their spectral analysis, as suggested by several
+statistical studies (e.g., Li 2022a). Recently, (Li 2022a) has made a
+great effort to collect a complete GRB sample in which all bursts
+were detected by Fermi/GBM with known redshift, and created a
+spectral parameter catalog based on their model-wise properties. He
+discovered that ∼ 5% (7/153) of the analyzed bursts were found to
+require a subdominant thermal component in their time-integrated
+spectral analysis, including GRB 110721A. Our results imply that
+high-degree polarization measurements may also be associated with
+a thermal component originating from photosphere emission. Our
+time-resolved spectral analysis, on the other hand, further suggests
+that two bursts exhibit the peak-KED pattern (GRB 110721A and
+GRB 160802A) while the other three bursts (GRB 110301A, GRB
+140206A, and GRB 160625B) display the KED-to-PFD transition
+pattern across the entire burst durations since all α indices are
+above the synchrotron limit early on (α >2/3), and then drop be-
+low the synchrotron limit later on (α <2/3), indicating a thermal-
+to-nonthermal transition signature. Interestingly, after carrying out
+a detailed time-resolved spectral analysis for a sample of the multi-
+pulsed bursts, Li (2019a) reported that the jet properties for a good
+fraction of the multi-pulsed bursts exhibit a transition from thermal
+to non-thermal component among pulses within a single burst, and
+claimed that such “transition" jet properties are clearly observed in
+four bursts (GRB 140206B, GRB 140329B, GRB 150330A, and
+GRB 160625B), and the polarization properties in those transition
+bursts would also differ.
+4.2 The Observed Parameter Correlations: Polarization
+Degrees versus Other Relevant Quantities
+The most interesting result that draws our attention is that all five
+bursts in our target sample have a relatively high-degree polarization
+measurement and are associated with the “HD” jet properties. In
+the following discussion, we, therefore, pay special attention to this
+interesting observation and its theoretical interpretation.
+5 Note that the results obtained for those lower-S spectra may not be robust,
+since the spectral fits would not be well determined due to lack of enough
+photons.
+Much evidence points towards the fact that correlation analysis
+plays a crucial role in the understanding of GRB physics as it pro-
+vides a crucial clue to revealing its nature (e.g., Amati et al. 2002;
+Yonetoku et al. 2004; Liang & Zhang 2005; Dainotti et al. 2008;
+Xu & Huang 2012; Zhang et al. 2012; Liang et al. 2015; Dain-
+otti & Amati 2018; Li 2022a,
+and references therein). Here, an
+attempt has been made to explore the correlations between the po-
+larization properties and several typical GRB observed quantities.
+For instance, the polarization degrees π correlated with (i) the cos-
+mological rest-frame peak energy (Ep,z) of the νFν prompt emission
+spectrum, (ii) the isotropic-bolometric-equivalent emission energy
+Eγ,iso, (iii) the magnetization parameter σ0, (iv) the blackbody tem-
+perature kT, (v) the redshift z, and (vi) the corresponding energy
+fluence Sγ. Using the same observed epoch during the prompt emis-
+sion phase, our target sample allows for a reasonable comparison.
+Our analysis includes the following steps. (1) For a given burst in our
+target sample, we first select the same time interval for the prompt
+emission data as for the polarization measurement. (2) We then at-
+tempt to perform a spectral fit to the selected prompt emission data
+using PL, BB, CPL, Band, PL+BB, CPL+BB, and Band+BB func-
+tions, respectively. The Band+BB model has an AIC/BIC-statistic
+improvement of at least 10 with respect to the Band-alone and other
+models for these bursts, which suggests Band+BB as the preferred
+model that would fit the data and a thermal component existing in
+the spectrum. Since the thermal flux ratio (FBB/Ftot) for these bursts
+is less than 50% (see Table 5), the thermal components thus are sub-
+dominant. Interestingly, Chattopadhyay et al. (2019) analyzed the
+prompt emission and polarization data of 11 bright bursts detected
+during the first year of operation of CZTI, and reported that of these
+bursts, four bursts (GRB 160106A, GRB 160509A, GRB 160802A,
+and GRB 160910A) in their spectral analysis showed a deviation
+from the Band model and an additional thermal blackbody is needed
+in order to model their spectrum more precisely. (3) With the spec-
+tral analysis done in Step (2), we are able to obtain the spectral peak
+energy (Ep,z) and the blackbody temperature (kT). Eγ,iso can also be
+calculated in the cosmological frame with a redshift known (GRB
+110721A and GRB 140206B), and with a k-correction applied by
+integrating the observed energy spectrum over 1 KeV/(1+z) to 10
+MeV/(1+z). We note here that for the remaining three bursts (GRB
+100826A, GRB 110301A, and GRB 160802A) with an unknown
+redshift, we use the Yonetoku relation (Yonetoku et al. 2004) to es-
+timate their redshift values (see Section 3.2). Using these hybrid-
+spectrum observed properties and following the method described
+in Gao & Zhang (2015) and Li (2020), we also calculate the mag-
+netization parameter σ0 for these bursts. Finally, with these Steps
+completed, we therefore present π-Ep,z (Fig.7a), π-Eγ,iso (Fig.7b),
+π-kTz (Fig.7c), π-σ0 (Fig.7d), π-z (Fig.7e), and π-Sγ (Fig.7f) plots
+in Figure 7. Interestingly, these scatter plots all seem to exhibit a
+monotonic power-law decay in their log-log space (except for the
+π-σ0 correlation), with a similar decay slope ranging from -0.40 to
+-0.20. Our results indicate that a higher Ep,z, Eγ,iso, and kTz tend to
+have a lower-degree polarization π. However, we should note that
+the sample size is too small to be reliable enough to support the de-
+rived conclusion, so the results may not be statistically significant.
+5 PHYSICAL IMPLICATION
+There are several parameters to impact on the degree of polariza-
+tion in GRBs including the geometry of the jet, its angular struc-
+ture, the bulk Lorentz factor of outflow material, the magnetic field
+configuration and observer’s point of view. Here, we consider an
+MNRAS 000, 1–?? (2023)
+
+Time-averaging Polarimetric and Spectral Properties of GRBs
+7
+ultra-relativistic axi-symmetric jet which lunched by a central en-
+gine weather a black hole or an rapidly rotating magnetar (e.g., Usov
+1992; Thompson 1994; Dai & Lu 1998; Wheeler et al. 2000; Zhang
+& Mészáros 2001; Liu et al. 2007; Metzger et al. 2008; Lei et al.
+2009; Metzger et al. 2011; Bucciantini et al. 2012; Lü & Zhang
+2014; Li et al. 2018). During the prompt emission, we have ultra-
+relativistic jet with bulk lorentz factor Γ ≫ 1 leading to strong beam-
+ing effect of GRB outflow materials where the Doppler factor can be
+approximated as
+δD ≈
+2Γ
+1+(Γ ˜θ)2 ,
+(6)
+Due to the relativistic beaming effect of GRB jets, the measured ra-
+diation energy of the bursts is smaller than its isotropic energy as
+Eγ = fbEγ,iso by a factor fb = ∆Ω/4π = (1 − cosθ j) ≈ θ2
+j/2 where
+θ j is the half opening angle of the ejecta. In principle, different
+GRBs can be viewed from different observing angles θobs with re-
+spect to the jet’s central axis. Only those observers whose line-of-
+sight (LOS) intersects the surface of the jet can detect the GRBs. In
+the ultra-relativistic regime, the observed emission mainly receives
+from a region that is limited to a cone with angular size ˜θ ≲ 1/Γ
+around LOS. At the early time prompt emission when the LOS in-
+tersects the jet surface, if θobs/θ j ≲ 1−(Γθ j)−1 and Γθ j ≳ O(10), the
+jet’s edge remains invisible to the observer Gill et al. (2018).
+In this case, the emission region can be approximated as an ex-
+panding thin spherical shell of width ∆ ≪ R/Γ2 (in the lab frame)
+in which particles cool relatively fast compared to the dynamical
+time scale of the system. As the GRB jet has slowed down signifi-
+cantly when the opening angle θ j ≃ Γ−1 then the jet break happens
+and the edge effects become important. The flux density measured
+by a distant observer from each fluid element in an infinite thin-shell
+approximation for the prompt emission is given by (Granot 2005)
+Fν(t) =
+(1+z)
+16π2d2
+L(z)
+�
+δ3
+DL′
+ν��(r)d ˜Ω,
+(7)
+where dL(z) is the luminosity distance of the source, and d ˜Ω =
+d ˜φd(cos ˜θ) is the solid angle with ˜θ and ˜φ as the polar angle and
+the azimuthal angle measured from the LOS, respectively. The co-
+moving spectral luminosity L′
+ν′(r) for the synchrotron emission is
+L′
+ν′(r) = L′
+ν′(R)
+�
+1−(ˆn′ · ˆB′)2� 1+α
+2 ,
+(8)
+where α = −dlog(Fν)/dlog(ν) is the spectral index, ˆn is the ob-
+server’s LOS in the comoving frame of the GRB jet and ˆB is the
+local direction of the magnetic field. The spectral luminosity L′
+ν′(R)
+in the comoving frame of the fluid in terms of frequency ν′ and the
+peak frequency ν′
+p at which the most of the power is emitted is given
+by
+L′
+ν′(R) = L′
+ν′p
+� ν′
+ν′p
+�−α
+,
+(9)
+here we consider a constant luminosity with a radius which the peak
+value L′
+ν′p. We assume to have synchrotron emission from accelerat-
+ing electrons in the magnetig field with isotropic velocity distribu-
+tion and the energy distribution as a power law ne ∝ γ−p.
+In our scenario, we assume that each pulse originates from a sin-
+gle thin shell and Γ can in principle change for different pulses. The
+state of polarization of a radiation field can be expressed in terms of
+the Stokes parameters I (intensity), Q and U (linear polarizations), V
+(circular polarization). In spite of linear polization, only a few mech-
+anisms can generate the high value of circular polarization in the
+usual scattering processes in GRBs, and the measured circular polar-
+ization has only been reported once in a GRB afterglow Wiersema
+et al. (2014). Therefore we will not consider circular polarization
+in this paper. Stokes parameters Q and U are differences in flux for
+two orthogonal directions on the sky which are coordinate dependent
+quantities (Rybicki & Lightman 2008; Westfold 1959), we define the
+local degree of linear polarization Π =
+�
+Q2 +U2/I where
+U
+I = Πsin2θp ,
+Q
+I = Πcos2θp ,
+θp = 1
+2 arctan
+�U
+Q
+�
+, (10)
+and θp is the polarization position angle (PA). The direction of the
+polarization vector in the synchrotron emission is orthogonal to the
+LOS of the observer ˆn and the local direction of the magnetic field
+ˆB in the jet,
+ˆΠ = (ˆn× ˆB)
+|ˆn× ˆB)|
+,
+(11)
+The polarization measurements can help in order to probe the
+magnetic field structure inside the shock wave. Moreover, the de-
+gree of the polarization depends on the GRB jet’s angular structure
+and the observer’s viewing angle from jet symmetry axis (Lazzati
+et al. 2004). The magnetic field structure in KED and PFD flows
+has a different origin and can be classified into three categories (Gill
+et al. 2018; Gill et al. 2021): (i) a locally ordered magnetic field
+(Bord) with angular coherent length θ j > θB ≳ 1/Γ, (ii) a toroidal
+magnetic field (Btrod) which has an ordered axisymmetric configura-
+tion in the transverse direction with respect to the jet (iii) a tangent
+magnetic field which could be in principle parallel (B∥) or perpen-
+dicular (B⊥) to the local fluid velocity. In the PFD the magnetic field
+is dynamically dominated and usually has a large coherence length
+such as Btrod which can be produced by a rotating central engine or
+in a high magnetized flow, other locally and globally ordered field
+configuration are also possible in this case. On the other hand in
+KED we may have a tangled magnetic field structure with B⊥ or/and
+B∥ components, however generating such an anisotropic field con-
+figurations in shock waves seems to be challenging (Gill & Granot
+2020). A globally ordered magnetic field may naturally be advected
+from near the central source, while the random magnetic fields gen-
+erated in the shock dissipation region (Kumar & Zhang 2015; Geng
+et al. 2018; Gill et al. 2018; Fan et al. 2008). The magnetic field
+structures that are generated at relativistic collision-less shocks, due
+to the two-stream instabilities, are expected to be tangled within the
+shock plane (Medvedev & Loeb 1999).
+The degree of linear polarization generated in the synchrotron
+emission from an isotropic electron distribution with power-law en-
+ergy spectrum, and for a given direction of magnetic field is given
+by (Rybicki & Lightman 2008; Westfold 1959)
+Πlin
+max =
+α+1
+α+5/3 =
+peff +1
+peff +7/3,
+(12)
+where peff = 2α + 1 is the effective power-law index of the electron
+distribution.
+peff =
+�
+�
+�
+2,
+νc < ν < νm,
+slow cooling
+p,
+νm < ν < νc,
+fast cooling
+p+1,
+ν > max(νc,νm),
+either fast or slow cooling
+(13)
+and, therefore
+Πlin
+max =
+�
+�
+�
+�
+�
+9/13,
+νc < ν < νm,
+fast cooling
+(p+1)
+(p+7/3),
+νm < ν < νc,
+slow cooling
+(p+2)
+(p+10/3),
+ν > max(νc,νm),either fast or slow cooling
+(14)
+MNRAS 000, 1–?? (2023)
+
+8
+Li & Shakeri
+This polarization may originate from a very small region (point-
+like emitter) in which the magnetic field has a specific orientation.
+Only in the case of ordered magnetic field with the coherence length
+comparable or larger than the visible surface of the emitting region,
+the highest value of the polarization Πlin
+max in Eq. (19) can be gen-
+erated. The photon index in the Synchrotron radiation is limited to
+−1/3 ⩽ α ≲ 3/2 which regarding Eq. (19) leads to the maximum
+degree of polarization 50% ≲ Π ≲ 75%. In the left panel of Fig-
+ure (8), we see predicted polarization values using this theoretical
+model with observed data using our target sample. Here α indices
+are obtained using the spectral analysis defined in §4. We find that
+the observed data are well distributed along the line predicted by this
+model.
+In general, the measured polarization is obtained by integrating
+the local stokes parameters over the flux of the GRB jet as
+Π(tf ) =
+¯
+Q(tf )
+I(t f ) =
+�
+dFν cos2θp
+�
+dFν
+,
+(15)
+assuming to have an axisymmetric flow and taking into account sym-
+metry consideration, we see that ¯U = 0 and consequently the instan-
+taneous total degree of the linear polarization is ¯Π = | ¯Q|/I. We per-
+form an integration over the equal time surface (EATS) for a single
+pulse
+�
+¯
+Q(t)dt/
+�
+I(t)dt in order to obtain pulse integrated polariza-
+tion of the prompt emission which leads to
+Πord
+Πmax =
+� ymax
+0
+dy(1+y)−2−α �
+dφΛ(y,φ)cos2θp
+� ymax
+0
+dy(1+y)−2−α �
+dφΛ(y,φ)
+,
+(16)
+The above formula is valid for the prompt emission from an ultra-
+relativistic thin-shell for an on-axis observer (θobs = 0) where ymax =
+(Γθmax)2 and θmax defined as the maximum angle from LOS Granot
+(2003a). The factor Λ(y,φ) is an average over the magnetic field
+orientations in the plane of the ejecta as
+Λ(y,φ) ≡ ⟨(1−(ˆn′ · ˆB′)2)
+1+α
+2 ⟩ ,
+(17)
+The polarization angle θp and Λ(y,φ) take different forms regarding
+the configuration of the magnetic field in the plane the GRB jet. In
+the case of an ordered magnetic field Bord we have :
+Λord(y,φ) ≈
+�
+(1−y
+1+y)2 cos2 φ+sin2 φ
+� 1+α
+2
+,
+(18)
+θp = φ+arctan
+�
+(1−y
+1+y)cotφ
+�
+.
+(19)
+The time-integrated linear polarization in the presence of an ordered
+magnetic field in the plane normal to the jet velocity is plotted as
+a function of the spectral index in the right panel of Fig. (8). As it
+is seen the polarization degree increases towards higher values of α
+and lower values of ymax which can cover the observed polarization
+of GRB 110721A, GRB 160802A, and GRB 110301A. Therefore
+for a configuration with the globally ordered magnetic field, high
+values of linear polarization even larger than 50% are obtainable.
+The degree of polarization for a magnetic field with locally tan-
+gled or random configuration is obtained by averaging over all direc-
+tions of the local magnetic field within the plane of the shock (Granot
+& Königl 2003a; Sari 1999; Gruzinov 1999; Nava et al. 2016). The
+presence of a random magnetic field leads to negligible values of net
+linear polarization measured by an on-axis observer. In the case of a
+random field behind the shock wave only if the observer is off-axis
+and the circular symmetry is broken, non-zero net polarization is
+measurable. The total linear polarization arising from the whole jet
+which is subjected to a random field with a direction perpendicular
+to the jet velocity is given by Granot (2003a)
+Π⊥
+Πmax =
+� y2
+y1 dy(1+y)−2−α sin[2Ψ1(y)]G(y,α)
+Θ(1−ζ)
+� y1
+0
+dy H(y,α)
+(1+y)α+2 +
+� y2
+y1 dy dy H(y,α)
+(1+y)α+2
+� π−Ψ(y)
+π
+�,
+(20)
+where Θ(1 − ζ) is the Heaviside step-function with ζ ≡ θobs/θ j as a
+parameter to define observer’s point of view, and
+G(y,α) = 1
+2π
+� π
+0
+dφ
+�(1−y)2
+(1+y)2 cos2 φ−sin2 φ
+��
+1− 4ycos2 φ
+(1+y)2
+� α−1
+2
+, (21)
+H(y,α) =
+� π
+0
+dφ
+�
+1− 4ycos2 φ
+(1+y)2
+� 1+α
+2
+,
+(22)
+cosΨ(y) = (1−ζ2)y j −y
+2ζ√yy j
+.
+(23)
+In above expressions y1,2 = (1∓ζ)2yj and y j = (Γθ j)2. The variation
+of the linear polarization in the presence of a random field configura-
+tion measured by an off-axis observer is displayed in Fig. (9). In the
+left panel, the spectral indices are selected to be consistent with aver-
+age values reported in Table (5) for our target sample and for y j = 10.
+In the right panel, yj is changed while α = 1, it is found that the ap-
+peared peak has a width in order of 1/√y j. From Fig. (9), we see that
+the polarization degree is limited to small values for ζ < 1 while it
+is sharply increased for ζ ≈ 1 and finally reaches to an asymptotic
+limit at ζ > O(1). It is seen that the Synchrotron radiation with B⊥
+can potentially generate wide range of polarization values from low
+levels to moderate values which cover observed values associated to
+our sample. In principle, various viewing angles θobs and different
+angular structures of the jet affect the measured fluence of GRBs.
+Note that the fluence significantly decreases for a top-hat jet view-
+ing from outside the jet’s sharp edge, so high levels of polarization
+in off-axis jets may only be obtainable in very close bursts. In fact,
+the detectibility of GRB polarization needs high-fluence sources and
+usually, the fluence rapidly drops below the detector threshold for a
+large off-axis observer.
+The time-resolved spectral analysis in §4.1 showed thermal to
+non-thermal (KED-to-PFD) transitions in our sample where a sub-
+dominant component of the thermal emission during bursts is ob-
+served. Observing hard values of the spectral indices during the
+bursts can be served as hints that LOS is not highly off-axis, since
+high latitude emission leads to a softer spectrum Lundman et al.
+(2013).
+As it was reported in §4.2, a correlation between the polariza-
+tion and the isotropic energy π-Eγ,iso (Fig.7b) has been observed
+within our sample, it is worth mentioning that higher polariza-
+tion values are recorded for closer bursts GRB 110301A (z=0.36),
+GRB 110721A (z=0.382), GRB 160802A (z=0.90) and lower val-
+ues for farther sources GRB 140206B (z=2.73) and GRB 100826A
+(z=2.3) (Fig.7e). The observed fluences of GRB 140206B and GRB
+100826A is higher than other sources (see Table. 4 and Fig.7f) and
+due to their higher redshifts ζ can not obtain large values, however,
+low values of ζ would be enough to reproduce their measured polar-
+izations.
+The local degree of linear polarization for a tangled or random
+field configuration for a thin ultrarelativistic shell modeling of the
+prompt emission by assuming α = 1 is obtained by averaging over
+all local magnetic field directions as
+Πlin
+rnd = Πlin
+max
+(b−1)sin2 θB
+2+(b−1)sin2 θB
+(24)
+MNRAS 000, 1–?? (2023)
+
+Time-averaging Polarimetric and Spectral Properties of GRBs
+9
+where b ≡ 2⟨B2
+∥⟩/⟨B2
+⊥⟩ denotes the anisotropy of the magnetic field
+distribution as the ratio of the parallel B∥ to the perpendicular B⊥
+components with respects to the shock direction, and θB is the an-
+gle between the LOS from the observer and the direction of the
+shock Sari (1999); Gruzinov (1999). In the case of a globally ordered
+magnetic field configuration aligned with the jet direction (B → B∥,
+b → ∞), Eq. (24) returns back to Eq. (19) and gives the maximum
+value of the linear polarization.
+The polarized emission may also originate from independent
+magnetic patches with various field orientation Li (2022a) where
+magnetic patches are locally coherent but distributed randomly in
+observed emission region. In this case the measured polarization
+from different patches is estimated as Π = Πmax/
+√
+N, where N is the
+number of magnetic patches or equivalently multiple pulses where
+the coherence length of the magnetic field is as large as the emis-
+sion region in a single pulse and observed polarization is an average
+over multiple pulses (Gruzinov & Waxman 1999; Granot & Königl
+2003b).
+The magnetic field which is generated within IS for KED jets has
+usually a coherence length much smaller than the angular size of the
+emission region which causes negligible net polarization. It has been
+shown that even by taking into account the angular structure of the
+flow the polarization is limited to Π ≲ 20% for photospheric emis-
+sion of a relativistically expanding fireball Ito et al. (2014); Lund-
+man et al. (2014a); Parsotan et al. (2020). The observed high val-
+ues of the polarization for GRB 110721A and GRB 160802A while
+they show the peak-KED pattern cannot be explained simply by the
+sub-photospheric dissipation model based on Comptonisation. Be-
+cause the multiple scatterings at large optical depths region leads to
+wash out the directionality of polarization vectors (Lundman et al.
+2018b). To explain the strong polarized signals, models invoking dis-
+sipation of ordered magnetic field are favored (Lyutikov et al. 2003;
+Zhang & Yan 2011b; McKinney & Uzdensky 2012). A structured jet
+photosphere model may also generate polarized photons via Comp-
+ton scattering but with a different energy-dependence compare to
+the synchrotron model in the ordered magnetic field (Chang et al.
+2014b,a, 2013).
+The Jitter radiation emitted by ultra-relativistic electrons acceler-
+ating in a small-scale random magnetic field (Medvedev 2000), can
+also generate a hard energy spectrum with the photon index as high
+as α = +0.5. Due to the random distribution of the magnetic field,
+jitter radiation is highly symmetric in the electron radiative plane,
+leading to the vanishing polarization degree for an on-axis observer
+(Mao & Wang 2013, 2017; Mao et al. 2018). The maximum level
+of polarization is obtainable when the emitting plane is viewed from
+the edge on, it can even reach up to 90% (Prosekin et al. 2016).
+However, for smaller off-axis viewing angles which can yield mea-
+surable fluences, jitter radiation causes almost negligible polariza-
+tion degree. Meanwhile, regardless of the viewing angle the Jitter
+radiation cannot produce the observed high degree of polarisation
+close to the spectral peak energy of the jet.
+To summarize, polarization features can be explained either by the
+synchrotron radiation in the ordered/random magnetic field (Granot
+2003b; Granot & Königl 2003b; Nakar et al. 2003), the jet structure
+(Lazzati & Begelman 2009), or the observer’s viewing angle with re-
+spect to the jet (Lazzati et al. 2004), even in the case of thermal radi-
+ation from the jet photosphere (Lundman et al. 2014b). For a hybrid
+spectrum which include thermal and non-thermal components, we
+expect to see relatively high values of the polarization in the prompt
+emission which can be produced by synchrotron emission mecha-
+nism in the ordered magnetic field of the jet, and for random field
+configurations only for off-axis observers (Gill et al. 2021). How-
+ever, the spectral properties of our target sample demonstrated that
+off-axis observations specially for the large viewing angle is not the
+case, and the observed values of the polarization most probably is
+a hint of the ordered magnetic field originating from the central en-
+gine. Since from PFD jets towards HD and KED jets, polarization
+washout effects are increased gradually due to thermal photons, we
+would expect that the inequality πKED ≲ πHD ≲ πPED is satisfied if
+other conditions are fixed for a given jet. Due to the different de-
+grees of polarization predicted by different emission models in var-
+ious energy bands, it is essential to have a high-sensitivity gamma-
+ray polarimeter with a wide band-pass to detect energy-dependent
+polarization signals and constrain different models (Zhang 2014).
+However, it should be noted that due to several free parameters in
+polarization models, upcoming more precise observations and theo-
+retical investigations are needed to discriminate between competing
+models in order to explain observed joint polarization and spectral
+properties.
+6 CONCLUSION
+Early polarization observations during the prompt emission phase
+play a crucial role in understanding the radiation mechanism and
+jet composition of GRBs. Observations over the past few decades
+suggest that the jet composition of GRBs may have diverse prop-
+erties. If the jet composition is matter-dominated (i.e., a fireball),
+the GRB prompt emission spectra would include a bright thermal
+component originating from the fireball photosphere. Alternatively,
+if the jet composition is Poynting-flux-dominated, the GRB prompt
+emission spectra would include a dominant non-thermal compo-
+nent originating from the synchrotron radiation. Moreover, if the jet
+composition is hybrid-dominated, the GRB prompt emission spec-
+tra would include a thermal component originating from the fire-
+ball photosphere and a non-thermal component originating from the
+synchrotron radiation. It is highly speculated that the prompt emis-
+sion is likely expected to be strongly polarized owing to its non-
+thermal origin. Consequently, a different level of polarization de-
+grees (πKED ≲ πHD ≲ πPED) during the prompt emission phase is nat-
+urally expected due to the different types of jet composition. In this
+paper, we have collected a GRB sample in which all the bursts de-
+tected by Fermi/GBM and whose polarization detection in the emis-
+sion region was also reported in the literature, containing five inter-
+esting bursts (GRB 100826A, GRB 110301A, GRB 110721A, GRB
+140206A, and GRB 160802A). Using the time-averaging polariza-
+tion observations and selecting the same epoch for the GBM data
+taken during the prompt emission phase, we then attempted to ex-
+plore the correlations between jet properties and polarization prop-
+erties of GRBs and aimed to confirm the validity of this correlation
+(πKED ≲ πHD ≲ πPED) from observations.
+We first performed a detailed time-averaged spectral analysis for
+each burst in our target sample by using several frequency-used GRB
+spectral models and selected the best one by using information crite-
+ria (AIC and BIC). The jet properties of GRBs can be classified into
+three categories based on their spectral analysis: the “KED”, “PFD”,
+and “HD” types. Using the spectral properties we then inferred their
+jet properties and discovered that all five bursts belong to the “HD”-
+jet type. The lack of the other two types of jets (KED and PED)
+prevents us from validating this correlation (πKED ≲ πHD ≲ πPED).
+Hopefully, upcoming instruments will provide high-sensitivity po-
+larization observations in the future, leading to well-sampled, well-
+studied data sets, enabling such statistical analysis.
+We next conducted a time-resolved spectral analysis for each in-
+MNRAS 000, 1–?? (2023)
+
+10
+Li & Shakeri
+dividual burst by dividing the emission period into multiple-time
+slices using the BBlocks method using the Band-alone model. Our
+refined time-resolved spectral analysis, on the other hand, further
+suggested that the “HD”-type has two subcategories: the peak-KED
+pattern and the KED-to-PFD transition pattern. In our attempt to as-
+sess the jet properties of GRBs using Band-α evolution, we discov-
+ered that two bursts exhibit the peak-KED pattern (GRB 110721A
+and GRB 160802A) whereas the other three bursts show the KED-
+to-PFD transition pattern (GRB 110301A, GRB 140206A, and GRB
+160625B). All five bursts found in the “HD”-type imply that the pho-
+tosphere emission may also be a possible mechanism to power the
+high-degree polarization observation.
+We also made an attempt to explore the correlations between the
+polarization properties and several typical GRB observed quantities.
+Using the same observed epoch during the prompt emission phase,
+our target sample allows for a reasonable comparison. The corre-
+lations we attempted to study included the polarization degrees π
+correlated with (i) the cosmological rest-frame peak energy (Ep,z)
+of the νFν prompt emission spectrum, (ii) the isotropic-bolometric-
+equivalent emission energy Eγ,iso, (iii) the magnetization parameter
+σ0, (iv) the blackbody temperature kT, (v) the redshift z, and (vi) the
+corresponding energy fluence Sγ. As a result, we discovered that a
+higher Ep,z, Eγ,iso, and kTz tend to have a lower-degree polarization
+π.
+Lastly, we discovered that all five bursts in our target sample have
+a relatively high-degree polarization detection that seems to corre-
+late with the “HD”-jet type. If it is an intrinsic characteristic of
+GRBs, this could provide a clue to studying the radiation mechanism
+and jet composition of GRBs. We have also discussed some physical
+interpretations of this interesting phenomenon. Since the configura-
+tion of the magnetic field inside the jet is one of the crucial param-
+eters to determine the polarization degree, we discussed two main
+configurations (i.e. ordered and random fields), and their connec-
+tion to the jet composition is clarified. We considered polarization
+patterns as a function of different dynamical parameters associated
+to the outflow materials, the spectral indices and the observer’s LOS
+with respect to the jet. Combining the spectral analysis and the polar-
+ization measurements allowed us to find out the detection of polar-
+ization values Π > 50% during prompt emission of GRB 160802A,
+GRB 110721A and GRB 110301A is a piece of strong evidence for
+the synchrotron emission mechanism in the presence of an ordered
+magnetic field which can be advected from the GRB central engine.
+Regarding the different properties of our target sample, we conclude
+that geometrical effects and large off-axis observations are unlikely
+responsible for the measured polarizations assuming random mag-
+netic fields within the jets.
+Finally, there are some caveats that are worth mentioning when
+applying our analysis. (i) Spectrum. We have resolved the jet proper-
+ties based on the low-energy spectrum. However, it may be difficult
+to classify jets as either KED or PFD jets based on the spectral index
+alone. Indeed, a photospheric quasi-thermal component would have
+a harder low-energy spectral index as compared to an optically-thin
+synchrotron, but that does not guarantee that the jet is KED (an ex-
+ample, see Gill et al. 2020). (ii) Polarization. The degree of polariza-
+tion ultimately probes the (local) structure of the B-field in the emis-
+sion region. An ordered field would necessarily yield high polariza-
+tion whereas a tangled field would yield a very small polarization. It
+is unclear, however, whether these field configurations are exclusive
+to a given jet configuration (or a particular level of magnetization).
+In addition, the angular structure of the jet also plays an important
+role in governing the observed polarization. Thus, due to the large
+range of model parameters, it is difficult to attribute a given level of
+polarization to a given jet composition. More discussion is provided
+in a recent review article (Gill et al. 2021). (iii) Different instrument
+analysis. Currently, it is not clear why different instruments, namely,
+POLAR, IKAROS-GAP, and ASTROSAT/CZTI, are finding differ-
+ent levels of polarization for a small sample of GRBs (Chattopad-
+hyay et al. 2019). There is no consensus. POLAR is finding a rather
+low-level polarization, which is consistent with zero within 3σ of
+their quoted central values, whereas both IKAROS and AstroSAT
+are finding much higher levels. Hard X-ray to soft gamma-ray po-
+larization measurements are very tricky and the analysis has to be
+carried out very carefully. As such, some of these measurements are
+probably not representative of GRBs and need to be further verified
+by future more precise instruments. (iv) Time-resolved polarization
+analysis. In the current analysis, none of the cases have shown time-
+resolved polarization measurements. Even though the GRBs in our
+target sample have time-resolved spectral indices, not having corre-
+sponding polarization measurements makes it difficult to ascertain
+the properties of the B-field and outflows.
+Acknowledgements. We thank Ramandeep Gill, Jonathan Gra-
+not, Rahim Moradi, Mi-Xiang Lan, Asaf Pe’er, Jin-Jun Geng,
+Christoffer Lundman, Remo Ruffini, and ICRANet members for
+many discussions on GRBs physics and phenomena. This research
+made use of the High Energy Astrophysics Science Archive Re-
+search Center (HEASARC) Online Service at the NASA/Goddard
+Space Flight Center (GSFC).
+Data availability. The data underlying this article will be shared
+on reasonable request to the corresponding author.
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+MNRAS 000, 1–?? (2023)
+
+12
+Li & Shakeri
+Table 1 A sample of GRB polarimetric observations
+GRB
+PD
+PA
+Energy band
+Time
+Significance
+Instrument
+Ref.
+z
+(Fermi ID)
+(π%)
+(◦)
+(t-t0)
+(σ)
+(For polarization)
+100826A(957)
+27±11
+159±18, 75±20
+γ-ray
+0-100s
+2.9σ
+IKAROS-GAP
+Yonetoku et al. (2011b)
+NA
+110301A(214)
+70±22
+73±11
+γ-ray
+0-7s
+3.7σ
+IKAROS-GAP
+Yonetoku et al. (2012a)
+NA
+110721A(200)
+84+16
+−28
+160±11
+γ-ray
+0-11s
+3.3σ
+IKAROS-GAP
+Yonetoku et al. (2012a)
+0.382
+140206A(275)
+> 28
+80±15
+γ-ray
+4-26s
+90% confidence
+INTEGRAL-IBIS
+Götz et al. (2014)
+2.73
+160802A(259)
+85±29
+∼-32
+hard X-rays
+0-20.34s
+∼3σ
+AstroSat-CZTI
+Chand et al. (2018)
+NA
+MNRAS 000, 1–?? (2023)
+
+Time-averaging Polarimetric and Spectral Properties of GRBs
+13
+Table 2 Estimated values of redshift using the Yonetoku relation
+GRB
+tstart∼tstop
+S
+Eobs
+p
+Fobs
+p
+kc
+Lp
+z
+(s)
+(keV)
+(erg cm−2 s−1)
+(erg s−1)
+(Estimated)
+100826957
+18.208∼22.288
+107.97
+459±20
+(1.42±0.10)×10−5
+0.85
+(1.27±0.10)×1053
+2.3
+110301214
+3.876∼4.126
+107.60
+126±6
+(1.36±0.16)×10−5
+1.02
+(1.46±0.17)×1053
+0.36
+160802259
+0.962∼1.171
+73.79
+385±39
+(3.04±0.80)×10−5
+0.95
+(3.05±0.80)×1053
+0.90
+MNRAS 000, 1–?? (2023)
+
+14
+Li & Shakeri
+Table 3 Comparison of AIC/BIC between the best model and other models
+GRB
+t1 ∼ t2
+AIC/BIC(1)
+AIC/BIC(2)
+AIC/BIC(3)
+AIC/BIC(4)
+AIC/BIC(5)
+AIC/BIC(6)
+AIC/BIC(7)
+(s)
+(PL)
+(BB)
+(CPL)
+(Band)
+(PL+BB)
+(CPL+BB)
+(Band+BB)
+100826A(957)
+0∼100
+11662/11670
+12225/12232
+6819/6830
+6706/6721
+7819/7834
+6693/6712
+6638/6661
+110301A(214)
+0∼7
+10969/10978
+20029/20038
+5342/5354
+5284/5301
+6124/6140
+5302/5323
+5254/5278
+110721A(200)
+0∼11
+7319/7327
+14194/14203
+5694/5707
+5541/5557
+7323/7340
+5524/5544
+5504/5528
+140206A(275)
+4∼26
+12649/12657
+20413/20421
+7617/7630
+7431/7448
+8549/8566
+7345/7366
+7293/7317
+160802A(259)
+0∼20.34
+6840/6847
+7480/7487
+3984/3994
+3898/3912
+4383/4397
+3839/3856
+3840/3861
+MNRAS 000, 1–?? (2023)
+
+Time-averaging Polarimetric and Spectral Properties of GRBs
+15
+Table 4 Global properties of the Sample
+GRB
+T90
+Fluence
+Detectors
+∆Tsrc
+[∆T(bkg,1),∆T(bkg,2)]
+Spectral model
+(Fermi ID)
+(s)
+(erg cm−2)
+(s)
+(s)
+(Preferred)
+100826A(957)
+84.993±0.724
+(1.64±0.01)×10−4
+n7(n8)b1
+(0 to 100)
+(-20 to -10, 200 to 250)
+Band+BB
+110301A(214)
+5.693±0.362
+(3.59±0.01)×10−5
+n7(n8)nbb1
+(0 to 7)
+(-20 to -10, 40 to 60)
+Band+BB
+110721A(200)
+21.822±0.572
+(3.70±0.01)×10−5
+(n6)n7n9b1
+(0 to 11)
+(-20 to -10, 40 to 60)
+Band+BB
+140206A(275)
+146.690±4.419
+(1.23±0.01)×10−4
+n0(n1)n3b0
+(4 to 26)
+(-40 to -20, 70 to 90)
+Band+BB
+160802A(259)
+16.384±0.362
+(6.84±0.01)×10−5
+(n2)b0
+(0 to 20.34)
+(-20 to -10, 60 to 80)
+Band+BB
+MNRAS 000, 1–?? (2023)
+
+16
+Li & Shakeri
+Table 5 Spectral Fit Results of the Sample with the Band+BB Model.
+GRB
+t1∼t2
+S
+K
+α
+Ep
+β
+K
+kT
+Flux
+Flux ratio
+(s)
+(Band)
+(Band)
+(Band)
+(Band)
+(BB)
+(BB)
+(FBB/Ftot)
+(s)
+(ph.s−1.cm−2.keV−1)
+(keV)
+(ph.s−1.cm−2.keV−1)
+(keV)
+(erg.cm−2.s−1)
+100826957
+0∼82
+102.7
+(3.04+0.16
+−0.16)×10−2
+-0.84+0.04
+−0.04
+518+46
+−46
+-2.28+0.07
+−0.07
+(5.54+1.97
+−1.93)×10−5
+21+2
+−2
+(3.20+0.34
+−0.31)×10−6
+0.03+0.03
+−0.03
+110301214
+0∼7
+276.8
+(3.73+0.34
+−0.30)×10−1
+-0.72+0.07
+−0.07
+114+2
+−3
+-2.87+0.08
+−0.07
+(1.14+0.57
+−0.49)×10−2
+7+1
+−1
+(5.90+0.73
+−0.69)×10−6
+0.04+0.03
+−0.03
+110721200
+0∼11
+114.5
+(3.01+0.09
+−0.09)×10−2
+-1.20+0.02
+−0.02
+1620+234
+−229
+-2.19+0.10
+−0.10
+(1.73+0.33
+−0.34)×10−5
+33+2
+−2
+(6.94+0.66
+−0.63)×10−6
+0.03+0.01
+−0.01
+140206275
+4∼26
+170.2
+(3.87+0.11
+−0.11)×10−2
+-1.06+0.02
+−0.02
+679+43
+−43
+-2.32+0.08
+−0.08
+(4.78+0.53
+−0.52)×10−5
+27+1
+−1
+(4.57+0.25
+−0.24)×10−6
+0.05+0.01
+−0.01
+160802259
+0∼20.34
+151.7
+(3.89+0.21
+−0.21)×10−2
+-1.00+0.03
+−0.03
+515+44
+−44
+-3.23+0.73
+−0.74
+(9.11+1.22
+−1.21)×10−5
+25+1
+−1
+(3.85+0.55
+−0.41)×10−6
+0.09+0.02
+−0.02
+MNRAS 000, 1–?? (2023)
+
+Time-averaging Polarimetric and Spectral Properties of GRBs
+17
+1
+0
+2
+4
+6
+8
+10
+redshift
+10
+1
+100
+101
+102
+103
+GRB 100826A
+Estimated value
+0
+2
+4
+6
+8
+10
+redshift
+10
+1
+100
+101
+102
+103
+GRB 110301A
+Estimated value
+0
+2
+4
+6
+8
+10
+redshift
+10
+1
+100
+101
+102
+103
+GRB 160802A
+Estimated value
+Figure 1. Estimated redshift using the Yonetoku relation for three bursts (GRB 100826A, GRB 110301A, and GRB 160802A). The yellow and cyan lines
+represent the left and right function of Eq.(5), and their intersection point (purple color) is the estimated value of redshift.
+MNRAS 000, 1–?? (2023)
+
+18
+Li & Shakeri
+1
+20
+0
+20
+40
+60
+80
+100
+120
+140
+Time (s)
+0
+20
+40
+60
+80
+100
+=27±11%
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+ (S > 20)
+ (10 < S < 20)
+ (S < 10)
+100826A
+101
+102
+103
+104
+105
+Photon Energy - keV
+100
+101
+102
+103
+104
+keV × [keV
+1S
+1cm
+2]
+Band fits
+Blackbody
+Band+Blackbody
+NAI7
+NAI8
+BGO1
+Figure 2. Left panel: prompt emission GBM light curve (overlaid in gray) and polarization observations in γ-ray/hard X-ray energy bands (cyan shaded area),
+as well as the temporal evolution of α based on the time-resolved spectral analysis. The horizontal dashed line represents the limiting value of α = −2/3 for
+electrons in the slow-cooling regime. Right panel: the spectral data and its best-fit model (Band+BB) during the time epoch (see Table 1 and Table 4) of the
+matching polarization observations.
+MNRAS 000, 1–?? (2023)
+
+Time-averaging Polarimetric and Spectral Properties of GRBs
+19
+1
+5.0
+2.5
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+Time (s)
+0
+20
+40
+60
+80
+100
+=70±22%
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+ (S > 20)
+ (10 < S < 20)
+ (S < 10)
+110301A
+101
+102
+103
+104
+105
+Photon Energy - keV
+100
+101
+102
+103
+104
+keV × [keV
+1S
+1cm
+2]
+Band fits
+Blackbody
+Band+Blackbody
+NAI7
+NAI8
+NAIb
+BGO1
+nai
+bgo
+Figure 3. Same as Figure 2 but for GRB 110301A.
+MNRAS 000, 1–?? (2023)
+
+20
+Li & Shakeri
+1
+15
+10
+5
+0
+5
+10
+15
+20
+25
+30
+Time (s)
+0
+20
+40
+60
+80
+100
+= (84+16
+28)%
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+ (S > 20)
+ (10 < S < 20)
+ (S < 10)
+110721A
+101
+102
+103
+104
+105
+Photon Energy - keV
+101
+102
+103
+104
+keV × [keV
+1S
+1cm
+2]
+Band fits
+Blackbody
+Band+Blackbody
+NAI6
+NAI7
+NAI9
+BGO1
+nai
+bgo
+Figure 4. Same as Figure 2 but for GRB 110721A.
+MNRAS 000, 1–?? (2023)
+
+Time-averaging Polarimetric and Spectral Properties of GRBs
+21
+1
+20
+0
+20
+40
+60
+80
+100
+120
+140
+Time (s)
+0
+20
+40
+60
+80
+100
+ (Upper limit)
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+ (S > 20)
+ (10 < S < 20)
+ (S < 10)
+140206A
+101
+102
+103
+104
+105
+Photon Energy - keV
+101
+102
+103
+104
+keV × [keV
+1S
+1cm
+2]
+Band fits
+Blackbody
+Band+Blackbody
+NAI0
+NAI1
+NAI3
+BGO0
+nai
+bgo
+Figure 5. Same as Figure 2 but for GRB 140206A.
+MNRAS 000, 1–?? (2023)
+
+22
+Li & Shakeri
+1
+20
+10
+0
+10
+20
+30
+40
+Time (s)
+0
+20
+40
+60
+80
+100
+= 85 ± 29
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+ (S > 20)
+ (10 < S < 20)
+ (S < 10)
+160802A
+101
+102
+103
+104
+105
+Photon Energy - keV
+101
+102
+103
+104
+keV × [keV
+1S
+1cm
+2]
+Band fits
+Blackbody
+Band+Blackbody
+NAI2
+BGO0
+nai
+bgo
+Figure 6. Same as Figure 2 but for GRB 160802A.
+MNRAS 000, 1–?? (2023)
+
+Time-averaging Polarimetric and Spectral Properties of GRBs
+23
+1
+101
+102
+103
+104
+Ep, z[keV]
+100
+101
+102
+103
+a
+100826A
+110301A
+110721A
+140206B
+160802A
+100
+101
+102
+103
+Power-law index= -0.41±0.24
+1051
+1052
+1053
+1054
+1055
+E , iso (erg)
+100
+101
+102
+103
+b
+100826A
+110301A
+110721A
+140206B
+160802A
+100
+101
+102
+103
+Power-law index= -0.20±0.04
+100
+101
+102
+103
+104
+KT,z[keV]
+100
+101
+102
+103
+c
+100826A
+110301A
+110721A
+140206B
+160802A
+100
+101
+102
+103
+Power-law index= -0.32±0.18
+100
+101
+102
+1+
+0
+100
+101
+102
+103
+d
+100826A
+110301A
+110721A
+140206B
+160802A
+100
+101
+102
+103
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+redshift
+100
+101
+102
+103
+e
+100826A
+110301A
+110721A
+140206A
+160802A
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+Fluence (erg cm
+2)
+100
+101
+102
+103
+f
+100826A
+110301A
+110721A
+140206A
+160802A
+Figure 7. Scatter plots of polarization degree π versus several other observed quantities: (a) the cosmological rest-frame peak energy (Ep,z) of the νFν prompt
+emission spectrum, (b) the isotropic-bolometric-equivalent emission energy Eγ,iso, (c) the magnetization parameter σ0, (d) the blackbody temperature kT, (e)
+the redshift z, and (d) the corresponding energy fluence Sγ. Data points with different colors indicate the different bursts in our target sample. The solid lines
+(grey) are the best fit using the power-law model with 2σ (95% confidence interval) error region (shadow area).
+MNRAS 000, 1–?? (2023)
+
+24
+Li & Shakeri
+1
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+100
+101
+102
+100826A
+110301A
+110721A
+160802A
+140206A
+100
+101
+102
+lin
+max =
++ 1
++ 5/3
+ymax=1
+ymax=2
+ymax=5
+ymax=10
+ymax=102
+110721A
+160802A
+110301A
+100826A
+140206A
+0.0
+0.5
+1.0
+1.5
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+α
+Π/Πmax
+Figure 8. Left: The maximum degree of the linear polarization applying synchrotron emission model (Πlin
+max Eq. (19)) with observed data using α indices based
+on a time-integrated spectral analysis. Right: Time integrated polarization degree in the presence of an ordered magnetic field Bord with in the plane of ejecta
+(Eq. (19)) measured by an on-axis observer (θobs = 0), the evolution of the polarization is plotted in terms of α for different values of ymax = (Γθmax)2.
+MNRAS 000, 1–?? (2023)
+
+Time-averaging Polarimetric and Spectral Properties of GRBs
+25
+1
+α=0.72
+α=0.84
+α=1
+α=1.2
+Random B⟂
+yj = 10
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+1.8
+2.0
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+ζ=θobs/θj
+Π/Πmax
+yj=103
+yj=102
+yj=10
+yj=1
+α = 1
+Random B⟂
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+1.8
+2.0
+0.0
+0.2
+0.4
+0.6
+ζ=θobs/θj
+Π/Πmax
+Figure 9. The time integrated polarization for a random magnetic field B⊥ which lies entirely in the plane of the shock (Eq. (20)) as a function of the off-axis
+parameter ζ = θobs/θ j for different values of spectral index α (left) and yj = (Γθ j)2 (right) as labeled.
+MNRAS 000, 1–?? (2023)
+

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@@ -0,0 +1,1569 @@
+ERA-Solver: Error-Robust Adams Solver
+for Fast Sampling of Diffusion Probabilistic Models
+Shengmeng Li 1 Luping Liu 2 Zenghao Chai 3 Runnan Li 1 Xu Tan 3
+Abstract
+Though denoising diffusion probabilistic models
+(DDPMs) have achieved remarkable generation
+results, the low sampling efficiency of DDPMs
+still limits further applications. Since DDPMs
+can be formulated as diffusion ordinary differ-
+ential equations (ODEs), various fast sampling
+methods can be derived from solving diffusion
+ODEs. However, we notice that previous sam-
+pling methods with fixed analytical form are not
+robust with the error in the noise estimated from
+pretrained diffusion models. In this work, we
+construct an error-robust Adams solver (ERA-
+Solver), which utilizes the implicit Adams nu-
+merical method that consists of a predictor and a
+corrector. Different from the traditional predictor
+based on explicit Adams methods, we leverage a
+Lagrange interpolation function as the predictor,
+which is further enhanced with an error-robust
+strategy to adaptively select the Lagrange bases
+with lower error in the estimated noise. Exper-
+iments on Cifar10, LSUN-Church, and LSUN-
+Bedroom datasets demonstrate that our proposed
+ERA-Solver achieves 5.14, 9.42, and 9.69 Fenchel
+Inception Distance (FID) for image generation,
+with only 10 network evaluations.
+1. Introduction
+In recent years, denoising diffusion probabilistic models
+(DDPMs) (Ho et al., 2020) have been proven to have poten-
+tial in data generation tasks such as text-to-image genera-
+tion(Poole et al., 2022; Gu et al., 2022; Kim & Ye, 2021;
+Chen et al., 2022), speech synthesis(Huang et al., 2021; Lam
+et al., 2022; Leng et al., 2022), and molecular conformation
+formation(Hoogeboom et al., 2022; Jing et al., 2022; Wu
+et al., 2022; Huang et al., 2022). They build a diffusion
+process to add noise into the sample and a denoising process
+to remove noise from the sample gradually. Compared with
+1Microsoft Cloud+AI 2Zhejiang University 3Microsoft Re-
+search Asia. Correspondence to: Xu Tan <xuta@microsoft.com>.
+Implicit Adams
+DPM-Solver
+ERA-Solver (Ours)
+Figure 1. We adopt the pretrained diffusion model from (Song
+et al., 2020a) and visualize the error between the estimated noise
+and ground-truth noise in training. We also provide the gener-
+ated samples from the pretrained model based on implicit Adams
+(traditional predictor-corrector method (Diethelm et al., 2002)),
+DPM-Solver(Lu et al., 2022a), and our error-robust Adams solver
+(ERA-Solver). Our solver is robust for the error from pretrained
+model so as to generate samples with better quality.
+generative adversarial networks (GANs)(Goodfellow et al.,
+2014) and variational auto-encoders (VAEs)(Child, 2021),
+DDPMs have achieved remarkable generation quality. How-
+ever, due to the properties of the Markov chain, the sampling
+process requires hundreds or even thousands of denoising
+steps. Such defects limit the wide applications of diffusion
+models. Thus, it is an urgent request for a fast sampling of
+DDPMs.
+There have already existed many works for accelerating
+sampling speed. Some works introduced an extra training
+stage, such as knowledge distillation method (Salimans &
+Ho, 2021), training sampler (Watson et al., 2021), or directly
+combining with GANs (Wang et al., 2022), to obtain a fast
+sampler. These methods require a cumbersome training
+process for each task, and are black-box samplers due to
+the lack of theoretical explanations. Denoising diffusion
+implicit model (DDIM) (Song et al., 2020a) and Score-
+SDE(Song et al., 2020b) revealed that the sampling can
+be reformulated as a diffusion ordinary differential equa-
+tion (ODE) solving process, which inspired many works to
+design learning-free fast samplers based on numerical meth-
+ods. PNDM (Liu et al., 2021) introduced the analytic form
+arXiv:2301.12935v1  [cs.LG]  30 Jan 2023
+
+70
+2
+60
+50
+40
+)03
+30
+20
+3
+10
+0
+0.9
+0.8
+0.7
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+time tError-Robust Adams Solver
+of a 4-order linear multistep method, which is also called ex-
+plicit Adams method (Małgorzata & Marciniak, 2002), and
+utilized the estimated noises observed at previous sampling
+steps to sample efficiently, with a warming initialization
+based on Runge-Kutta methods (Butcher, 1996). DPM-
+Solver(Lu et al., 2022a) introduced exponential integrator
+from ODE literature (Atkinson et al., 2011) and required ex-
+tra network evaluations to observe intermediate noise terms
+for approximating Taylor expansion (Mohazzabi & Becker,
+2017) in the sampling process.
+The existing learning-free methods (Liu et al., 2021; Lu
+et al., 2022a; Song et al., 2020a) are based on the assump-
+tion that the learned network has high accuracy in noise
+estimations across all the sampling steps. However, we no-
+tice that the noise estimated from the neural network is not
+accurate enough and the error exists at almost every time
+t, especially when time t approaches 0, as shown in Fig.
+1. Existing sampling methods are not able to be robust for
+noise estimation errors, since they consist of an analytic
+sampling scheme with fixed coefficients to ensure sampling
+convergence. For example, explicit Adams (Małgorzata &
+Marciniak, 2002) consists of the analytic form with fixed
+coefficients as formulated in Eq. 9.
+In this paper, we aim at designing an error-robust diffusion
+ODE solver to speed up the sampling process of DDPMs
+while achieving good sampling quality. To this end, we
+focus on implicit Adams solver (Małgorzata & Marciniak,
+2006), a kind of numerical ODE solver, which involves
+unobserved terms to achieve high-order precision and con-
+vergence. In existing ODE literature(Atkinson et al., 2011),
+predictor-corrector has been introduced to perform implicit
+Adams solver, which avoids solving the implicit equation.
+Explicit Adams usually acts as the predictor to predict the
+unobserved term. However, traditional predictor-corrector
+still suffers from the inaccurate estimation of diffusion noise
+at each sampling step since it is composed of fixed coeffi-
+cients, which is shown in Fig. 1. Instead of utilizing explicit
+Adams as the predictor, we adopt the Lagrange interpola-
+tion function (Sauer & Xu, 1995) that interpolates several
+Lagrange function bases as the predictor. We maintain a
+buffer of estimated noises observed at previous sampling
+steps during sampling and adaptively select those estimated
+noises with low estimation error as the Lagrange function
+bases, to ensure accurate interpolation result and thus an ac-
+curate predictor. In this way, we can obtain a diffusion ODE
+sampler with not only high convergence (thanks to the im-
+plicit Adams method(Małgorzata & Marciniak, 2006)) but
+also good error robustness (thanks to the adaptive strategy
+for the Lagrange interpolation function).
+However, it is not easy to select noises with low estima-
+tion error as the Lagrange function bases. That is because,
+unlike the training stage, there exist no reference noises
+at the sampling stage to judge how accurate the estimated
+noise is. Thus, we further propose an approach to roughly
+measure the accuracy of the estimated noise by calculating
+the difference between the noise obtained by the predictor
+(as the prediction) and the noise observed at the previous
+sampling step (as the reference). Based on this measure-
+ment, we propose a selection strategy for the buffer which
+adaptively chooses estimated noises that are more accurate
+to construct the Lagrange function bases, so as to result in a
+more accurate predictor, and thus a better ODE solver.
+Our contributions can be summarized as follows:
+• We are the first to explore the potential of numerical im-
+plicit Adams methods (Małgorzata & Marciniak, 2006)
+to solve diffusion ODEs. We propose an error-robust im-
+plicit Adams Solver (ERA-Solver), which is training-free
+and can be extended to pretrained DDPMs conveniently.
+• We propose to adopt the Lagrange interpolation function
+(Sauer & Xu, 1995) that selectively interpolates several
+Lagrange function bases with low errors of estimated
+noises to ensure the ERA-solver is robust to the error in
+estimated noise.
+• Experiment results show that we achieve better results
+on LSUN-Bedroom, LSUN-Church, and Cifar10 datasets
+when compared with previous methods in 10 function
+evaluations. On LSUN-Church and LSUN-Bedroom, we
+achieve state-of-the-art FID results of 7.39 and 5.39 with
+more function evaluations, compared with the previous
+best results of 7.74 and 6.46.
+2. Preliminary
+2.1. Denoising Diffusion Probabilistic Models
+Denoising diffusion probabilistic models (DDPMs)(Ho
+et al., 2020) have demonstrated their great generation poten-
+tial on various applications, such as text-to-image synthesis
+(Poole et al., 2022; Gu et al., 2022; Kim & Ye, 2021), image
+inpainting (Lugmayr et al., 2022; Liu et al., 2022; Kawar
+et al., 2022), speech synthesis (Huang et al., 2021; Lam
+et al., 2022; Leng et al., 2022), and molecular conformation
+generation (Hoogeboom et al., 2022; Jing et al., 2022; Wu
+et al., 2022; Huang et al., 2022). It involves a diffusion
+process to gradually add noise to data, and a parameterized
+denoising process to reverse the diffusion process, sampling
+through gradually removing the noise from random noise.
+The diffusion process is modeled as a transition distribution:
+q(xt|xt−1) := N(xt; √αtxt−1, (1 − αt)I),
+(1)
+where α1, ..., αT are fixed parameters. With the transition
+distribution above, noisy distribution conditioned on clean
+data x0 can be formulated as follows:
+q(xt|x0) = N(xt; √αtx0, (1 − αt)I),
+(2)
+
+Error-Robust Adams Solver
+where ¯αt = �t
+s=1 αs. When t is large enough, the Markov
+process will converge to a Gaussian steady-state distribution
+N(0, I).
+DDPMs also build the reverse process, i.e., the sampling
+process, to the diffusion process above. Similar to VAEs,
+DDPMs introduce the variational bound to derive optimiza-
+tion objectives. After simplification (Luo, 2022), the main
+objective of training optimization is formulated as a KL
+divergence:
+Lt−1 = DKL(q(xt−1|x0, xt)||pθ(xt−1|xt)),
+(3)
+where q(xt−1|x0, xt) has an analytical form derived from
+the bayesian formula and can be expressed as a Gaussian
+distribution parameterized with µt−1 =
+1
+√αt (xt −
+1−αt
+√1−¯αt ϵ)
+and σt−1 = 1−¯αt−1
+1−¯αt (1 − αt).
+To simplify the optimization loss, the parameterized denois-
+ing distribution is formulated as follows:
+pθ(xt−1|xt) = N(xt−1; µθ(xt), σ2
+t−1I)
+µθ(xt) =
+1
+√αt
+(xt − 1 − αt
+√1 − ¯αt
+ϵθ(xt, t)),
+(4)
+where ϵθ(xt, t) estimates the noise in noisy sample xt and
+is called estimated noise in our paper. The variance σ2
+t−1 is
+constant. In this way, the objective function Eq. 3 can be
+rewritten as following:
+Lt−1 = Eq[
+1
+2σ2
+t−1
+||µt−1 − µθ(xt)||2]
+= Ex0,ϵ[
+1
+2σ2
+t−1
+||
+1
+√αt (xt − 1 − αt
+√1 − ¯αt
+ϵ) − µθ(xt)||2]
+= Ex0,ϵ[
+1
+2σ2
+t−1
+(1 − αt)2
+αt(1 − ¯αt)||ϵ − ϵθ(xt, t)||2].
+(5)
+Though DDPMs have good theoretical properties, they suf-
+fer from sampling efficiency. It usually requires hundreds
+or even thousands of network evaluations, which limits var-
+ious downstream applications for DDPMs. There already
+existed methods (Watson et al., 2021; Lyu et al., 2022; Lam
+et al., 2022; Salimans & Ho, 2021) which depend on extra
+training stage to derive fast sampling methods. The training-
+based methods usually require tremendous training costs
+for different data manifolds and tasks, which inspires many
+works to explore a training-free sampler based on numerical
+methods.
+2.2. Numerical Methods for Fast Sampling
+Denoising diffusion implicit model (DDIM) (Song et al.,
+2020a) is a classic training-free approach for fast sampling,
+which introduces a non-Markovian process that allows the
+sampling with any number of evaluation steps. We formu-
+late the iteration time steps for solving diffusion ODEs as
+{ti}N
+i=0, where t0 is the beginning time and tN is the end
+time. The sequence of sampling steps maintains the prop-
+erty that ti>ti+1 and tN = 0. In particular, the last iterate
+xtN represents the final generated sample. The denoising
+iteration process of DDIM can be formulated as follows:
+xti+1 = �¯αti+1(xt − √1 − ¯αtiϵθ(xti, ti)
+√¯αti)
++
+�
+1 − ¯αti+1 − σ2
+tiϵθ(xti, ti) + σtiz.
+(6)
+Furthermore, DDIM shares the same training objective with
+DDPMs, which means the fast sampling scheme can be
+applied to any pre-trained models. When the variance σ2
+ti is
+set to 0, the sampling process becomes deterministic, which
+can be regarded as the solving process of diffusion ODEs.
+We use a term ϵti for the ease of description so that the
+solving formula of diffusion ODE can be formulated (Song
+et al., 2020a) as follows:
+ϵti = ϵθ(xti, ti),
+(7)
+xti+1 =
+√¯αti+1
+√¯αti
+xti
++ (
+�
+1 − ¯αti+1 −
+�
+¯αti+1(1 − ¯αti)
+√¯αti
+)ϵti.
+(8)
+Many numerical solvers, such as Runge-Kutta (Butcher,
+1996) method and linear multistep method (Wells, 1982),
+can be applied to solve diffusion ODEs. PNDM(Liu et al.,
+2021) combined Runge-Kutta and linear multistep method
+so as to solve the manifold problem of diffusion ODEs.
+Essentially, the linear multistep method can be regarded
+as explicit Adams (Małgorzata & Marciniak, 2002), which
+utilized previously estimated noises to calculate the ϵti. The
+ϵti in Eq. 7 is reformulated as following:
+ϵti = 1
+24(55ϵθ(xti, ti) − 59ϵθ(xti−1, ti−1)
++ 37ϵθ(xti−2, ti−2) − 9ϵθ(xti−3, ti−3)).
+(9)
+DPM-Solver(Lu et al., 2022a) introduced the method of
+exponential integrator from ODE literature (Atkinson et al.,
+2011) to eliminate the discretization errors of the linear term
+in the sampling process. Furthermore, DPM-Solver pro-
+posed to utilize Taylor expansion (Mohazzabi & Becker,
+2017) to approximate ϵti and contributed its analytical
+forms for solving diffusion ODEs.
+We notice that the estimated noises of pretrained diffusion
+models are not precise across all sampling time, especially
+when time ti is close to 0. Previous numerical methods
+with analytical forms will suffer from the noise estimation
+error of pretrained DDPMs. In this paper, we propose an
+error-robust solver based on Adams numerical methods,
+adaptively selecting noises with low estimation error.
+
+Error-Robust Adams Solver
+Eq.(8)
+Eq.(17)
+Corrector 
+Eq.(11) 
+Predictor
+ Lagrange Buffer
+Initial indexs
+Selected Indexs
+ERA-Solver
+ 
+Push
+Selected
+Not Selected
+Initial Indexs
+Error-Robust Selection
+Figure 2. The pipeline of ERA-Solver. The sampling scheme is based on the predictor-corrector method for implicit Adams. Our predictor
+is robust to the errors of the estimated noises from pretrained models. The sampling starts from normal Gaussian noise xt0 and performs a
+denoising scheme (from xti to xti+1) iteratively to get the final generated image.
+3. ERA-Solver
+In this section, we first point out that the error in the esti-
+mated noise ϵθ(xti, ti) by the network θ limits the previ-
+ous numerical fast samplers and introduce implicit Adams
+numerical solver (Sec. 3.1). Then, we apply predictor-
+corrector sampling and leverage a Lagrange interpolation
+function as the predictor (Sec. 3.2). We design an error
+distance to measure the accuracy of the estimated noise and
+enhance the proposed predictor with an error-robust strategy
+to adaptively select the Lagrange bases with lower noise
+estimation error (Sec. 3.3). The whole sampling process is
+shown in Fig. 2.
+3.1. Implicit Adams Methods
+The sampling of DDPMs starts from a prior noise distribu-
+tion xt0 ∼ N(0, I), and iteratively denoises xti to xti+1
+until time t reaches 0. In the sampling process, the most
+time-consuming step is network evaluation. Assuming we
+have a pretrained noise estimation model ϵθ, we need to
+achieve good generation quality with as few evaluation times
+as possible.
+We notice that the noise estimation error exists across every
+sampling time, especially when time ti is close to 0. It limits
+previous numerical high-order solvers (Liu et al., 2021; Lu
+et al., 2022a; Song et al., 2020a) since they are based on
+the assumption that the network has no estimation errors.
+Previous solvers usually involved observed noise terms to
+achieve the analytic form of the sampling scheme that is
+critical for sampling convergence. Thus, they are sensitive
+to the errors of estimated noises from pretrained models.
+In this paper, we explore the potential of implicit Adams
+solver (Małgorzata & Marciniak, 2006). Different from
+explicit Adams (Eq. 9), implicit Adams involves the unob-
+served noise term and the ϵti in Eq. 7 is reformulated as
+follows:
+ϵti = 1
+24(9ϵθ(xti+1, ti+1) + 19ϵθ(xti, ti)
+− 5ϵθ(xti−1, ti−1) + ϵθ(xti−2, ti−2)).
+(10)
+It can be noticed that xti+1 can be observed only when
+ϵti is achieved, while the ϵti contains unobserved term
+ϵθ(xti+1, ti+1), which makes it challenging to solve implicit
+equations and may need more time-consuming iteration
+steps. This greatly limits the implicit Adams method to be a
+fast solver for diffusion ODEs.
+Fortunately, in numerical ODE literature, the sampling effi-
+ciency of implicit Adams can be improved with a predictor-
+corrector sampling scheme (Diethelm et al., 2002). Specifi-
+cally, the predictor makes a rough estimation of unobserved
+term ¯ϵθ(xti+1, ti+1) and the corrector derives the precise
+xti+1, which can reformulate Eq.10 as follows:
+ϵti = 1
+24(9¯ϵθ(xti+1, ti+1) + 19ϵθ(xti, ti)
+− 5ϵθ(xti−1, ti−1) + ϵθ(xti−2, ti−2)).
+(11)
+The traditional predictor-corrector utilizes explicit Adams
+(Eq. 9) to perform predictor to make xti+1 observed so as
+to derive ¯ϵθ(xti+1, ti+1). However, it still suffers from the
+fixed form that is not robust to the noise estimation errors,
+which can be observed in Fig. 1.
+3.2. Predictor with Lagrange Interpolation Function
+In this paper, we propose to utilize noises observed at pre-
+vious sampling steps and construct the Lagrange interpo-
+lation function as the predictor to predict unobserved term
+ϵθ(xti+1, ti+1). In this way, we can design an adaptive
+strategy to select Lagrange function bases to construct the
+error-robust predictor.
+Specifically, we maintain a buffer about all previously esti-
+mated noises, which have been observed and need no extra
+
+Error-Robust Adams Solver
+Selected
+Not Selected
+Sampling Process
+Figure 3. Error-robust selection process. ∆ϵ is calculated based on
+Eq. 15 instead of the training loss in Fig. 1. The sampling NFE is
+set to 20.
+computations, and its corresponding time:
+{(tn, ϵθ(xtn, tn)), n = 0, 1, .., i}.
+(12)
+The maintained buffer is also called the Lagrange buffer
+in this paper. Assume that the interpolation order is k, the
+selected function bases to construct the Lagrange function
+can be written as {(tτm, ϵθ(xtτm, tτm)), m = 0, ...k − 1}.
+The corresponding Lagrange interpolation function can be
+formulated as:
+lm(t) =
+k−1
+�
+l=0,l̸=m
+( t − tτl
+tτm − tτl
+),
+Lϵ(t) =
+k−1
+�
+m=0
+lm(t) ∗ ϵθ(xtτm , tτm),
+(13)
+where τl belongs to the maintained Lagrange buffer and
+has already been observed. At time ti+1, we can derive an
+estimation about ϵθ(xti+1, ti+1):
+¯ϵθ(xti+1, ti+1) = Lϵ(ti+1).
+(14)
+With this prediction, we apply the corrector process in Eq.
+11 to get the ϵti and Eq. 8 to get the denoised sample xti+1.
+It can be noticed that the proposed predictor makes use of
+the observed noise estimations and involves no network eval-
+uations. Furthermore, the Lagrange bases in the predictor
+can be adaptively selected from those noise estimations with
+low errors, which is more error-robust. We introduce this
+error-robust selection strategy in the next subsection.
+3.3. Error-Robust Selection Strategy
+In this part, our goal is to design an error-robust selection
+strategy for the maintained Lagrange buffer. When the
+interpolation order is k, the intuitive selection approach is
+to make a fixed selection of the last k estimated noises from
+the maintained Lagrange buffer, which means τm = i − m
+in Eq. 13. However, we notice that the noise estimation
+Algorithm 1 ERA-Solver
+1: Input: {ti}N
+i=0, k, ϵθ
+2: Instantiate: xt0 ∼ N(0, I), buffer Ω = ∅, ∆ϵ = λ
+3: Ω = Ω ∪ {(t0, ϵθ(xt0, t0))}
+4: for i in 0, 1, · · · , N − 1 do
+5:
+if i<k − 1 then
+6:
+Derive xti+1 based on Eq. 8 and ϵθ(xti, ti)
+7:
+Ω = Ω ∪ {(ti+1, ϵθ(xti+1, ti+1))}
+8:
+else
+9:
+Calculate {¯τm}k−1
+m=0 via Eq. 16
+10:
+Calculate {τm}k−1
+m=0 via Eq. 17 and ∆ϵ
+11:
+Derive Lagrange function Lϵ via Eq. 13 and τm
+12:
+¯ϵθ(xti+1, ti+1) ← Lϵ(ti+1)
+13:
+Calculate ϵti via Ω, ¯ϵθ(xti+1, ti+1), and Eq. 11
+14:
+Derive xti+1 based on Eq. 8 and ϵti
+15:
+Ω = Ω ∪ {(ti+1, ϵθ(xti+1, ti+1))}
+16:
+Update ∆ϵ via Eq. 15 and ¯ϵθ(xti+1, ti+1)
+17:
+end if
+18: end for
+19: return xtN
+error tends to increase as time ti approaches 0 (Fig. 1), in
+which case the fixed selection strategy may aggregate the
+noise estimation errors from Lagrange buffer and make the
+constructed Lagrange function inaccurate for prediction at
+time ti+1. It motivates us to seek a reasonable measure of
+the error in estimated noise so as to select those noises with
+low estimation error for Lagrange interpolation.
+Error Measure for Estimated Noise.
+Since there exists
+no ground-truth noise in sampling process, it is hard to mea-
+sure the error in estimated noise. To this end, we propose to
+utilize the observed noise term ϵθ(xti, ti) as the target noise
+and the predicted noise term ¯ϵθ(xti, ti) from last sampling
+step as the estimated noise to calculate the approximation
+error, which can be seen in Fig. 2. It can be formulated as
+follows:
+∆ϵ = ||ϵθ(xti, ti) − ¯ϵθ(xti, ti)||2.
+(15)
+ϵθ(xti, ti) is observed based on xti, which is achieved via
+the ¯ϵθ(xti, ti) and Eq. 8. When the error of estimated noise
+from pretrained models increases, it tends to be hard for
+¯ϵθ(xti, ti) to approximate ϵθ(xti, ti). As shown in Fig.3,
+our error measure in the sampling process shares the same
+trend as the error of estimated noises in the training process
+(Fig. 1), which demonstrates the rationality of the proposed
+error measure.
+Selection Strategy.
+The high-level idea of our error-
+robust selection strategy is that we tend to choose those
+estimated noises from Lagrange buffer with low errors mea-
+sured by Eq. 15 as the Lagrange bases. When sampling
+
+14
+12
+10
+8
+3
+6
+4
+2
+0
+0.8
+0.7
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+time tiError-Robust Adams Solver
+Table 1. Generation quality measured by FID ↓ on LSUN-Church, varying the number of function evaluation (NFE).
+Sampling method \ NFE
+5
+10
+12
+15
+20
+40
+50
+100
+DDIM (Song et al., 2020a)
+56.74
+19.62
+15.77
+13.31
+11.75
+10.24
+10.1
+9.97
+FON (Liu et al., 2021)
+\
+\
+\
+21.32
+10.35
+9.44
+9.70
+9.83
+PNDM (Liu et al., 2021)
+\
+\
+\
+9.31
+7.74
+9.49
+9.89
+10.38
+DPM-Solver-2 (Lu et al., 2022a)
+310.49
+23.01
+16.56
+13.68
+11.59
+10.48
+10.60
+10.49
+DPM-Solver-fast (Lu et al., 2022a)
+346.38
+19.81
+13.35
+11.52
+10.64
+10.37
+10.19
+10.49
+ERA-Solver
+32.76
+9.42
+8.15
+7.41
+7.39
+8.26
+8.50
+9.52
+at time ti, the length of the Lagrange buffer is i + 1. We
+initialize k indexes uniformly to cover the whole buffer:
+�τm = (i/k) ∗ m, m = 1, 2, .., k
+(16)
+Then we utilize the power function as an index translator.
+We parameterize the power function with the error measure
+(Eq. 15) so that the translation of initial indexes can be
+formulated as:
+τm = ⌊(�τm/i)∆ϵ/λ ∗ i⌋.
+(17)
+where λ is a hyperparameter to adjust the scale.
+As shown in Fig. 3, our selection focus more on the be-
+ginning of the Lagrange buffer, which tends to be more
+accurate, when the error of estimated noises increases. In
+this way, our selection strategy of buffer makes our La-
+grange interpolation function more accurate in an adaptive
+way, which contributes to an error-robust Adams solver.
+3.4. Overall Sampling Algorithm
+In this part, we summarize our sampling algorithm. Given a
+pretrained diffusion model ϵθ, we sample from a prior noise
+xt0 ∼ N(0, I) and iteratively denoise from xti to xti+1
+until time ti reaches 0 and xtN is our final generated sample.
+The sampling algorithm is based on the predict-corrector
+method. The k represents the Lagrange interpolation order.
+For the initialization of Lagrange buffer, the first k sampling
+steps are based on DDIM sampling scheme. The details of
+the sampling process can be found in Alg. 1.
+4. Experiment
+In this section, we demonstrate that, as a training-free sam-
+pler, ERA-Solver is able to accelerate the sampling process
+and be robust for the error from existing pretrained diffusion
+models on various datasets.
+4.1. Experiment Setting
+We conduct sampling experiments mainly based on three
+datasets: Cifar10 (32 × 32) (Krizhevsky et al., 2009),
+LSUN-Church (256 × 256), (Yu et al., 2015), LSUN-
+Bedroom (256 × 256) (Yu et al., 2015), and Celeba (64
+× 64) (Liu et al., 2015). We adopt the pretrained models
+from DDIM (Song et al., 2020a) on three datasets above to
+evaluate our method.
+We adopt Fenchel Inception Distance (FID) (Heusel et al.,
+2017) as the evaluation metric to test the generation quality
+of all sampling methods. All evaluation results are based on
+50k generated samples.
+When sampling on LSUN-Bedroom and LSUN-Church,
+we set tN = 10−4 (withdraw the denoising trick (Song
+et al., 2020b) at the final step) and adopt a uniform timestep
+scheme (select ti uniformly) with discrete-time pretrained
+models. We set λ = 5.0 in Eq. 17 emprically. We set k = 4
+for LSUN-Church and k = 3 for LSUN-Bedroom.
+When sampling on Cifar10, we follow DPM-Solver (Lu
+et al., 2022a) and evaluate on both tN = 10−3 and tN =
+10−4 settings. We adopt logSNR timestep scheme applied
+in DPM-Solver (Lu et al., 2022a). It has been demonstrated
+(Lu et al., 2022a) that logSNR timestep scheme is more
+suitable for Cifar10, which causes lower discretization er-
+rors. We set λ = 15.0 in Eq. 17 and k = 4 emprically. The
+experiment details and results on Celeba can be found in
+Appendix. B.
+4.2. Comparison with Previous Fast Sampling Methods
+In this subsection, we show the comparison results of our
+ERA-Solver and other training-free sampling methods. We
+first compare ERA-Solver with previous fast sampling meth-
+ods on LSUN-Church and LSUN-Bedroom datasets. We
+directly use the code released in (Liu et al., 2021; Lu et al.,
+2022a) to implement PNDM (Liu et al., 2021), FON(Liu
+et al., 2021), and DPM-Solver(Lu et al., 2022a) methods to
+generate the samples for evaluation. We compare the sam-
+pling quality of the training-free sampling methods based on
+5, 10, 12, 15, 20, 40, 50, and 100 NFEs. Note that PNDM
+and FON (Liu et al., 2021) use 4-order Runge-Kutta for
+the first three sampling steps and each step costs 4 NFEs,
+which means their methods need at least 13 NFEs to per-
+form sampling. Thus, their FID results are missing on 5, 10,
+and 12 NFEs. For DPM-Solver, we adopt a 2-order setting
+(DPM-Solver-2) and its fast scheme (DPM-Solver-fast) for
+comparison.
+
+Error-Robust Adams Solver
+Table 2. Generation quality measured by FID ↓ on LSUN-Bedroom, varying the number of function evaluation (NFE).
+Sampling method \ NFE
+5
+10
+12
+15
+20
+40
+50
+100
+DDIM (Song et al., 2020a)
+73.18
+19.81
+15.24
+11.57
+9.24
+6.95
+6.76
+6.84
+FON (Liu et al., 2021)
+\
+\
+\
+15.89
+7.85
+6.08
+6.16
+6.81
+PNDM (Liu et al., 2021)
+\
+\
+\
+11.59
+7.19
+6.46
+6.87
+7.77
+DPM-Solver-2 (Lu et al., 2022a)
+354.12
+20.95
+16.98
+13
+10.17
+8.91
+8.74
+8.62
+DPM-Solver-fast (Lu et al., 2022a)
+376.69
+19.03
+10.47
+8.79
+8.26
+7.35
+7.52
+7.31
+ERA-Solver
+21.69
+10.44
+9.53
+8.10
+6.99
+5.45
+5.39
+5.72
+Table 3. Generation quality measured by FID ↓ on Cifar10, varying the number of function evaluation (NFE).
+Sampling method \ NFE
+5
+10
+12
+15
+20
+40
+50
+100
+DDPM(Ho et al., 2020)
+\
+278.67
+246.29
+197.63
+137.34
+\
+32.63
+\
+Analytic-DDPM (Bao et al., 2022)
+\
+35.03
+27.69
+20.82
+15.35
+\
+7.34
+\
+Analytic-DDIM (Bao et al., 2022)
+\
+14.74
+11.68
+9.16
+7.20
+\
+4.28
+\
+DDIM(Song et al., 2020a)
+40.41
+13.58
+11.02
+8.92
+6.94
+4.92
+4.73
+3.65
+PNDM(Liu et al., 2021)
+\
+\
+\
+11.8
+5.7
+3.47
+3.31
+3.35
+DPM-Solver-fast (Lu et al., 2022a) (tN = 10−3)
+269.36
+6.37
+4.65
+3.78
+4.28
+3.80
+4.03
+3.88
+DPM-Solver-fast (Lu et al., 2022a) (tN = 10−4)
+330.08
+11.32
+7.31
+4.75
+3.80
+3.51
+3.54
+3.44
+ERA-Solver (tN = 10−3)
+32.45
+5.14
+4.38
+3.86
+3.79
+3.97
+3.91
+4.0
+ERA-Solver (tN = 10−4)
+43.31
+6.16
+4.84
+4.2
+3.84
+3.45
+3.42
+3.51
+LSUN-Church.
+The comparison results on LSUN-
+Church are shown in Tab. 1. When tested on extremely
+few NFE like 5, our ERA-Solver obtains better FID 32.76,
+achieving 42.2% improvement compared with DDIM (Song
+et al., 2020a). When tested on a few NFE like 10 and 15, our
+ERA-Solver obtains 9.42 and 7.41 FID results, achieving
+51.9% and 20.4% improvements compared with the previ-
+ous best results of 19.62 and 9.31. When tested on larger
+NFE, our ERA-Solver achieves state-of-the-art generation
+quality with FID 7.39, compared with the previous best
+result of 7.74.
+LSUN-Bedroom.
+The comparison results on LSUN-
+Bedroom are shown in Tab. 2. When tested on extremely
+small NFE like 5, our ERA-Solver obtains better FID 21.69,
+achieving 70.3% improvement compared with 73.18 from
+DDIM (Song et al., 2020a). When tested on small NFEs like
+10 and 12, our ERA-Solver obtains 10.44 and 9.53 FID re-
+sults, achieving 45.1% and 14.5% improvements compared
+with the previous best results of 19.03 and 10.47. When
+tested on larger NFE, our ERA-Solver achieves state-of-the-
+art generation quality with FID 5.39, compared with the
+previous best result of 6.08.
+Cifar10.
+The comparison results on Cifar10 are shown
+in Tab. 3. Our ERA-Solver achieves better results at 5,
+10, 12, 20, and 40 NFEs. When evaluated on 10 NFE, our
+ERA-Solver obtains 5.14 FID result, achieving a 19.15%
+improvement compared with the previous best result of 6.37.
+Fixed
+NFE=10
+NFE=20
+NFE=30
+ERS
+Figure 4. Generation quality comparison between error-robust se-
+lection strategy (ERS) with λ = 5.0 and fixed strategy (Fixed)
+that select the last k estimated noises fixedly based on 5-order
+Lagrange interpolation function.
+4.3. Ablation Study
+Effectiveness of error-robust selection strategy.
+We
+compare our error-robust selection strategy and fixed strat-
+egy which selects the last k estimated noises (τm = i − m
+in Eq. 13) based on various orders of Lagrange interpolation
+function in the predictor. The results can be seen in Tab .4.
+We further visualize the samples generated from two strate-
+gies based on a 5-order Lagrange interpolation function, the
+results of which are shown in Fig. 4. It can be concluded
+that our error-robust strategy is more effective than intuitive
+strategies, especially when Lagrange function order is high.
+
+Error-Robust Adams Solver
+Table 4. FID comparison between error-robust selection strategy
+(ERS) and fixed strategy (fixed) that select the last k estimated
+noises fixedly based on various Lagrange function orders (k =
+3, 4, 5, 6). All experiments are conducted on LSUN-Church.
+Method\ NFE
+10
+15
+20
+40
+50
+ERA-Solver-3 fixed
+9.83
+8.52
+8.72
+9.58
+9.69
+ERS
+10.2
+8.03
+7.63
+8.11
+8.37
+ERA-Solver-4 fixed 10.56
+8.72
+8.99
+9.89
+9.92
+ERS
+9.42
+7.41
+7.39
+8.26
+8.5
+ERA-Solver-5 fixed
+26.7
+30.36 31.58
+10.09
+10.12
+ERS 10.85
+7.48
+7.28
+8.26
+8.64
+ERA-Solver-6 fixed 63.91 191.69 315.6 298.22 182.44
+ERS 13.79
+8.41
+7.41
+8.43
+8.7
+Effectiveness of error measure.
+We compare different
+power scales based on the proposed error measure and var-
+ious constants. The results can be seen in Fig. 5. It can
+be observed that the selection strategy based on our error
+measure has better performance than those based on the
+constant scale in general.
+5. Discussion
+Why FID gets worse when NFE >50.
+We can notice
+that our solver performs worse when NFE increases to 50,
+especially on LSUN dataset (Tab. 2 and Tab. 1). It can be
+understood that FID results fluctuate around a value, which
+is verified on PNDM (Liu et al., 2021).
+Why ERA-Solver performs worse on Cifar10.
+From
+Tab. 3, we can notice that ERA-Solver performs slightly
+worse than DPM-Solver and PNDM when NFE increases,
+which is different from the results on LSUN. The reason
+behind it is the resolution of data. Cifar10 has a very low res-
+olution (32 × 32) while LSUN has a higher resolution (256
+× 256), which means the LSUN dataset is more informative
+than Cifar10. Thus, the model tends to have lower train-
+ing error (noise estimation error) when trained on Cifar10,
+leading to the slightly worse performance of ERA-Solver.
+Rationality of selection function.
+In this paper, we de-
+sign the selection function for the error robustness of the
+sampling solver. The changing trend of noise estimation
+error with timestep (Fig. 1) inspires us to choose the power
+function as an error-robust selection function and experi-
+ments demonstrate its effectiveness. There may exist a better
+selection function or a learning-based approach to selecting
+Lagrange functions and we leave it for future work.
+Why ERA-Solver has better error robustness.
+We no-
+tice that the previous numeric solvers depend on the assump-
+tion that the pretrained diffusion model achieves accurate
+10
+15
+20
+25
+30
+35
+40
+45
+50
+NFE
+9.5
+10.0
+10.5
+11.0
+11.5
+12.0
+12.5
+13.0
+FID
+ERS = 3.0
+ERS = 5.0
+ERS = 8.0
+Constant 0.2
+Constant 0.5
+Constant 0.8
+Constant 1.0
+Constant 2.0
+Figure 5. Generation quality measured by FID ↓. We compare our
+error-aware scale (∆ϵ/λ in Eq. 17) and constant scale (replace
+∆ϵ/λ with a constant) to demonstrate the effectiveness of the pro-
+posed error measure based on the 3-order Lagrange interpolation
+function. All FID results are evaluated on LSUN-Church with
+5000 generated samples.
+noise estimation. In this paper, we demonstrate that there
+exists an obvious estimation error (Fig. 1) from pretrained
+model, which limits the sampling efficiency of previous fast
+solvers.
+Combined with the Lagrange interpolation as the predictor
+and an error-robust selection strategy, we show that the Im-
+plicit Adams solver has the potential to be error-robust so as
+to achieve better sampling efficiency. The predictor based on
+Lagrange interpolation makes sure our solver has the conver-
+gence property of the numerical solver (predictor-corrector
+sampling scheme). The error-robust selection strategy adap-
+tively introduces relatively accurate estimated noises and
+mitigates the error accumulation in the sampling process.
+More analysis can be found in Appendix. C.
+6. Conclusion
+In this paper, we propose an error-robust Adams solver
+(ERA-Solver) that consists of a predictor and a corrector.
+We leverage the Lagrange interpolation function to perform
+the predictor, and further propose an error measure for the
+sampling process and an error-robust strategy to enhance
+the predictor. Experiments demonstrate that ERA-Solver
+achieves better generation quality on LSUN-Church and
+LSUN-Bedroom datasets at all NFEs. Our ERA-Solver
+might be misused with powerful diffusion models (Rombach
+et al., 2022; Kim & Ye, 2021) to generate adverse fake
+content. We will restrict the usage of our solver and share it
+with the fake detection community.
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+on Learning Representations, 2021.
+Sauer, T. and Xu, Y. On multivariate lagrange interpolation.
+Mathematics of computation, 64(211):1147–1170, 1995.
+Song, J., Meng, C., and Ermon, S. Denoising diffusion
+implicit models. arXiv:2010.02502, 2020a.
+Song, Y., Sohl-Dickstein, J., Kingma, D. P., Kumar,
+A., Ermon, S., and Poole, B.
+Score-based genera-
+tive modeling through stochastic differential equations.
+arXiv:2011.13456, 2020b.
+von Platen, P., Patil, S., Lozhkov, A., Cuenca, P., Lambert,
+N., Rasul, K., Davaadorj, M., and Wolf, T. Diffusers:
+State-of-the-art diffusion models. https://github.
+com/huggingface/diffusers, 2022.
+Wang, Z., Zheng, H., He, P., Chen, W., and Zhou, M.
+Diffusion-gan: Training gans with diffusion.
+arXiv
+preprint arXiv:2206.02262, 2022.
+Watson, D., Chan, W., Ho, J., and Norouzi, M. Learning fast
+samplers for diffusion models by differentiating through
+sample quality. In International Conference on Learning
+Representations, 2021.
+Wells, D. R. Multirate linear multistep methods for the so-
+lution of systems of ordinary differential equations. Uni-
+versity of Illinois at Urbana-Champaign, 1982.
+Wu, L., Gong, C., Liu, X., Ye, M., and Liu, Q. Diffusion-
+based molecule generation with informative prior bridges.
+arXiv preprint arXiv:2209.00865, 2022.
+Yu, F., Seff, A., Zhang, Y., Song, S., Funkhouser, T., and
+Xiao, J.
+Lsun: Construction of a large-scale image
+dataset using deep learning with humans in the loop.
+arXiv:1506.03365, 2015.
+A. Ablation Study on Cifar10
+In this part, we provide ablation experiments on Cifar10
+dataset. We replace our error-robust selection strategy with
+the fixed selection strategy that fixedly selects the last k
+estimated noises previously saved in the Lagrange buffer at
+every sampling step. The results are shown in Tab. 5. From
+the table, we can see that our selection strategy achieves
+better FID generation results.
+We also conduct ablation studies for proposed error mea-
+sures in the sampling process. We parameterize the power
+function with various constants instead of our error mea-
+sure. The comparison results are shown in Fig. 6. Our
+error measure can enhance the selection strategy with better
+generation results in general.
+B. Additional Results on Celeba
+The comparison results on Celeba are shown in Tab. 6.
+When tested on a few NFEs like 10 and 12, our ERA-Solver
+obtains 5.06 and 3.67 FID results, achieving 26.8% and
+14.4% improvements compared with the previous best re-
+sults of 6.92 and 4.20. It can also be observed that FID result
+of ERA-Solver converges at about NFE 15. Compared with
+DPM-Solver which converges at NFE 36, ERA-Solver has
+better convergence speed, while achieving comparable gen-
+eration quality (FID 2.75 vs 2.71).
+C. Analysis of Error Robustness
+Since data distributions like LSUN are high-dimensional
+and complex, it is difficult to seek a ground truth for the pre-
+dicted noise in the sampling process. Thus, we demonstrate
+the error robustness of ERA-Solver from another perspec-
+tive. Assume that we achieve a generated sample xgen
+0
+from
+a solver, we can utilize the diffusion process to add noise
+to the sample, remapping the sample obtained from the de-
+noising process back to the noise space to derive xgen
+t
+. We
+measure the error robustness of the solver as the following:
+||ϵ − ϵθ(xgen
+t
+, t)||2,
+(18)
+The estimated noise from pretrained model can be seen
+as the stepping direction of the noisy sample (Song et al.,
+2020b). If a solver is not error-robust, its generated samples
+will deviate from the original generation path in the sam-
+pling process. Since the generation process can be seen as
+the reverse process of the diffusion process, the deviation of
+generated samples will increase the error in Eq. 18.
+We select normal Implicit Adam solver (Małgorzata &
+Marciniak, 2006), DPM-Solver (Lu et al., 2022a), and ERA-
+Solver for comparisons and generate a batch of samples to
+calculate Eq. 15 separately. Three solvers share the same
+sampling NFE, random seed, and the pretrained model ϵθ.
+
+Error-Robust Adams Solver
+Table 5. FID comparison between error-robust selection strategy
+(ERS) and fixed strategy (fixed) based on various Lagrange func-
+tion orders (k = 3, 4, 5, 6).
+Method\ NFE
+10
+15
+20
+50
+ERA-Solver-3
+fixed
+5.95
+4.62
+4.24
+4.0
+ERS
+5.79
+4.31
+4.07
+3.94
+ERA-Solver-4
+fixed
+6.4
+4.46
+4.1
+3.98
+ERS
+5.14
+3.86
+3.79
+3.91
+ERA-Solver-5
+fixed
+17.21
+15.11
+17.47
+3.99
+ERS
+6.26
+3.73
+3.69
+3.98
+ERA-Solver-6
+fixed
+36.34
+51.58
+83.39
+118.38
+ERS
+19.26
+4.16
+3.73
+4.04
+10
+15
+20
+25
+30
+35
+40
+45
+50
+NFE
+4.0
+4.5
+5.0
+5.5
+6.0
+6.5
+FID
+ERS = 10.0
+ERS = 5.0
+ERS = 8.0
+Constant 0.2
+Constant 0.5
+Constant 0.8
+Constant 1.0
+Constant 1.2
+Constant 1.5
+Figure 6. Generation quality measured by FID ↓. We compare our
+error-aware scale (∆ϵ/λ) and constant scale based on the 4-order
+Lagrange function. All FID results are evaluated on Cifar10 with
+50k generated samples.
+The results are shown in Fig. 7, from which we can see that
+ERA-Solver achieves lower error in general.
+D. Qualitative Results
+We sample and visualize the generated samples from LSUN-
+Church and LSUN-Bedroom datasets. We evaluate DDIM,
+DPM-Solver, and our method based on 5, 8, 10, 12, 15, and
+20 NFEs. Since PNDM (Liu et al., 2021) requires at least 13
+NFEs to perform sampling, we only compared our method
+with DDIM (Song et al., 2020a) and DPM-Solver-fast (Lu
+et al., 2022a) methods. We set λ = 5.0 and apply a 4-order
+Lagrange interpolation function (k = 4) for comparison.
+The comparison results are shown in Fig. 9 and Fig. 10.
+Compared with DDIM, our ERA-Solver can obtain sharper
+textures, benefiting from its high precision of the implicit
+Adams scheme. Compared with DPM-Solver-fast, our ERA-
+Solver can avoid excessive contrast and exposure, achieving
+more natural texture details.
+We also provide generated samples based on the 4-order
+ERA-Solver with λ = 5.0 on Cifar10, as shown in Fig. 8.
+E. Results on Stable Diffusion
+We conduct generation comparison based on large-scale
+latent diffusion mode, i.e., Stable Diffusion (Rombach et al.,
+2022). We apply the improved version (Lu et al., 2022b) of
+DPM-Solver for better conditional generation. The PNDM
+and improved DPM-Solver are all encapsulated in diffusers
+(von Platen et al., 2022). We directly apply them by diffusers
+to make sampling. For ERA-Solver, we set k = 4 and
+λ = 10.0. The results are shown in Fig. 11 and Fig. 12. It
+can be observed that ERA-Solver can generate promising
+images when NFE is 15, which is faster than DPM-Solver
+and PNDM.
+We also provide computation time for each sampling pro-
+cess, as shown in Tab. 7. It can be observed that at NFE
+15, ERA-Solver consumes slightly more time (0.08s) than
+DPM-Solver, which can be contributed to the cost of main-
+taining the Lagrange buffer. The cost of the Lagrange buffer
+will increase with NFE increasing. However, ERA-Solver
+has been able to generate exquisite images on NFE 15. Thus,
+the computation cost of the Lagrange buffer can be ignored.
+
+Error-Robust Adams Solver
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+time t
+0
+50
+100
+150
+200
+250
+300
+350
+||
+(xgen
+t
+, t)||2
+DPM-Solver
+ERA-Solver (Ours)
+Normal Implicit Adams
+Figure 7. The error comparison between three fast solvers. For ERA-Solver, we set k = 4 and λ = 5.0. The random seed and pretrained
+model are shared with three solvers.
+Table 6. Generation quality measured by FID ↓ on Celeba, varying the number of function evaluation (NFE).
+Sampling method \ NFE
+5
+10
+12
+15
+20
+40
+50
+100
+DDIM (Song et al., 2020a)
+24.26
+13.4
+13.23
+11.63
+9.62
+6.87
+6.08
+4.5
+Analytic-DDPM (Bao et al., 2022)
+\
+28.99
+25.27
+21.80
+18.14
+\
+11.23
+\
+Analytic-DDIM (Bao et al., 2022)
+\
+15.62
+13.90
+12.29
+10.45
+\
+6.13
+\
+PNDM (Liu et al., 2021)
+\
+\
+\
+10.91
+7.59
+4.06
+3.45
+2.97
+DPM-Solver-fast (Lu et al., 2022a)
+\
+6.92
+4.20
+3.05
+2.82
+2.71 (NFE = 36)
+ERA-Solver
+23.32
+5.06
+3.67
+2.99
+2.75
+2.69
+2.73
+2.72
+Figure 8. Generated samples from ERA-Solver-4 (20 NFE).
+Table 7. Computation time of sampling on Stable Diffusion, vary-
+ing different solvers and NFE.
+Sampling Method \ NFE
+15
+25
+50
+PNDM (Liu et al., 2021)
+2.69
+3.74
+6.13
+DPM-Solver (Lu et al., 2022a)
+1.86
+2.76
+5.30
+ERA-Solver
+1.94
+3.05
+6.01
+
+Error-Robust Adams Solver
+DDIM
+DPM-
+Solver-
+fast​
+Ours
+NFE=5
+NFE=8
+NFE=10
+NFE=12
+NFE=15
+NFE=20
+DDIM
+DPM-
+Solver-
+fast​
+Ours
+DDIM
+DPM-
+Solver-
+fast​
+Ours
+Figure 9. Generation quality comparison with 5, 8, 10, 12, 15, and 20 NFEs on LSUN-Church dataset.
+
+Error-Robust Adams Solver
+DDIM
+DPM-
+Solver-
+fast​
+Ours
+NFE=5
+NFE=8
+NFE=10
+NFE=12
+NFE=15
+NFE=20
+DDIM
+DPM-
+Solver-
+fast​
+Ours
+DDIM
+DPM-
+Solver-
+fast​
+Ours
+Figure 10. Generation quality comparison with 5, 8, 10, 12, 15, and 20 NFEs on LSUN-Bedroom dataset.
+
+Error-Robust Adams Solver
+PNDM
+DPM-Solver
+Ours
+NFE=15
+NFE=25
+NFE=50
+Prompt: Cute and adorable ferret wizard, wearing coat and suit
+Figure 11. Samples using the pretrained Stable-Diffusion (Rombach et al., 2022) with a classifier-free guidance scale 7.5 (the default
+setting), varying different solvers and NFEs. The main part of input prompt is: “Cute and adorable ferret wizard, wearing coat and suit”.
+
+Error-Robust Adams Solver
+NFE=15
+NFE=25
+NFE=50
+PNDM
+DPM-Solver
+Ours
+Figure 12. Samples using the pretrained Stable-Diffusion (Rombach et al., 2022) with a classifier-free guidance scale 7.5 (the default
+setting), varying different solvers and NFEs. The main part of input prompt is: “A beautiful mansion beside a waterfall in the woods”.
+

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+arXiv:2301.02233v1  [math.OA]  5 Jan 2023
+THE STABLE EXOTIC CUNTZ ALGEBRAS ARE HIGHER-RANK
+GRAPH ALGEBRAS
+JEFFREY L. BOERSEMA AND SARAH L. BROWNE AND ELIZABETH GILLASPY
+Abstract. For each odd integer n ≥ 3, we construct a rank-3 graph Λn with involution γn
+whose real C∗-algebra C∗
+R (Λn, γn) is stably isomorphic to the exotic Cuntz algebra E
+R
+n. This
+construction is optimal, as we prove that a rank-2 graph with involution (Λ, γ) can never
+satisfy C∗
+R (Λ, γ) ∼ME E
+R
+n, and the first author reached the same conclusion for rank-1 graphs
+(directed graphs) in [Boe17, Corollary 4.3]. Our construction relies on a rank-1 graph with
+involution (Λ, γ) whose real C∗-algebra C∗
+R (Λ, γ) is stably isomorphic to the suspension SR.
+In the Appendix, we show that the i-fold suspension SiR is stably isomorphic to a graph
+algebra iff −2 ≤ i ≤ 1.
+1. Introduction
+For every odd integer n ≥ 3, the (complex) Cuntz algebra On has two real forms: the real
+Cuntz algebra O
+R
+n and the exotic Cuntz algebra En. While the existence of En follows from the
+classification of simple purely infinite real C∗-algebras [Boe06, BRS11], the non-constructive
+nature of the existence portion of this classification theorem [Boe06, Theorem 1] means that
+we know very little about En beyond its K-theory. In particular, until now there has been
+no construction or representation of En in terms of familiar C∗-algebraic objects.
+In this paper, we give an explicit realization of the stabilized exotic Cuntz algebras KR ⊗R
+En as higher-rank graph algebras associated to rank-3 graphs with involution. Given the
+extensive literature on the properties of higher-rank graph C∗-algebras, we anticipate that
+this concrete description will facilitate an improved understanding of these elusive algebras.
+Higher-rank graphs, or k-graphs, are a k-dimensional generalization of directed graphs
+which were introduced by Kumjian and Pask in [KP00]. Many of the properties of (complex)
+directed graph C∗-algebras, such as their K-theory [RS04] and their ideal structure [BHRS02,
+HS04], are visible from the graph.
+While the structure of k-graph C∗-algebras is more
+intricate than that of graph C∗-algebras, k-graph C∗-algebras also encompass a broader
+range of examples. Indeed [RSS15], every complex UCT Kirchberg algebra is a direct limit
+of 2-graph C∗-algebras. The real C∗-algebra C∗
+R(Λ, γ) of a higher-rank graph with involution
+(Λ, γ) was recently introduced by the first and third authors in [BG22]. In that paper, the
+authors also generalized the work of [Eva08] and [Boe17] to describe a spectral sequence
+which converges to the CR K-theory of these real C∗-algebras.
+The main result (Theorem 4.3) of the present paper, that the exotic Cuntz algebra is stably
+isomorphic to the C∗-algebra of a 3-graph with involution, is the best possible in terms of
+the rank of Λ. In [Boe17], the first author made an extensive analysis of the K-theory of the
+real C∗-algebra C∗
+R(Λ, γ) of a rank-1 graph (directed graph) with involution. 1 In particular,
+[Boe17, Corollary 4.3] establishes that the exotic Cuntz algebra cannot be isomorphic or
+stably isomorphic to the real C∗-algebra C∗
+R(Λ) of a directed graph Λ, or to the real C∗-
+algebra C∗
+R(Λ, γ) of a graph with involution, since KO7(En) = Z2 but KO7(C∗
+R(Λ, γ)) is
+always torsion-free. Theorem 3.1 below uses the K-theory spectral sequence for real higher-
+rank graph C∗-algebras ([BG22, Section 3]) to show that En ̸∼ME C∗
+R(Λ, γ)) for any rank-2
+1This class of C∗-algebras includes the real C∗-algebras C∗
+R (Λ) of a directed graph, introduced in [Boe14],
+as C∗
+R (Λ) ∼= C∗
+R (Λ, γtriv).
+1
+
+2
+JEFFREY L. BOERSEMA AND SARAH L. BROWNE AND ELIZABETH GILLASPY
+graph with involution (Λ, γ).
+However, we construct in Theorem 4.3 a family of rank-3
+graphs with involution (Λn, γn) such that C∗
+R(Λn, γn) ∼= En ⊗R KR.
+Our construction combines the directed graphs with involution (En, γn) of [Boe17, Exam-
+ple 6.2], which satisfy C∗
+R(En, γn) ∼ME S6En, with a directed graph with involution (Λ, γ)
+such that (Proposition 4.1) C∗
+R(Λ, γ) is a real Kirchberg algebra which is KK-equivalent
+to the suspension algebra SR ∼= C0((0, 1)).
+To be precise, the 3-graph Λn which gives
+C∗
+R(Λn, γn) ∼= En ⊗R KR is a product graph, Λn = En × Λ × Λ.
+Prompted by the graph with involution of Proposition 4.1, we consider in Section 5 the
+question of which suspensions SiR are KK-equivalent to the real C∗-algebra of a graph with
+involution. For −2 ≤ i ≤ 1 we exhibit an example of a graph with involution (Λ, γ) such that
+C∗
+R(Λ, γ) is KK-equivalent to SiR, and we show in Proposition 5.2 that SiR ̸∼KK C∗
+R(Λ, γ)
+if 2 ≤ i ≤ 5. (However, we can realize these suspensions as 2-graph or 3-graph algebras, by
+taking products of the graphs which do realize suspensions of R.)
+Many key questions remain open for further investigation about the class of real C∗-
+algebras that can be obtained using higher-rank graphs. For example, it is still unknown
+whether or not En itself can be realized as a rank-k graph-with-involution algebra. Similarly,
+it remains unknown which real Kirchberg algebras can be realized by higher-rank graphs
+with involution (as opposed to inductive limits of such objects); we would particularly like
+to find a K-theoretic characterization of such algebras.
+Acknowledgments: E.G. was partially supported by NSF grant 1800749.
+2. Preliminaries
+2.1. Higher-rank graphs.
+Definition 2.1. [KP00, Definition 1.1] A higher-rank graph of rank k, or a k-graph, is
+a countable small category Λ equipped with a degree functor d: Λ → Nk such that, if a
+morphism λ ∈ Λ satisfies d(λ) = m + n, then there exist unique morphisms µ, ν ∈ Λ such
+that λ = µν, d(µ) = m and d(ν) = n.
+Write ei for the standard ith basis vector of Nk.
+The morphisms of degree ei can be
+advantageously viewed as the “edges of color i” in Λ. In this perspective, if e is an edge of
+color i and f is an edge of color j, their composition ef ∈ Λ satisfies
+d(ef) = ei + ej = ej + ei,
+so we must be able to rewrite ef = f ′e′ for some morphisms e′, f ′ ∈ Λ with d(f ′) = ej and
+d(e′) = ei.
+Indeed, by [HRSW13, Theorems 4.4 and 4.5], a k-graph can be equivalently thought of
+as arising from a directed graph G, with k colors of edges and with a factorization rule on
+multicolored paths. That is, given any two colors (“red” and “blue”) and any two vertices
+v, w in G, the factorization rule identifies each red-blue path ef from v to w with an equivalent
+blue-red path f ′e′ from v to w.
+We would like the quotient of the space G∗ of directed paths in G by the equivalence
+relation ∼ generated by the factorization rule to be a k-graph.
+For this to occur, the
+factorization rule must also satisfy certain consistency conditions which ensure that, for each
+path in G∗, its equivalence class under ∼ corresponds to a k-dimensional hyper-rectangle;
+see [EFG+21, Theorem 2.3] for more details. As our work in this paper does not depend on
+these consistency conditions, we will not reproduce them here. That said, we remark that in
+
+THE STABLE EXOTIC CUNTZ ALGEBRAS ARE HIGHER-RANK GRAPH ALGEBRAS
+3
+a rank-1 graph, the factorization rule is nonexistent, and so a 1-graph is precisely the space
+of paths of a directed graph.
+Let Λ be a k-graph. Given n ∈ Nk and objects v, w ∈ Λ, we write
+(1)
+Λn = {λ ∈ Λ : d(λ) = n}.
+By the factorization rule, for every λ ∈ Λ, there are unique v, w ∈ Λ0 with vλ = λw =
+vλw = λ. That is, we can identify Λ0 with the objects of Λ. If λ = vλw, we write v = r(λ)
+and w = s(λ). Thus, expanding on Equation (1), we have
+vΛn = {λ ∈ Λ : r(λ) = v and d(λ) = n}
+Λnw = {λ ∈ Λ : s(λ) = w and d(λ) = n},
+(2)
+as well as the obvious variations such as vΛnw.
+A k-graph Λ has k adjacency matrices Mi ∈ MΛ0(N), which are given by
+(3)
+Mi(v, w) = #vΛeiw.
+In the graphical picture, λ ∈ Λ(n1,...,nk) means that λ represents the ∼-equivalence class of
+a path with ni edges of color i, for each 1 ≤ i ≤ k. That is, Λ0 consists of the length-0 paths,
+ie, the vertices. Then Mi(v, w) is the number of edges of color i from vertex w to vertex v.
+If Λ1 is a k1-graph and Λ2 is a k2-graph, then [KP00, Proposition 1.8] their (Cartesian)
+product Λ1 ×Λ2 is a (k1 +k2)-graph; the degree functor is given by d(λ1, λ2) = (d(λ1), d(λ2).
+We have (Λ1 × Λ2)0 = Λ0
+1 × Λ0
+2 and s(λ1 × λ2) = (s(λ1), s(λ2)).
+In this paper we will focus on k-graphs which are row-finite and source-free (or have no
+sources). We say a k-graph Λ is row-finite if |vΛn| < ∞ for all n ∈ Nk and v ∈ Λ0. The
+k-graph has no sources if vΛn ̸= ∅ for all v, n. It is straightforward to check that if Λ1, Λ2
+are row-finite and source-free, then so is Λ1 × Λ2.
+For a row-finite source-free k-graph Λ, its (complex) C∗-algebra is the universal C∗-algebra
+generated by a Cuntz–Krieger Λ-family.
+Definition 2.2. [KP00, Definition 1.5] Given a row-finite source-free k-graph Λ, a Cuntz–Krieger
+Λ-family is a collection {tλ}λ∈Λ of partial isometries in a C∗-algebra A which satisfy the fol-
+lowing conditions:
+(CK1) For each v ∈ Λ0, tv is a projection, and tvtw = δv,wtv.
+(CK2) For each λ ∈ Λ, t∗
+λtλ = ts(λ).
+(CK3) For each λ, µ ∈ Λ, tλtµ = tλµ.
+(CK4) For each v ∈ Λ0 and each n ∈ Nk, tv =
+�
+λ∈vΛn
+tλt∗
+λ.
+We define C∗(Λ) to be the universal (complex) C∗-algebra generated by a Cuntz–Krieger
+family, in the sense that for any Cuntz–Krieger Λ-family {tλ}λ∈Λ, there is a surjective ∗-
+homomorphism C∗(Λ) → C∗({tλ}λ).
+We write {sλ}λ∈Λ for the generators of C∗(Λ). By using the Cuntz–Krieger relations,
+one can compute that C∗(Λ) = span{sλs∗
+µ : s(λ) = s(µ)}. Corollary 3.5(iv) of [KP00] also
+establishes that C∗(Λ1 × Λ2) ∼= C∗(Λ1) ⊗ C∗(Λ2); the isomorphism takes s(λ1,λ2) to sλ1 ⊗ sλ2.
+We will use one more ingredient – an involution – to construct the real C∗-algebras asso-
+ciated to higher-rank graphs.
+Definition 2.3. An involution γ on a k-graph Λ is a degree-preserving functor γ : Λ → Λ
+which satisfies γ ◦ γ = id Λ.
+
+4
+JEFFREY L. BOERSEMA AND SARAH L. BROWNE AND ELIZABETH GILLASPY
+As established in [BG22, Lemma 2.4], the real C∗-algebra associated to a k-graph Λ and
+an involution γ : Λ → Λ is
+(4)
+C∗
+R(Λ, γ) = spanR{zsλs∗
+µ + zsγ(λ)s∗
+γ(µ) | z ∈ C, λ, µ ∈ Λ}.
+Equivalently, we have C∗
+R(Λ, γ) = {a ∈ C∗(Λ) | �γ(a) = a∗}, where �γ is the antimultiplicative
+C∗-involution uniquely determined by �γ(sλ) = s∗
+γ(λ). For any involution γ on Λ, we have
+C∗(Λ) ∼= C ⊗R C∗
+R(Λ, γ).
+2.2. CRT K-theory. In our work, we will use the full united K-theory K
+CRT(A) (introduced
+in [Boe02]) as well as the abbreviated variation K
+CR(A) which contains just the real and
+complex parts. Theorem 10.2 of [BRS11] shows that the category of real purely infinite
+simple C∗-algebras, whose complexifications are simple and in the UCT class, is classified up
+to isomorphism by either of these invariants. We tend to use K
+CR(A) since it is simpler and
+usually sufficient, but we will also need to use K
+CRT(A) on occasion since that is the context
+in which we have the K¨unneth formula. Specifically, recall that for a real C∗-algebra A,
+K
+CR(A) = {KO∗(A), KU∗(A)}
+K
+CRT(A) = {KO∗(A), KU∗(A), KT∗(A)}
+where KO∗(A) is the standard 8-periodic real K-theory for a real C∗-algebra and KU∗(A) =
+K∗(C ⊗C A) is the 2-periodic K-theory of the complexification of A. Meanwhile KT∗(A) is
+the 4-periodic self-conjugate K-theory. These invariants also include the additional CR and
+CRT -module structure. In particular for K
+CR(A) there are natural transformations
+ri : KUi(A) → KOi(A)
+induced by the standard inclusion C → M2(R)
+ci : KOi(A) → KUi(A)
+induced by the standard inclusion R → C
+ψi : KUi(A) → KUi(A)
+induced by conjugation C → C
+ηi : KOi(A) → KOi+1(A)
+induced by multiplication by η ∈ KO1(R) = Z2
+ξi : KOi(A) → KOi+1(A)
+induced by multiplication by ξ ∈ KO4(R) = Z.
+This additional structure tends to aid in the computations of KO∗(A) because the natural
+transformations satisfy the relations
+rc = 2
+cr = 1 + ψ
+2η = 0
+rψ = r
+ψ2 = id
+η3 = 0
+ψc = c
+ψβU = −βUψ
+ξ = rβ2
+Uc
+and they fit into a long exact sequence
+(5)
+· · ·
+rβ−1
+U
+−−−→ KOi(A)
+η−→ KOi+1(A)
+c−→ KUi+1(A)
+rβ−1
+U
+−−−→ KOi−1(A)
+η−→ · · ·
+These two invariants K
+CR(A) and K
+CRT(A) contain the same information by results of
+[Hew96] (summarized, for example, in [BRS11, Proposition 2.5]).
+2.3. K-theory for higher-rank graphs. For the real C∗-algebra C∗
+R(Λ, γ) of a higher-
+rank graph with involution, [BG22, Theorem 3.10] establishes the existence of a spectral
+sequence {Er, dr} of CR-modules that converges to K
+CR(C∗
+R(Λ, γ)). The complex part of this
+spectral sequence (Er
+p,q)
+U coincides with the Evans spectral sequence [Eva08] and converges
+to KU∗(C∗
+R(Λ, γ)) = K∗(C∗(Λ)). The real part of this spectral sequence (Er
+p,q)
+O converges to
+KO∗(C∗
+R(Λ, γ)).
+
+THE STABLE EXOTIC CUNTZ ALGEBRAS ARE HIGHER-RANK GRAPH ALGEBRAS
+5
+The E2 page of the spectral sequence arises from the homology of a certain chain complex C
+based on the combinatorial information of Λ and γ. We will use the spectral sequence only in
+the rank-1 and rank-2 cases, where these chain complexes have the following straightforward
+descriptions which we recall from [BG22, Theorems 3.14 and 3.15]. First let Λ0
+f be the set
+of vertices fixed by the involution γ and let Λ0
+g ⊔ Λ0
+h be any partition of Λ0\Λ0
+f satisfying
+γ(Λ0
+g) = Λ0
+h and γ(Λ0
+h) = Λ0
+g. Then let A = K
+CR(R)Λ0
+f ⊕ K
+CR(C)Λ0
+g.
+For a rank-1 graph with involution the chain complex C is given by
+0 → A
+∂1
+−→ A → 0,
+where ∂1 = ρ1. For a rank-2 graph with involution the chain complex C is given by
+0 → A
+∂2
+−→ A2 ∂1
+−→ A → 0,
+where ∂1 =
+�
+ρ1
+ρ2�
+and ∂2 =
+�
+−ρ2
+ρ1
+�
+. Here the maps ρi are determined by the adjacency
+structure of Λ as follows. For 1 ≤ i ≤ k, the complex part (ρi)
+U
+0 : ZΛ0 → ZΛ0 is represented
+by the matrix Bi = I − Mt
+i , where Mi is the adjacency matrix of the graph Λ for the edges
+of degree ei, and (ρi)
+U
+1 = 0 for all i. The real parts of this map (ρi)
+O
+j , for 0 ≤ j ≤ 7, can
+similarly be determined for each i by some variations of Bi as shown in Table 3 of [BG22]
+and Theorem 4.4 of [Boe17], which we also reproduce here as Table 1.
+complex part
+0
+
+
+B11
+B12
+B12
+B21
+B22
+B23
+B21
+B23
+B22
+
+
+Z|Λ0
+f | ⊕ Z|Λ0
+g| ⊕ Z|Λ0
+h → Z|Λ0
+f| ⊕ Z|Λ0
+g| ⊕ Z|Λ0
+h
+1
+0
+0
+real part
+0
+�
+B11
+2B12
+B21
+B22 + B23
+�
+Z|Λ0
+f| ⊕ Z|Λ0
+g| → Z|Λ0
+f| ⊕ Z|Λ0
+g|
+1
+B11
+Z
+|Λ0
+f |
+2
+→ Z
+|Λ0
+f|
+2
+2
+�
+B11
+B12
+0
+B22 − B23
+�
+Z
+|Λf|
+2
+⊕ Z|Λ0
+g| → Z
+|Λf|
+2
+⊕ Z|Λ0
+g|
+3
+0
+0
+4
+�
+B11
+B12
+2B21
+B22 + B23
+�
+Z|Λ0
+f| ⊕ Z|Λ0
+g| → Z|Λ0
+f| ⊕ Z|Λ0
+g|
+5
+0
+0
+6
+B22 − B23
+Z|Λ0
+g| → Z|Λ0
+g|
+7
+0
+0
+Table 1. Chain complex maps for real K-theory
+For reference, the groups of K
+CR(R) and K
+CR(C) are shown below. The natural trans-
+formations η, c, r, and ψ that are part of the structure of united K-theory are uniquely
+determined from these groups and the long exact sequence (5); they are also shown in Ta-
+bles 1 and 2 of [BG22]. In particular we note that for KO∗(R), the map ηi is non-trivial
+exactly for i = 0, 1.
+
+6
+JEFFREY L. BOERSEMA AND SARAH L. BROWNE AND ELIZABETH GILLASPY
+0
+1
+2
+3
+4
+5
+6
+7
+KO∗(R)
+Z
+Z2
+Z2
+0
+Z
+0
+0
+0
+KU∗(R)
+Z
+0
+Z
+0
+Z
+0
+Z
+0
+0
+1
+2
+3
+4
+5
+6
+7
+KO∗(C)
+Z
+0
+Z
+0
+Z
+0
+Z
+0
+KU∗(C)
+Z2
+0
+Z2
+0
+Z2
+0
+Z2
+0
+3. Non-Existence of a Rank-2 Graph with Involution
+Theorem 3.1. Let n be an odd integer, n ≥ 3. There does not exist a (row-finite, source-
+free) rank-2 graph with involution (Λ, γ) such that K
+CR(C∗
+R(Λ, γ)) ∼= K
+CR(En).
+Proof. Suppose that (Λ, γ) is a row-finite, source-free rank-2 graph with involution and that
+K
+CR(C∗
+R(Λ, γ)) ∼= K
+CR(En). For reference we reproduce the groups of K
+CR(En) here:
+0
+1
+2
+3
+4
+5
+6
+7
+KO∗(En)
+Z2(n−1)
+Z2
+Z2
+0
+Z(n−1)/2
+0
+Z2
+Z2
+KU∗(En)
+Zn−1
+0
+Zn−1
+0
+Zn−1
+0
+Zn−1
+0
+Note that since KU7(En) = 0 and since im η6 = ker c7 from the long exact sequence (5) relat-
+ing KO∗(A) and KU∗(A), we see immediately that η6: Z2 → Z2 must be an isomorphism.
+Now, we consider only the real part of the spectral sequence from [BG22, Theorem 3.15].
+This is a spectral sequence converging to KO∗(C∗
+R(Λ, γ)), where the E2-page consists of the
+homology of the chain complex
+(6)
+0 → A
+∂2
+−−→ A2
+∂1
+−−→ A → 0
+where A ∼= KO∗(R)Λ0
+f ⊕ KO∗(C)Λ0
+g as described in Section 2.3.
+In particular, the spectral sequence has three non-zero columns (for 0 ≤ p ≤ 2) and
+is periodic in q (with period 8). Furthermore, a quick examination of the structure of A
+(cf. Figure 1) reveals that Ai = 0 for i = 3, 5, 7 and also that Ai is free for i = 0, 4, 6.
+Consequently, E2
+p,q = 0 for q = 3, 5, 7. Moreover, since E2
+2,q = ker ∂2 is a subgroup of Ai, we
+must have E2
+2,q free for q = 0, 4, 6. Indeed, E∞
+2,q = ker d2
+2,q is a subgroup of E2
+2,q, so E∞
+2,q must
+also be free for q = 0, 4, 6. On the other hand, KOi(C∗
+R(Λ, γ)) is finite in all degrees, so it
+must be that E∞
+2,q = 0 for q = 0, 4, 6.
+
+THE STABLE EXOTIC CUNTZ ALGEBRAS ARE HIGHER-RANK GRAPH ALGEBRAS
+7
+From what we have said so far, the E∞
+p,q page of the spectral sequence is as follows:
+...
+...
+...
+7
+0
+0
+0
+6
+∗
+∗
+0
+5
+0
+0
+0
+4
+∗
+∗
+0
+3
+0
+0
+0
+2
+∗
+∗
+∗
+1
+∗
+∗
+∗
+0
+∗
+∗
+0
+q/p
+0
+1
+2
+This E∞ page identifies a filtration of KO∗(C∗
+R(Λ, γ)), in which the subquotients of KOj(C∗
+R(Λ, γ))
+appear along the diagonal p + q = j of the E∞ page.
+By hypothesis we have KO0(C∗
+R(Λ, γ)) = Z2(n−1). However, the only non-zero group along
+the diagonal p + q = 0 is E∞
+0,0. Thus E∞
+0,0 = Z2(n−1). Similarly, since KO7(C∗
+R(Λ, γ)) = Z2
+and KO6(C∗
+R(Λ, γ)) = Z2, we must have E∞
+1,6 = Z2 = E∞
+0,6:
+...
+...
+...
+7
+0
+0
+0
+6
+Z2
+Z2
+0
+5
+0
+0
+0
+4
+∗
+∗
+0
+3
+0
+0
+0
+2
+∗
+∗
+∗
+1
+∗
+∗
+∗
+0
+Z2(n−1)
+∗
+0
+q/p
+0
+1
+2
+Now, we consider the natural transformation η: A → A of degree 1. Because A is a
+direct sum of copies of K
+CR(R) and K
+CR(C), which data already includes the map η, the
+natural transformation η : KO∗(C∗
+R(Λ, γ)) → KO∗+1(C∗
+R(Λ, γ)) exists at the level of the
+chain complex (6), passes to a map on the E2-page, then to a map on the E∞-page, and
+finally converges to the map η: KOi(C∗
+R(Λ, γ)) → KOi+1(C∗
+R(Λ, γ)) described in Section 2.2.
+This means that the map η on KO∗(C∗
+R(Λ, γ)) respects the filtration of KOi(C∗
+R(Λ, γ))
+associated with the spectral sequence; and the resulting maps on the subquotients are the
+same as that on the E∞ page.
+In particular, the diagonals of the E∞-page that yield
+KO6(C∗
+R(Λ, γ)) and KO7(C∗
+R(Λ, γ)) give the commutative diagram below. The vertical η
+maps on the left and right come from A as described above.
+0
+� E∞
+0.6
+�
+η
+�
+KO6(C∗
+R(Λ, γ))
+�
+η
+�
+E∞
+1,5
+�
+η
+�
+0
+0
+� E∞
+0.7
+� KO7(C∗
+R(Λ, γ))
+� E∞
+1,6
+� 0
+⇐⇒
+0
+� Z2
+�
+η
+�
+Z2
+�
+η
+�
+0
+�
+η
+�
+0
+0
+� 0
+� Z2
+� Z2
+� 0
+
+8
+JEFFREY L. BOERSEMA AND SARAH L. BROWNE AND ELIZABETH GILLASPY
+Since the nonzero horizontal maps must be isomorphisms, the commutative diagram forces
+the vertical map η6: KO6(C∗
+R(Λ, γ)) → KO7(C∗
+R(Λ, γ)) in the center of the diagram to be
+zero, which contradicts the known value of η in KO∗(E
+R
+n).
+□
+4. Existence of a Rank-3 Graph with Involution
+In this section, we will construct a 3-graph Λ with involution γ, by taking products of
+1-graphs with involution, such that C∗
+R(Λ, γ) is stably isomorphic to E
+R
+n. In what follows, we
+use the following convention for suspension of graded modules. If H = {Hi} is a Z-graded
+group, then ΣH is a Z-graded group with (ΣH)i = Hi+1. Similarly, Σ−1H is a Z-graded
+group with (Σ−1H)i = Hi−1. This convention is consistent with K-theory and suspensions
+of C∗-algebras: K∗(SnA) = ΣnK∗(A).
+For the proof of the following proposition, we will need a few more preliminaries. For a
+graph Λ, a subset X ⊆ Λ0 is hereditary if whenever v ∈ X and vΛw ̸= ∅, then w ∈ X. A cycle
+in Λ is a path e1e2 · · ·en with r(e1) = s(en) but, for all 1 ≤ i < n, we have s(ei) ̸= r(ei+1).
+We say that a cycle has an entrance if there exists 1 ≤ j ≤ n and an edge f ̸= ej with
+r(f) = r(ej). If the only hereditary subsets of Λ0 are ∅ and Λ0, and every cycle in Λ has an
+entrance, then [Szy01, Theorem 12] C∗(Λ) is simple. Furthermore, if every cycle in Λ has an
+entrance, then Λ is aperiodic.
+Proposition 4.1. There exists a 1-graph Λ and involution γ such that K
+CR(C∗
+R(Λ, γ)) ∼=
+ΣK
+CR(R). Furthermore, C∗
+R(Λ, γ) is simple and purely infinite.
+Proof. Let Λ be the 1-graph below (which extends infinitely in both directions) and let γ be
+the non-trivial involution, which fixes the vertices and edges of the infinite branch on the
+left and swaps the vertices and edges of the two infinite branches on the right in the obvious
+way. (In fact γ is the only non-trivial involution on Λ.)
+•
+�❅
+❅
+❅
+❅
+❅
+❅
+❅
+�
+•
+�❅
+❅
+❅
+❅
+❅
+❅
+❅
+�
+•
+�❅
+❅
+❅
+❅
+❅
+❅
+❅
+�
+•
+�
+�❅
+❅
+❅
+❅
+❅
+❅
+❅
+•
+�
+�❅
+❅
+❅
+❅
+❅
+❅
+❅
+•
+�
+�❅
+❅
+❅
+❅
+❅
+❅
+❅
+•�
+�
+�⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+•
+�
+�⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+•
+�
+�⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+•
+�
+�
+�
+�
+•
+�
+�⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+•
+�
+�⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+•
+�
+�⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+•
+�
+�⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+�
+�❅
+❅
+❅
+❅
+❅
+❅
+❅
+�
+•
+�❅
+❅
+❅
+❅
+❅
+❅
+❅
+•
+�
+�❅
+❅
+❅
+❅
+❅
+❅
+❅
+•
+�
+�❅
+❅
+❅
+❅
+❅
+❅
+❅
+• �
+�
+�
+•
+�⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+�
+•
+�⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+�
+•
+�⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+�
+We begin by showing that C∗
+R(Λ, γ) is simple and purely infinite. It is straightforward to
+check that Λ has no nontrivial hereditary subsets, and that every cycle has an entrance, so
+simplicity of the complex algebra C∗(Λ) follows from [Szy01, Theorem 12]. Note further that
+for every vertex v ∈ Λ0, there is a vertex w with vΛw ̸= ∅ for some vertex w which supports
+a loop. As Λ is aperiodic, one easily checks that the conditions of [KP00, Proposition 4.9]
+are satisfied, and so C∗(Λ) is purely infinite. Consequently, [BRS11, Theorem 3.9] implies
+that the real C∗-algebra C∗
+R(Λ, γ) is also simple and purely infinite.
+We now show that KU0(C∗
+R(Λ, γ)) = 0 and KU1(C∗
+R(Λ, γ)) = Z. (This is the same as
+calculating K∗(C∗(Λ)) and does not involve the involution γ.) Let M be the adjacency matrix
+for Λ (so Mv,w is the number of edges from w to v). Then KU0(C∗
+R(Λ, γ)) ∼= coker (I − Mt),
+
+THE STABLE EXOTIC CUNTZ ALGEBRAS ARE HIGHER-RANK GRAPH ALGEBRAS
+9
+which can be interpreted as saying that KU0(C∗
+R(Λ, γ)) is generated by vertex projection
+classes [pv], which are subject only to relations of the form
+[pv] =
+�
+w∈Λ0
+Mv,w[pw] .
+Let v be one of the vertices of Λ that has a loop, and let w ̸= v be the vertex for which there
+is an edge from w to v (in each case there is a unique such w). Then the formula above
+gives the relation [pv] = [pv] + [pw], which implies that [pw] = 0. If [pw] = 0 we will say
+that w is a zero vertex. Now if w is a zero vertex and there is only one edge to w, say from
+vertex u, then it follows that u is also a zero vertex. More generally, if w is a zero vertex
+and all the edges to w are known to emanate from zero vertices except possibly one edge
+from vertex u, then it follows that u is also a zero vertex. Using these principles, it is now
+straightforward to work through the graph and to find that every vertex is a zero vertex.
+Hence KU0(C∗
+R(Λ, γ)) = 0.
+We know that KU1(C∗
+R(Λ, γ)) ∼= ker(I − Mt), which is to say that
+(7)
+KU1(C∗
+R(Λ, γ)) ∼= NΛ :=
+�
+α: Λ0 → Z | α(v) =
+�
+w∈Λ0
+Mw,v α(w)
+�
+.
+Let v be one of the vertices of Λ that has a loop, and let w ̸= v be the vertex for which
+there is an edge from v to w (in each case there is a unique such w). Then we have the
+relation α(v) = α(v) + α(w), which implies that α(w) = 0 for any α ∈ NΛ. If α(w) = 0 for
+all α ∈ NΛ we will say that w is a null vertex. Now if w is a null vertex and there is only one
+edge emanating from w, say to vertex u, then it follows that u is also a null vertex. More
+generally, if w is a null vertex and all the edges from w are known to point to null vertices
+except possibly one edge to vertex u, then u is also a null vertex. Using these principles,
+it is now straightforward to work through the graph and to find that every vertex is a null
+vertex, except for the six vertices labelled u, v, w, x, y, z shown below.
+v•
+�❇
+❇
+❇
+❇
+❇
+❇
+❇
+❇
+�
+u•
+�❇
+❇
+❇
+❇
+❇
+❇
+❇
+❇
+�
+•
+�❅
+❅
+❅
+❅
+❅
+❅
+❅
+�
+•
+�
+�❅
+❅
+❅
+❅
+❅
+❅
+❅
+•
+�
+�❅
+❅
+❅
+❅
+❅
+❅
+❅
+•
+�
+�❅
+❅
+❅
+❅
+❅
+❅
+❅
+w•�
+�
+�②
+②
+②
+②
+②
+②
+②
+②
+•
+�
+�⑤
+⑤
+⑤
+⑤
+⑤
+⑤
+⑤
+⑤
+•
+�
+�⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+•
+�
+�
+�
+�
+•
+�
+�⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+•
+�
+�⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+•
+�
+�⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+•
+�
+�④
+④
+④
+④
+④
+④
+④
+④
+�
+�❈
+❈
+❈
+❈
+❈
+❈
+❈
+❈
+�
+x•
+�❉
+❉
+❉
+❉
+❉
+❉
+❉
+❉
+•
+�
+�❆
+❆
+❆
+❆
+❆
+❆
+❆
+❆
+•
+�
+�❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+• �
+�
+�
+y•
+�⑥
+⑥
+⑥
+⑥
+⑥
+⑥
+⑥
+⑥
+�
+z•
+�⑥
+⑥
+⑥
+⑥
+⑥
+⑥
+⑥
+⑥
+�
+•
+�⑧
+⑧
+⑧
+⑧
+⑧
+⑧
+⑧
+⑧
+�
+Using Equation (7) and the fact that the unlabeled vertices are null vertices, we see that
+any α ∈ NΛ must satisfy the equations
+0 = α(u) + α(w)
+α(w) = α(v) + α(x)
+0 = α(w) + α(x)
+α(x) = α(w) + α(y)
+0 = α(x) + α(z)
+
+10
+JEFFREY L. BOERSEMA AND SARAH L. BROWNE AND ELIZABETH GILLASPY
+Solving this system over Z, we find that α(u) is a free variable and that
+α(v) = −2α(u), α(w) = −α(u), α(x) = α(u), α(y) = 2α(u), and α(z) = −α(u) .
+Thus NΛ ∼= Z. Hence KU∗(C∗
+R(Λ, γ)) = K∗(C∗(Λ)) = (0, Z).
+Turning to the real K-theory, we now prove that KO∗(C∗
+R(Λ, γ))) = (Z2, Z2, 0, Z, 0, 0, 0, Z).
+First, we show that the real and complex E2 = E∞ page of the Evans spectral sequence for
+C∗
+R(Λ, γ) is as follows.
+E2
+p,q
+(8)
+real part
+...
+...
+...
+7
+0
+0
+6
+0
+Z
+5
+0
+0
+4
+0
+0
+3
+0
+0
+2
+0
+Z
+1
+Z2
+0
+0
+Z2
+0
+q/p
+0
+1
+complex part
+...
+...
+...
+7
+0
+0
+6
+0
+Z
+5
+0
+0
+4
+0
+Z
+3
+0
+0
+2
+0
+Z
+1
+0
+0
+0
+0
+Z
+q/p
+0
+1
+(9)
+We have already discussed the complex part of this spectral sequence. For the real part we
+will only discuss the computations for the rows corresponding to j = −1, 0, 1. As we will see,
+this is enough to determine KO∗(C∗
+R(Λ, γ)). The other rows can be computed using similar
+methods and we include them in the table above for completeness, but we will neither need
+nor discuss them.
+First, the spectral sequence for a 1-graph with involution always vanishes in row j = −1,
+since the chain complex vanishes in that degree. To compute row j = 1 of the spectral
+sequence, we refer to [BG22, Theorem 3.14] and Table 1 above, which indicates that E2
+0,1
+and E2
+1,1 are the cokernel and kernel of the map
+(∂1)1 = I − Mt
+11 : Z
+Λ0
+f
+2
+→ Z
+Λ0
+f
+2
+where Λ0
+f is the set of fixed vertices of (Λ, γ) and M11 is the restriction of the incidence
+matrix to those vertices. So it suffices to consider the graph consisting of the fixed points of
+Λ, shown here.
+•
+�
+�❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+•
+�
+�❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+•x
+�
+�❆
+❆
+❆
+❆
+❆
+❆
+❆
+❆
+•y
+�
+�❈
+❈
+❈
+❈
+❈
+❈
+❈
+❈
+�
+•
+�
+�⑧
+⑧
+⑧
+⑧
+⑧
+⑧
+⑧
+⑧
+•
+�
+�⑧
+⑧
+⑧
+⑧
+⑧
+⑧
+⑧
+⑧
+•
+�
+�⑥
+⑥
+⑥
+⑥
+⑥
+⑥
+⑥
+⑥
+•
+�
+�⑥
+⑥
+⑥
+⑥
+⑥
+⑥
+⑥
+•z
+�
+Using this graph, and the same sort of analysis that we did in the complex case, we find that
+coker (I − Mt
+11) = Z2. More precisely, working modulo 2 we find that [pv] = 0 for all vertices
+in Λ0
+f except those labeled x, y and z in the graph above and that [px] = [py] = [pz] ̸= 0. We
+also find easily that ker(I − Mt
+11) = 0.
+Now, for j = 0, we need to find the cokernel and kernel of the map (∂1)0 which we will
+do using Table 1. First recall the partition Λ0 = Λ0
+f ⊔ Λ0
+g ⊔ Λ0
+h where Λ0
+f is the set of fixed
+
+THE STABLE EXOTIC CUNTZ ALGEBRAS ARE HIGHER-RANK GRAPH ALGEBRAS
+11
+vertices (the branch on the left of Λ), Λ0
+g is the set of vertices of the “upper right” branch of
+Λ, and Λ0
+h is the set of vertices of the “lower right” branch. With this structure on Λ, the
+(infinite) matrix B = I − Mt can be written in block form as
+B = I − Mt =
+���
+
+B11
+B12
+B12
+B21
+B22
+B23
+B21
+B23
+B33
+
+
+where, for example, B12 keeps track of edges from vertices in Λ0
+f to vertices in Λ0
+g. Using
+Table 1, we see that
+(∂1)0 : ZΛ0
+f ⊕ ZΛ0
+g → ZΛ0
+f ⊕ ZΛ0
+g
+is given by
+(∂1)0 =
+�
+B11
+2B12
+B21
+B22 + B23
+�
+.
+We will use a new graph Λ′ to analyze this map. The graph Λ′, shown below, is obtained
+from Λ by keeping the vertices from Λ0
+f and Λ0
+g. For each edge in Λ from a vertex in Λ0
+g to
+a vertex in Λ0
+f, we create a corresponding edge in Λ′ and for each edge in Λ from a vertex
+in Λ0
+f to a vertex in Λ0
+g we create 2 corresponding edges in Λ′. Also, for each edge from a
+vertex v = γ(u) ∈ Λ0
+h to a vertex w ∈ Λ0
+g we obtain an edge in Λ′ from u to w.
+•x
+�
+�❆
+❆
+❆
+❆
+❆
+❆
+❆
+❆
+�
+⑥⑥⑥⑥⑥⑥⑥⑥
+•z
+�
+�❈
+❈
+❈
+❈
+❈
+❈
+❈
+❈
+�
+⑥⑥⑥⑥⑥⑥⑥⑥
+•
+�❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+�
+•
+�❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+�
+�
+•
+�
+•
+�
+•y
+(2)
+�
+�
+•w
+�
+�
+�⑤
+⑤
+⑤
+⑤
+⑤
+⑤
+⑤
+⑤
+•
+�
+�⑧
+⑧
+⑧
+⑧
+⑧
+⑧
+⑧
+⑧
+•
+�
+�
+�
+By construction the adjacency matrix M′ for the graph Λ′ satisfies
+I − (M′)t =
+�
+B11
+2B12
+B21
+B22 + B23
+�
+.
+Therefore, we can use the graph Λ′ to find the cokernel and kernel of (∂1)0. Using the same
+logic and terminology we used when calculating the complex K-theory, we see that w is
+a zero vertex because it emits an edge to a vertex v which supports a loop, and the edge
+from w to v is the only non-loop edge which points to v. Indeed, every vertex along the
+bottom row of the graph Λ′ except y is a zero vertex. The fact that these zero vertices
+(with the exception of w) only receive edges from one (potentially) nonzero vertex on the
+top row of Λ′ implies that every vertex in the top row except x and z are also zero vertices.
+Now, w is a zero vertex, but since there are two edges from y to w we obtain the relation
+[pw] = [pw] + 2[py] which implies that 2[py] = 0, but [py] ̸= 0. Finally, from the relations
+[px] = −[py] and [pz] = [py] we conclude that coker (∂1)0 = Z2.
+To compute ker(∂1)0, we also proceed as in the computations for the complex case: All
+of the vertices in the bottom row of Λ′, save w, are null vertices. Moreover, if a null vertex
+v emits n edges to a single potentially non-null vertex u, we must have nα(u) = 0 for any
+α ∈ NΛ′, and as α(u) ∈ Z we conclude that u must also be null. It follows that w is null, as
+are all of the vertices in the top row of Λ′. That is, ker(∂1)0 = {0}.
+Now, with the three rows that we’ve identified, the spectral sequence (8) implies that
+KO0(C∗
+R(Λ, γ)) ∼= KO1(C∗
+R(Λ, γ)) ∼= Z2. We claim that using this we can compute KOi(C∗
+R(Λ, γ))
+for 2 ≤ i ≤ 7 using the long exact sequence (5) and other aspects of CR-structure. The
+fact that KU6(C∗
+R(Λ, γ)) = KU0(C∗
+R(Λ, γ)) = 0 implies that η0 is injective, and hence an
+
+12
+JEFFREY L. BOERSEMA AND SARAH L. BROWNE AND ELIZABETH GILLASPY
+isomorphism, and also that η−1 is surjective. Since KU2(C∗
+R(Λ, γ)) = 0 it follows that η1 is
+surjective. Thus if KO2(C∗
+R(Λ, γ)) has a non-zero element, then it would have to be in the
+image of η1 ◦ η0 ◦ η−1 = η3. But η3 = 0 for all real C∗-algebras. Thus KO2(C∗
+R(Λ, γ)) = 0.
+Since KO2(C∗
+R(Λ, γ)) = 0, the long exact sequence implies that c3: KO3(C∗
+R(Λ, γ)) →
+KU3(C∗
+R(Λ, γ)) = Z is injective. This forces KO3(C∗
+R(Λ, γ)) = Z. Moreover, as im r1 =
+ker η1 = Z2 we must have r1 : Z → Z2 the unique nonzero map. Since im c3 ∼= ker r1, we
+conclude that c3 is multiplication by 2. The relation rc = 2 then implies that r3 = 1.
+Continuing this process using the long exact sequence, we compute that KO∗(C∗
+R(Λ, γ))) =
+(Z2, Z2, 0, Z, 0, 0, 0, Z). The module maps η, r, c, ψ are then completely determined by these
+groups and the long exact sequence (5); that is, K
+CR(C∗
+R(Λ, γ)) and hence K
+CRT(C∗
+R(Λ, γ))
+coincide with ΣK
+CRT(R).
+□
+Lemma 4.2. Suppose that (Λ1, γ1) and (Λ2, γ2) are higher-rank graphs with involutions.
+Then (Λ1 × Λ2, γ1 × γ2) is a higher-rank graph with involution and
+C∗
+R(Λ1 × Λ2, γ1 × γ2) ∼= C∗
+R(Λ1, γ1) ⊗R C∗
+R(Λ2, γ2) .
+Proof. Assume that Λ1 and Λ2 have rank k1 and k2 respectively.
+From [KP00, Proposi-
+tion 1.8] the product Λ1 × Λ2 is a graph of rank k1 + k2, with degree functor d(λ1, λ2) =
+d(λ1) + d(λ2). Furthermore, there is an involution γ on Λ1 × Λ2 defined by γ(λ1, λ2) =
+(γ1(λ1), γ2(λ2)).
+From [KP00, Corollary 3.5], there is an isomorphism φ: C∗(Λ1 × Λ2) → C∗(Λ1) ⊗ C∗(Λ2)
+defined by φ(s(λ1,λ2)) = sλ1 ⊗sλ2. To finish the proof, we need only show that φ preserves the
+real structures (4) of C∗(Λ1 × Λ2) and C∗(Λ1) ⊗ C∗(Λ2) which are induced by the graphical
+involutions γi. This is straightforward:
+φ(�γ(s(λ1,λ2)) = φ(s∗
+(γ1(λ1),γ2(λ2))))
+= s∗
+γ1(λ1) ⊗ s∗
+γ2(λ2)
+= �γ1(sλ1) ⊗ �γ2(sλ2)
+= �
+γ1 ⊗ γ2(sλ1 ⊗ sλ2)
+= �
+γ1 ⊗ γ2(φ(s(λ1,λ2))) .
+□
+Theorem 4.3. Let n be an odd integer, n ≥ 3. There exists a rank-3 graph with invo-
+lution (Λn, γn) such that C∗
+R(Λn, γn) ∼= K
+R ⊗R E
+R
+n.
+Furthermore, there exists a projection
+p ∈ C∗
+R(Λn, γn) such that pC∗
+R(Λn, γn)p ∼= E
+R
+n.
+Proof. Let (Λ, γ) be the 1-graph given by Proposition 4.1 and let (En, γn) be the finite 1-
+graph with involution from Example 6.2 of [Boe17]. Then both C∗
+R(Λ, γ) and C∗
+R(En, γn) are
+simple and purely infinite and we have
+K
+CR(C∗
+R(Λ, γ)) ∼= ΣK
+CR(R)
+and
+K
+CR(C∗
+R(En, γn)) ∼= Σ6K
+CR(En) .
+This implies by [BRS11, Proposition 2.1] that
+K
+CRT(C∗
+R(Λ, γ)) ∼= ΣK
+CRT(R)
+and
+K
+CRT(C∗
+R(En, γn)) ∼= Σ6K
+CRT(En) .
+Let (Λn, γn) be the product rank-3 graph with involution
+(Λn, γn) = (Λ, γ) × (Λ, γ) × (En, γn).
+
+THE STABLE EXOTIC CUNTZ ALGEBRAS ARE HIGHER-RANK GRAPH ALGEBRAS
+13
+Lemma 4.2 then implies that
+C∗
+R(Λn, γn) ∼= C∗
+R(Λ, γ) ⊗R C∗
+R(Λ, γ) ⊗R C∗
+R(En, γn).
+Now, K
+CRT(C∗
+R(Λ, γ)) is a free CRT -module, since it is isomorphic to a suspension of K
+CRT(R)
+(see [Boe02, Section 2.1]). Therefore the K¨unneth formula for the K-theory of real C∗-
+algebras (Proposition 3.5 and Theorem 4.2 of [Boe02]) gives
+K
+CRT(C∗
+R(Λn, γn)) ∼= K
+CRT(C∗
+R(Λ, γ)) ⊗CRT K
+CRT(C∗
+R(Λ, γ)) ⊗CRT K
+CRT(C∗
+R(En, γn))
+∼= Σ2K
+CRT(R) ⊗CRT Σ6K
+CRT(En)
+∼= K
+CRT(En) .
+Note that C∗
+R(Λn, γn) is a stable, simple, purely infinite, real C∗-algebra, thanks to Propo-
+sition 4.1 and [Boe17, Example 6.2]. We also know that KR ⊗R En is a a stable, simple,
+purely infinite, real C∗-algebra, because its complexification K ⊗ On is simple and purely
+infinite (see Theorem 3.9 of [BRS11]). Thus the first statement of the theorem follows by
+the classification of real Kirchberg algebras, [BRS11, Theorem 10.2, Part (1)].
+To prove the second statement, by [BRS11, Proposition 3.13] there is a projection p ∈
+C∗
+R(Λn, γn) such that [p] is a generator of KO0(C∗
+R(Λn, γn)) = Z2(n−1). Then
+K
+CR(pC∗
+R(Λn, γn)p) ∼= K
+CR(C∗
+R(Λn, γn)) ∼= K
+CR(En)
+(where the first isomorphism is by [Boe06, Proposition 9]). Furthermore the class of the
+identity [p] ∈ KO0(pC∗
+R(Λ, γ)p) ∼= Z2(n−1) corresponds under this isomorphism to the class
+of the identity [1] ∈ KO0(En) ∼= Z2(n−1). Therefore by [BRS11, Theorem 10.2, Part (2)], we
+have pC∗
+R(Λ, γ)p ∼= En.
+□
+5. Appendix – real Kirchberg suspension algebras
+In the previous section, we introduced a graph with involution for which K
+CR(C∗(Λ, γ)) ∼=
+ΣK
+CR(R). We consider this algebra as a sort of real Kirchberg suspension, since it is a real
+purely infinite simple stable nuclear C∗-algebra satisfying the UCT, and with the same KK-
+type as the suspension algebra SR ∼= C0((0, 1), R). By repeatedly taking the product of this
+graph with itself, which corresponds to repeatedly tensoring this algebra with itself, we can
+obtain a higher-rank graph, the real C∗-algebra of which is a real Kirchberg algebra with the
+same KK-type as SiR for any i. These tensor products will be higher-rank graph algebras
+of rank i. It is natural to ask which of these suspensions can be obtained from a 1-graph
+with involution. In this section, we will answer this question completely, providing a full
+characterization of the integers i (mod 8) for which there exists a 1-graph with involution
+(Λ, γ) such that K
+CR(C∗(Λ, γ)) ∼= ΣiK
+CR(R) ∼= K
+CR(SiR). For the positive results, we will
+exhibit directly the appropriate graph or graph with involution.
+Proposition 5.1. For each i = {−2, −1, 0, 1} there exists a 1-graph with involution (Λ, γ)
+such that K
+CR(C∗(Λ, γ)) ∼= ΣiK
+CR(R). Furthermore, C∗
+R(Λ, γ) is simple and purely infinite.
+Sketch of proof. For each i we show below a graph or graph with involution that satisfies
+K
+CR(C∗(Λ, γ)) ∼= ΣiK
+CR(R). The K-theory calculations, not shown, are carried out using
+the same techniques as in the proof of Proposition 4.1.
+
+14
+JEFFREY L. BOERSEMA AND SARAH L. BROWNE AND ELIZABETH GILLASPY
+i = −2. The graph Λ is shown below; we equip it with the non-trivial involution γ which
+interchanges the right-hand branches.
+• �
+❅
+❅
+❅
+❅
+❅
+❅
+❅
+�⑦⑦⑦⑦⑦⑦⑦
+�
+• �
+❅
+❅
+❅
+❅
+❅
+❅
+❅
+�⑦⑦⑦⑦⑦⑦⑦
+�
+• �
+❅
+❅
+❅
+❅
+❅
+❅
+❅
+�⑦⑦⑦⑦⑦⑦⑦
+�
+•
+�
+�⑦⑦⑦⑦⑦⑦⑦
+�
+❅
+❅
+❅
+❅
+❅
+❅
+❅
+•
+�
+�⑦⑦⑦⑦⑦⑦⑦
+�
+❅
+❅
+❅
+❅
+❅
+❅
+❅
+•
+�
+�⑦⑦⑦⑦⑦⑦⑦
+�
+❅
+❅
+❅
+❅
+❅
+❅
+❅
+•�
+�
+� •
+� •
+� •
+�
+�
+� •
+�
+�
+•
+� •
+� •
+�
+�⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+�
+�❅
+❅
+❅
+❅
+❅
+❅
+❅
+•
+� •
+� •
+� •
+�
+�
+•
+�
+⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+�❅❅❅❅❅❅❅
+�
+•
+�
+⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+�❅❅❅❅❅❅❅
+�
+•
+�
+⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+�❅❅❅❅❅❅❅
+�
+i = −1. The graph Λ is shown below, with trivial involution γ = id .
+•
+�❅
+❅
+❅
+❅
+❅
+❅
+❅
+�
+•
+�❅
+❅
+❅
+❅
+❅
+❅
+❅
+�
+•
+�❅
+❅
+❅
+❅
+❅
+❅
+❅
+�
+•
+�❅
+❅
+❅
+❅
+❅
+❅
+❅
+�
+•
+�⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+•
+�
+�⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+•
+�
+�⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+•
+�
+�⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+⑦
+•
+�
+�
+�
+i = 0. The graph Λ is shown below, with trivial involution γ = id .
+•
+�⑦⑦⑦⑦⑦⑦⑦
+�
+�
+❅
+❅
+❅
+❅
+❅
+❅
+❅
+•
+�⑦⑦⑦⑦⑦⑦⑦
+�
+�
+❅
+❅
+❅
+❅
+❅
+❅
+❅
+•
+�⑦⑦⑦⑦⑦⑦⑦
+�
+�
+❅
+❅
+❅
+❅
+❅
+❅
+❅
+•
+�⑦⑦⑦⑦⑦⑦⑦
+�
+�
+❅
+❅
+❅
+❅
+❅
+❅
+❅
+•
+� •
+� •
+� •
+� •
+�
+�
+i = 1. The graph Λ is shown below with non-trivial involution γ, as in Proposition 4.1.
+•
+�❅
+❅
+❅
+❅
+❅
+❅
+❅
+�
+•
+�❅
+❅
+❅
+❅
+❅
+❅
+❅
+�
+•
+�❅
+❅
+❅
+❅
+❅
+❅
+❅
+�
+•
+�
+�❅
+❅
+❅
+❅
+❅
+❅
+❅
+•
+�
+�❅
+❅
+❅
+❅
+❅
+❅
+❅
+•
+�
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+For the i = −2 graph, one can determine all of the groups KOi(C∗
+R(Λ, γ)) from the
+associated spectral sequence, except for KO2(C∗
+R(Λ, γ)). In that case, the spectral sequence
+KO2(C∗
+R(Λ, γ)) has the filtration 0 → Z → KO2(C∗
+R(Λ, γ)) → Z2 → 0.
+Although this
+filtration by itself does not determine KO2(C∗
+R(Λ, γ)), the long exact sequence (5) forces
+KO2(C∗
+R(Λ, γ)) = Z. Moreover, the module maps r, c, η, ψ are uniquely determined by (5).
+□
+
+THE STABLE EXOTIC CUNTZ ALGEBRAS ARE HIGHER-RANK GRAPH ALGEBRAS
+15
+Proposition 5.2. For 2 ≤ i ≤ 5, there does not exist a 1-graph (Λ, γ) with involution such
+that K
+CR(C∗
+R(Λ, γ)) ∼= ΣiK
+CR(R).
+Proof. Suppose that (Λ, γ) is a graph with involution and K
+CR(C∗
+R(Λ, γ)) ∼= ΣiK
+CR(R).
+The real Pimsner-Voiculescu sequence (or equivalently, the real Evans spectral sequence)
+for K
+CR(C∗(Λ, γ)) implies that KO−1(C∗
+R(Λ, γ)) and KO−3(C∗
+R(Λ, γ)) are free abelian groups.
+But recall that KO1(R) ∼= KO2(R) ∼= Z2. Thus the group (Σ2KO(R))−1 = KO1(R) has
+torsion, implying that K
+CR(C∗
+R(Λ, γ)) ≇ Σ2KO
+CR(R), hence i ̸= 2. Similarly, the groups
+(Σ3KO(R))−1, (Σ4KO(R))−3, and (Σ5KO(R))−3 have torsion, showing that i ̸= 3, 4, 5.
+□
+References
+[BG22]
+Jeffrey L. Boersema and Elizabeth Gillaspy, K-theory for real k-graph C∗-algebras, Ann. K-
+Theory 7 (2022), no. 2, 395–440.
+[BHRS02]
+T. Bates, J.-H. Hong, I. Raeburn, and W. Szyma´nski, The ideal structure of the C∗-algebras of
+infinite graphs, Illinois J. Math. 46 (2002), no. 4, 1159–1176.
+[Boe02]
+Jeffrey L. Boersema, Real C∗-algebras, united K-theory, and the K¨unneth formula, K-Theory
+26 (2002), no. 4, 345–402.
+[Boe06]
+, The range of united K-theory, J. Funct. Anal. 235 (2006), no. 2, 701–718.
+[Boe14]
+, The K-theory of real graph C∗-algebras, Rocky Mountain J. Math. 44 (2014), 397–417.
+[Boe17]
+, The real C∗-algebra of a graph with involution, M¨unster J. Math. 10 (2017), 485–521.
+[BRS11]
+Jeffrey L. Boersema, Efren Ruiz, and P. J. Stacey, The classification of real purely infinite simple
+C∗-algebras, Doc. Math. 16 (2011), 619–655. MR 2837543
+[EFG+21] C. Eckhardt, K. Fieldhouse, D. Gent, E. Gillaspy, I. Gonzales, and D. Pask, Moves on k-graphs
+preserving Morita equivalence, Canad. J. Math. (2021), to appear, arXiv:2006.13441.
+[Eva08]
+D.G. Evans, On the K-theory of higher rank graph C∗-algebras, New York J. Math. 14 (2008),
+1–31.
+[Hew96]
+Beatrice Hewitt, On the homotopical classification of KO-module spectra, Ph.D. thesis, Univer-
+sity of Illinois at Chicago, 1996.
+[HRSW13] R. Hazlewood, I. Raeburn, A. Sims, and S.B.G. Webster, Remarks on some fundamental results
+about higher-rank graphs and their C∗-algebras, Proc. Edinb. Math. Soc. (2) 56 (2013), no. 2,
+575–597.
+[HS04]
+Jeong Hee Hong and Wojciech Szyma´nski, The primitive ideal space of the C∗-algebras of infinite
+graphs, J. Math. Soc. Japan 56 (2004), 45–64.
+[KP00]
+A. Kumjian and D. Pask, Higher rank graph C∗-algebras, New York J. Math. 6 (2000), 1–20.
+[RS04]
+Iain Raeburn and Wojciech Szyma´nski, Cuntz-Krieger algebras of infinite graphs and matrices,
+Trans. Amer. Math. Soc. 356 (2004), no. 1, 39–59.
+[RSS15]
+E. Ruiz, A. Sims, and A. P. W. Sørensen, UCT-Kirchberg algebras have nuclear dimension one,
+Adv. Math. 279 (2015), 1–28.
+[Szy01]
+Wojciech Szyma´nski, Simplicity of Cuntz-Krieger algebras of infinite matrices, Pacific J. Math.
+199 (2001), no. 1, 249–256.
+

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@@ -0,0 +1,1894 @@
+Matching linear algebra and tensor code to
+specialized hardware accelerators
+Pablo Antonio Martínez
+pabloantonio.martinezs@um.es
+University of Murcia
+Murcia, Spain
+Jackson Woodruff
+J.C.woodruff@sms.ed.ac.uk
+University of Edinburgh
+Edinburgh, United Kingdom
+Jordi Armengol-Estapé
+jordi.armengol.estape@ed.ac.uk
+University of Edinburgh
+Edinburgh, United Kingdom
+Gregorio Bernabé
+gbernabe@um.es
+University of Murcia
+Murcia, Spain
+José Manuel García
+jmgarcia@um.es
+University of Murcia
+Murcia, Spain
+Michael F.P. O’Boyle
+mob@inf.ed.ac.uk
+University of Edinburgh
+Edinburgh, United Kingdom
+Abstract
+Dedicated tensor accelerators demonstrate the importance
+of linear algebra in modern applications. Such accelerators
+have the potential for impressive performance gains, but
+require programmers to rewrite code using vendor APIs — a
+barrier to wider scale adoption. Recent work overcomes this
+by matching and replacing patterns within code, but such
+approaches are fragile and fail to cope with the diversity of
+real-world codes.
+We develop ATC, a compiler that uses program synthesis
+to map regions of code to specific APIs. The mapping space
+that ATC explores is combinatorially large, requiring the
+development of program classification, dynamic analysis,
+variable constraint generation and lexical distance matching
+techniques to make it tractable.
+We apply ATC to real-world tensor and linear algebra
+codes and evaluate them against four state-of-the-art ap-
+proaches. We accelerate between 2.6x and 7x more programs,
+leading to over an order of magnitude performance improve-
+ment.
+ACM Reference Format:
+Pablo Antonio Martínez, Jackson Woodruff, Jordi Armengol-Estapé,
+Gregorio Bernabé, José Manuel García, and Michael F.P. O’Boyle.
+2023. Matching linear algebra and tensor code to specialized hard-
+ware accelerators. In Proceedings of the 32nd ACM SIGPLAN Interna-
+tional Conference on Compiler Construction (CC ’23), February 25–26,
+2023, Montréal, QC, Canada. ACM, New York, NY, USA, 13 pages.
+https://doi.org/10.1145/3578360.3580262
+1
+Introduction
+Linear algebra is a fundamental building block of many of
+today’s critical applications; from weather modeling [13] to
+CC ’23, February 25–26, 2023, Montréal, QC, Canada
+© 2023 Copyright held by the owner/author(s).
+This is the author’s version of the work. It is posted here for your personal
+use. Not for redistribution. The definitive Version of Record was published in
+Proceedings of the 32nd ACM SIGPLAN International Conference on Compiler
+Construction (CC ’23), February 25–26, 2023, Montréal, QC, Canada, https:
+//doi.org/10.1145/3578360.3580262.
+ubiquitous DNN [22] workloads. Its importance is reflected
+in the large number of accelerator libraries and hardware
+devices devoted to fast linear algebra. These range from
+specialized devices such as Google’s TPU [34] to the tensor
+cores on NVIDIA [12] among many others [5, 8, 25, 31, 33].
+While such devices promise significant performance for an
+important class of applications [19], their uptake is limited by
+their programmability [24]. Typically, these accelerators and
+libraries are accessed via calls to specialized APIs, meaning
+existing code has to be rewritten. Given the volume [35]
+and variety [39] of existing legacy code, such rewriting is a
+significant undertaking [19].
+The combined importance of linear algebra acceleration
+and the difficulty of rewriting legacy code to accelerators has
+led to recent work which attempts to automate the process.
+These techniques search user code for matrix multiplica-
+tions using constraints [20, 28] or polyhedral analyses [9]
+and replace regions of code with appropriate API calls or
+instructions.
+However, as we show in Section 8.1, these approaches are
+fragile. Constraints capture only a limited set of program
+patterns and small variations in the user code defeat them.
+While they work well on curated benchmarks, they perform
+poorly on real-world code [20, 63], defeated by function calls,
+optimized code and inline assembler.
+Neural classification (e.g. [18]) can effectively detect code
+despite these challenges. However, it does not provide a path
+to acceleration, but requires further steps. These include gen-
+erating variable mappings and checking for equivalence [63]
+which has shown promising results for Fourier Transforms.
+However, one of the key challenges in matching code to
+APIs is the cost of searching for user program variables that
+map to API formal parameters. As the width of the API and
+complexity of the user program increase, this becomes com-
+binatorially expensive. As we show in Section 8.3 existing
+approaches [63] fail to scale to the challenges that linear
+algebra APIs present.
+1
+arXiv:2301.11659v1  [cs.PL]  27 Jan 2023
+
+CC ’23, February 25–26, 2023, Montréal, QC, Canada
+P.A. Martínez, J. Woodruff, J. Armengol-Estapé, G. Bernabé, J.M. García, M.F.P. O’Boyle
+We present ATC, a compiler that applies program synthe-
+sis to compile general user-code to linear algebra accelerators.
+We identify and solve key challenges
+enabling the detect/synthesize paradigm to scale to the
+more complex APIs of linear algebra acceleration. In addi-
+tion, ATC employs a trained platform predictor to determine
+whether acceleration is profitable or not.
+We applied our approach to 50 GitHub GEMM and 15
+convolution projects and discovered between 2.6 and 7x more
+linear operators compared to KernelFaRer [20], IDL [28],
+Polly [30] or FACC[63]. This resulted in more than an order
+of magnitude performance improvement.
+This paper makes the following contributions:
+• We present ATC, which maps matrix multiplication
+and convolution programs to hardware accelerators,
+up to 7x more frequently than existing techniques.
+• We introduce novel heuristics to reduce the mapping
+search space by four orders of magnitude.
+• We develop novel dynamic analyses to determine higher-
+level information about variables, enabling synthesis
+without costly whole-program analyses.
+2
+Motivation
+Figure 1. Example application of API replacement. The
+above program is taken from the parboil benchmark [56], a
+widely-used benchmark suite, which is transformed into a
+call to an optimized matrix-multiplication accelerator API.
+Figure 2. GEMM code optimized for AVX2 found on GitHub
+consisting of 120 lines of hand-optimized intrinsics and how
+ATC matches the code to the accelerator API
+2.1
+Exisiting Match and replace
+IDL and KernelFaRer. Both aim to detect linear algebra
+operations in user programs and replace them with an appro-
+priate accelerator library call. To illustrate this consider the
+code in Figure 1. This shows a straight-forward matrix mul-
+tiplication program fragment, from the parboil benchmark
+suite [56]. They aim to detect this matrix-multiplication and
+replace it with a call to the library, shown at the bottom of
+the diagram.
+To replace code with an API call they have to both de-
+tect the code performing a matrix multiplication and also
+determine which user program variables correspond to the
+arguments of the API call. Both approaches are able to detect
+that this is a matrix multiplication, and can determine the
+mapping between user variables and API parameters.
+2.2
+Examples of complex GEMM programs
+Unfortunately, in practice, user code can be complex such
+that code structure or pattern-based approaches inevitably
+fail.
+As an example, consider the code found on GitHub shown
+in Figure 2 which implements a matrix-multiplication al-
+gorithm (only a fragment of the 120 lines of user code are
+shown here). The code structure is complex and difficult to
+understand as it makes extensive use of inline assembler in-
+trinsics which defeats the code structure analysis approaches
+of IDL and KernelFaRer, preventing acceleration.
+2
+
+Matching linear algebra and tensor code to specialized hardware accelerators
+CC ’23, February 25–26, 2023, Montréal, QC, Canada
+OJCLONE
+DATASET
++ GEMM
+PROGRAMS
++ CONV
+PROGRAMS
+NEURAL EMBEDDINGS
+FUNCTION
+CANDIDATES
+F1
+F2
+F3
+... FN
+IO
+EQUIVALENCE
+IO
+DETECTION
+ACCELERATABLE
+FUNCTION
+MATCHES
+GENERATION
+MISMATCH
+SOLVER
+PROGRAM
+SYNTHESIS
+SET OF MATCHES
+MATCHES HEURISTCS
+MATCH ALGORITHM
+LEVENSHTEIN
+VALID MATCHES
+PERFORMANCE
+ANALYSIS
+ACCELERATOR
+SAMPLING
+FUNCTION
+SAMPLING
+SVM CLASSIFIER
+PROGRAM CLASSIFICATION
+ACCELERATOR API
+USER CODE
+ACCELERATED CODE
+Acceleratable Candidate Detection
+IO Detection
+Matches Generation
+Matches
+Reduction
+Profitability Detection
+Figure 3. ATC compiler architecture
+2.3
+Our approach - ATC
+Rather than relying on code structure to guide detection,
+ATC uses behavioral equivalence to determine if a section of
+code is a linear algebra operation. Firstly, ATC uses neural
+program classification [18] to detect that the code in Figure
+2 is probably a GEMM. It then searches variable matches to
+determine the potential source and output arrays. As the
+search space is combinatorially large, we introduce scal-
+able, algorithm-independent heuristics (which we discuss in
+Section 5) that keep the number of mappings manageable.
+Next, ATC generates different input values for the arrays
+and records the output. After generating many randomized
+inputs, it observes that it has the equivalent behavior to the
+corresponding API and is able to replace the AVX2 code with
+the GEMM call at the bottom of Figure 2.
+Legality. Now, IO behavioral equivalence is not proof that
+a section of code is a particular linear algebra operation -
+similarly IDL and KernelFaRer do not prove equivalence. For
+proof, bounded model checking based on Kleene [14] can be
+deployed. In practice, as demonstrated in our experimental
+section, IO equivalence gives no false positives. For further
+guarantees, we can ask for programmer sign-off or employ
+model checking.
+Profitable. Once we have detected and can replace a sec-
+tion of code with an accelerator call, we need to determine if
+it is profitable to do. Due to hardware evolution, we do not
+use a hard-wired heuristic to determine profitability. Instead,
+we learn, off-line, a simple predictive model to determine if
+the target accelerator is faster than a CPU implementation.
+The model is called at runtime, determining if offloading is
+worthwhile.
+FACC. Behavioral equivalence is also employed in FACC
+[63]. Unfortunately, it is restricted to FFTs and one-dimensional
+arrays, and cannot detect the replacement in Figure 1. There-
+fore, we extended FACC to FACC* to consider GEMMs and
+multi-dimensional arrays. This, however, exposes its weak
+variable binding model which is combinatorial in the number
+of user array variables and their dimensionality. Furthermore,
+it relies on program synthesis to determine the length of ar-
+rays, which scales poorly to problems with many potential
+length parameters for arrays such as GEMM.
+FACC also relies on brittle inter-procedural liveness analy-
+ses to determine the liveness status of variables. This restricts
+it to running only at link time, rendering it invalid for use
+in shared libraries. We will see in Section 8 that the com-
+bination of these issues results in excessively large search
+spaces.
+3
+System overview
+Figure 3 gives a system flow overview of ATC. We first de-
+tect regions of code that are likely to be linear algebraic
+operations using a neural program classifier. The classifier is
+trained ahead of time, based on programs that are equivalent
+to the accelerator and prior examples of linear algebra code.
+Once candidate code sections have been identified, we ap-
+ply program analysis to match user program variables with
+the particular API formal parameters. Given the combina-
+torially large search space, we develop novel techniques to
+make the problem tractable.
+For each candidate matching, we generate multiple data
+inputs, execute the user code section and record the output
+values. If the input/output pairs correspond to the input/out-
+put behavior of the accelerator API, we can say they are
+behavioral equivalent and candidates for replacement.
+3
+
+CC ’23, February 25–26, 2023, Montréal, QC, Canada
+P.A. Martínez, J. Woodruff, J. Armengol-Estapé, G. Bernabé, J.M. García, M.F.P. O’Boyle
+While candidate user code may be replaceable with a call
+to an accelerator API, it may not be profitable. Therefore, we
+employ a simple ML classifier, trained offline, and invoked
+at runtime to see if acceleration is appropriate for the user
+code for the runtime known array sizes.
+3.1
+Neural Program Classification
+To detect potentially acceleratable parts of a program, we use
+prior work in neural program classification [18]. A network is
+trained with multiple instances of different program classes.
+We use the OJClone dataset [43], which includes 105 classes
+of different programs, and add examples of the programs that
+we want to detect e.g. GEMMs and convolutions, gathered
+from benchmark suite repositories other than GitHub.
+At compile time a new candidate program is divided into
+functions, which are presented to the neural classifier. The
+classifier assigns each function in the program a probability
+of belonging to a certain class. We consider the most proba-
+ble class, which in the case of a GEMM or convolution is then
+considered for variable matching and eventual code replace-
+ment as described in the following sections. Classification
+is fast (≤ 1.5 sec) and has negligible impact on compilation
+time (see Section 8.3).
+4
+Variable Matching
+To check if a section of user code is behaviorally equivalent
+to the API, we have to match up the user program variables
+with API formal parameters. We first detect what variables
+are livein/liveout (Section 4.1) and then the dimensions of
+arrays (Section 4.2).
+4.1
+Detecting livein and liveout variables
+Detecting livein and liveout variables via standard static
+analysis is
+straightforward for well-structured programs but fails for
+more diverse real-world codes, which may use assembly code
+or intrinsic functions.
+ATC uses dynamic analysis to determine which variables
+are livein and liveouts inside a function. In C, variables are
+passed by value so non-pointers variables are always livein.
+In the case of pointers (or arrays), we generate random inputs
+with arbitrary sizes. If the values in memory change after
+executing the program, the array is considered liveout.
+This allows us to detect which variables are livein or live-
+out, but not both livein and liveout at the same time. We gen-
+erate a new random input for liveout variables and re-execute
+the function. If the output differs from the first execution, it
+is both livein and liveout. We implement this algorithm as a
+just-in-time compiler pass in LLVM [37].
+4.2
+Detecting the dimensions of arrays
+Detecting arrays length enables offloading of appropriately-
+sized regions of codes, so it is a critical step in ATC. For
+Load/StoreInst
+Array A?
+Out of
+bound?
+Replace Load/Store
+to/from index 0
+Perform the
+instruction
+Exit with
+error code
+YES
+NO
+YES
+NO
+Figure 4. Dimension detection algorithm overview for a
+target example array called A.
+Algorithm 1 Dimensions detection algorithm
+1: for arr in function do
+2:
+fakeLoadAndStoresExcept(𝑎𝑟𝑟)
+3:
+replaceLoadAndStores(𝑎𝑟𝑟)
+4:
+repeat
+5:
+𝑐 = getNextCombination(𝑎𝑟𝑟)
+6:
+ffi_call(𝐴,𝑉 )
+7:
+if not failed then
+8:
+𝑓 𝑜𝑢𝑛𝑑 = 𝑇𝑟𝑢𝑒
+9:
+end if
+10:
+until not found
+11:
+Add 𝑐 to 𝐶
+12: end for
+13: return 𝐶
+some programs, lengths can be found using static analysis
+(e.g. [49]), but this fails in more complex cases. We use run-
+time analysis to determine which program variables define
+array size using a modified form of runtime array bound
+checking. For each set of variables that could define an ar-
+ray’s size (typically, from the argument list), we set such
+variables to a fixed value. We then execute the user code that
+is modified to check runtime array accesses.
+First, the compiler selects a target array to find its size.
+Then, to generate the modified program, we tweak the load
+and store instructions in the user program, replacing them
+with custom function calls in the IR. If a load or store does
+not access the array we are interested in, we modify it to
+load and store at a constant, safe location. If it does, the
+instruction is replaced with a function call that will check at
+runtime if the access is out of bounds. If so, the program exits
+with a custom error code. If not, we have found a valid array
+size. The basic idea is depicted in Figure 4. This is used by
+our JIT analysis as shown in Algorithm 1 and implemented
+in LLVM.
+This way, the compiler can assign different input sizes to a
+given array and check the exit code. Therefore, the compiler
+iterates over all the possible dimensions combinations until
+one of the executions does not end with the custom error exit
+code. That means that the program was completed without
+any illegal access to the target array, which indicates that it
+is the right dimension of the array.
+4
+
+Matching linear algebra and tensor code to specialized hardware accelerators
+CC ’23, February 25–26, 2023, Montréal, QC, Canada
+Algorithm 2 Automatic matching algorithm
+1: function dimsMatch(𝑓 1𝑎, 𝑓 2𝑎, 𝑝,𝑛)
+2:
+𝑆 = ∅
+3:
+𝑖𝑑𝑥 ← 0
+4:
+for 𝑎𝑟𝑔𝑠1 in f1a do
+5:
+𝑎𝑟𝑔𝑠2 = f2a[p[idx]]
+6:
+Add {𝑎𝑟𝑔𝑠1,𝑎𝑟𝑔𝑠2} to 𝑆
+7:
+𝑖𝑑𝑥 ← 𝑖𝑑𝑥 + 1
+8:
+end for
+9:
+return Size(S) = 𝑛
+10: end function
+11:
+12: function outMatch(𝑓 1𝑜, 𝑓 2𝑜, 𝑝)
+13:
+idx = IndexOf(𝑓 2𝑜, 1)
+14:
+return IndexOf(𝑝, idx) = IndexOf(𝑓 1𝑜, 1)
+15: end function
+16:
+17: function findMatchings(𝑓 1𝑎, 𝑓 2𝑎, 𝑓 1𝑜, 𝑓 2𝑜,𝑛)
+18:
+𝐵 = ∅
+19:
+for p in permutations(0...𝑛) do
+20:
+if dimsMatch(f1a, f2a, p) and
+21:
+outMatch(f1o, f2o, p) then
+22:
+Add 𝑝 to 𝐵
+23:
+end if
+24:
+end for
+25:
+return 𝐵
+26: end function
+5
+Reducing the matchings search space
+To match code to APIs, the compiler generates different can-
+didates for the variable to formal parameter mappings and
+then tests them using IO equivalence. For small APIs, all map-
+pings can be explored, but the combinatorial cost makes it
+prohibitive for real-world accelerator APIs. We develop tech-
+niques that reduce the mapping space by exploiting arrays
+information and human coding styles.
+5.1
+Exploiting array information
+Using array dimensions (Section 4.2), we can reduce the num-
+ber of possible matches that must be checked, as assigning
+one array to another means that the dimensions of each array
+must line up.
+5.1.1
+Automatic matching algorithm. We first gener-
+ate all 𝑛! permutations of the 𝑛 array variables to 𝑛 parame-
+ters mapping. We discard all permutations where variable
+livenesses do not match. Then for each candidate user array
+and parameter array pair, we generate the constraints defin-
+ing how their dimensions match. If we find contradictory
+constraints for any permutation, we discard it. The algorithm
+is shown in Algorithm 2.
+5.1.2
+Automatic Matching Algorithm: Example. To il-
+lustrate this, Figure 5 shows an example where we have two
+X(x0*x1)
+Y(x1*x2)
+Z(x2*x0)
+A(y0*y1)
+B(y1*y2)
+C(y2*y0)
+[0,1,2] 
+A:
+U:
+x0 -> y0
+x1 -> y1
+x2 -> y2
+Liveout A:[0,0,1] U:[0,0,1]
+[1,0,2] 
+x0 -> y1
+x1 -> y2
+x1 -> y0
+x2 -> y1
+x2 -> y2
+x0 -> y0
+[2,0,1] 
+x0 -> y2
+x1 -> y0
+x2 -> y1
+Figure 5. Example application of the matching algorithm.
+The right match is found the algorithm automatically. Per-
+mutations in red means they are invalid, while the green
+permutation means valid.
+functions with three 2D arrays each. First, the algorithm
+generates all the permutations between 0 and 𝑛 − 1 (𝑛 = 3 in
+this example). Then, for each permutation, it tries matching
+each variable in every array in the user code with the cor-
+responding variable in the array of the API (here we show
+only three of the six possible permutations).
+In the first case (with the permutation [0, 1, 2]), the algo-
+rithm tries matching the array variables of the user program
+𝑋,���,𝑍 with API parameters 𝐴, 𝐵,𝐶 . We then examine each
+of the variables defining each of the corresponding arrays.
+Comparing 𝑋 and 𝐴 gives a match of 𝑥0 → 𝑦0 and 𝑥1 → 𝑦1.
+For the second array variable 𝑌 and API parameter 𝐵, we
+have 𝑥1 → 𝑦1 and 𝑥2 → 𝑦2 and for the third variable pair
+𝑍,𝐶 we have 𝑥2 → 𝑦2 and 𝑥0 → 𝑦0. All of these are con-
+sistent with 𝑛=3 constraint, which satisfies the condition
+(dimsMatch in Algorithm 2). Liveout information is also sat-
+isfied so this permutation is added as a potential mapping.
+In the second permutation [1, 0, 2], where 𝑋,𝑌,𝑍 maps
+to 𝐵,𝐴,𝐶, the constraints are inconsistent e.g. 𝑥1 → 𝑦2 and
+𝑥1 → 𝑦0 leading to 6 ≥ 3, so it is not a valid match. In the
+third and last example, constraints are equal to 𝑛, but the
+liveout arrays do not match. Thus, the only valid match is
+the one found in the first permutation.
+5.2
+Using argument names
+Programs are developed by humans, so we can assume that
+the functions that humans write follow common patterns.
+We exploit this by analyzing the argument names of the API
+and the user program to find lexical similarities.
+To compare argument names, we use the Levenshtein dis-
+tance [38] to compute the distance between each of the user
+programs and API arguments. Figure 6 shows the definition
+of the Levenshtein distance, which calculation is based on
+the minimal number of modifications needed to transform
+one word into another, representing how close are those
+5
+
+CC ’23, February 25–26, 2023, Montréal, QC, Canada
+P.A. Martínez, J. Woodruff, J. Armengol-Estapé, G. Bernabé, J.M. García, M.F.P. O’Boyle
+𝑙𝑒𝑣(𝑎,𝑏) =
+
+
+|𝑎|
+if |𝑏| = 0,
+|𝑏|
+if |𝑎| = 0,
+𝑙𝑒𝑣(𝑡𝑎𝑖𝑙(𝑎),𝑡𝑎𝑖𝑙(𝑏))
+if 𝑎[0] = 𝑏[0],
+1 + 𝑚𝑖𝑛
+
+
+𝑙𝑒𝑣(𝑡𝑎𝑖𝑙(𝑎),𝑏)
+𝑙𝑒𝑣(𝑎,𝑡𝑎𝑖𝑙(𝑏))
+𝑙𝑒𝑣(𝑡𝑎𝑖𝑙(𝑎),𝑡𝑎𝑖𝑙(𝑏))
+otherwise
+(1)
+Figure 6. Levenshtein recursive definition
+gemm_api(float* tc_A , float* tc_B , float* tc_C ,
+int tc_m , int tc_n , int tc_k ,
+int tc_lda , int tc_ldb , int tc_ldc ,
+float
+tc_alpha , float
+tc_beta) {
+gemm(int M, int N, int K, float
+alpha ,
+float *A, int lda , float *B, int ldb ,
+float beta , float *C, int ldc) {
+tc A
+...
+tc lda
+...
+M
+1
+...
+3
+...
+N
+1
+...
+3
+...
+K
+1
+...
+3
+...
+alpha
+4
+...
+3
+...
+A
+0
+...
+2
+...
+lda
+2
+...
+0
+...
+........
+.......
+...
+.......
+...
+Figure 7. Levenshtein distance calculation for the arguments
+of the tensor core API (above) and an example user program.
+words. After computing the distance, the compiler selects
+the combination that minimizes the Levenshtein distance.
+Figure 7 shows an application example of the Levenshtein
+distance to a real case of GEMM matching. For calculating
+the distance, we strip the API suffix (tc_) and convert all
+names to lowercase. Results show that the most probable
+mapping for tc_A is A in the user code, and for tc_lda is
+lda, which are the right matches.
+5.3
+IO generation
+Once we have a candidate match we generate random inputs
+of different sizes and test for input-output (IO) equivalence.
+We use 30 inputs of varying sizes. Although IO behavioral
+equivalence is not proof, we can increase the number of tests
+for increased confidence. No existing technique such as IDL
+or KernelFaReR can prove that a matched piece of code is
+provably equivalent to an API and therefore rely on user
+sign-off.
+5.3.1
+Behavioral Equivalence and the Limits of Veri-
+fication. ATC, like prior work on floating-point accelera-
+tors [63], uses behavioral equivalence. The downside of this
+strategy is that it requires programmer sign-off to make any
+substitution. However, due to the complexities of verifying
+floating-point programs [63], verification of such liftings are
+some way off.
+In summary, the key challenges that all competing tech-
+niques face are:
+• Floating-point numbers often raise challenges in theo-
+rem provers as they are challenging to reason about.
+• Floating-point functions may have different accuracies
+in different input ranges, meaning that the obvious
+checks of correctness (even within bounds) are difficult
+to apply.
+The backend of ATC is not tied to using behavioral equiva-
+lence. As we will see, the use of such behavioral equivalence
+results in no false positives. Further development of theorem
+prover technologies would mean that the weak behavioral
+equivalence in ATC could easily be replaced with a theorem
+prover guaranteeing correctness and enabling automatic
+transformations.
+6
+Automatic profitability detection
+We assume that user code runs faster when replaced by a
+platform-specific library. The question is whether it is best
+to run on a CPU or accelerator version (XPU) of the library.
+This in turn depends on the input size, which is only known
+at runtime. We use a predictive model based on empirical
+data to enable accurate predictions as platforms and libraries
+evolve by retraining the model.
+SVM. We use the well-known support vector machine
+(SVM) classifier with a polynomial kernel of degree 3 with
+gamma=1 and𝐶=100. We sample the CPU and the accelerator
+with a common dataset of input sizes, which produces a
+dataset that is small enough to be processed in less than
+five minutes, but large enough to be highly accurate. Data
+is labeled with 0 or 1 meaning that the CPU or the XPU is
+faster. The model is then trained and deployed at runtime,
+when matrix sizes are known, The training phase is done
+only once, at “factory time”, and the resulting model when
+deployed has negligible (≤ 0.3𝑚𝑠𝑒𝑐) runtime overhead (see
+Section 8.2).
+7
+Setup
+We evaluate GEMM and convolution acceleration on special-
+ized platforms. For GEMM, we used an Intel i7-11700 (CPU)
+with an NVIDIA Quadro RTX 5000 (tensor cores) (XPU). For
+convolution, we used the Google Cloud Platform (GCP) ser-
+vices equipped with a TPUv3 with 8 TPU cores. Compilation
+benchmarks in Section 8.3 are executed in an AMD EPYC
+7413.
+The Intel/NVIDIA platform runs CentOS 8.3 with kernel
+4.18.0. LLVM was downloaded from the official Git repository,
+using commit 329fda3. User codes were compiled using gcc
+11.2.0 with -O3 -march=native flags. We used cuBLAS 11.2
+and MKL 2020.2.254 for compiling codes to the XPU and
+CPU, respectively. For compiling convolution programs to
+6
+
+Matching linear algebra and tensor code to specialized hardware accelerators
+CC ’23, February 25–26, 2023, Montréal, QC, Canada
+Algorithm
+Code
+LoC
+Nº Args
+Optimizations
+Constraints
+C struct?
+Direct
+1
+35
+12
+None
+None
+No
+2
+36
+10
+OpenMP
+FW = FH = 3
+No
+3
+34
+8
+OpenMP
+FW = FH = 3
+No
+4
+43
+11
+None
+FW = FH = 3
+No
+5
+39
+8
+OpenMP
+FW = FH = 3
+No
+6
+76
+16
+None
+N = 1
+No
+7
+209
+18
+Vectorized
+N = 1
+Yes
+8
+102
+12
+None
+None
+No
+9
+42
+16
+None
+None
+No
+im2col+
+gemm
+10
+189
+15
+None
+N = 1
+Yes
+11
+286
+15
+BLAS
+N = 1
+Yes
+12
+179
+17
+BLAS
+FW = FH
+Yes
+Winograd
+13
+687
+17
+Intrinsics + OpenMP
+FW = FH = 3
+No
+14
+254
+12
+None
+N = 1
+Yes
+15
+782
+12
+Intrinsics + OpenMP
+FW = FH = 3
+No
+Table 1. List of convolution codes
+the CPU, we used oneDNN v1.96. The TPU system runs
+Debian 10 with kernel 4.19.0-14.
+7.1
+User code
+We explored GitHub looking for C and C++ GEMM codes,
+analyzing more than 400 programs from which we selected
+50 programs. We discarded the rest of them because of wrong
+implementations, compilation errors or duplicated code. The
+final list of programs is shown in Table 8. We categorize
+the codes as follows: Naive: naive implementations with the
+traditional 3-loop structure; Naive Parallel: as Naive but with
+simple outer loop parallelization; Unrolled: naive implemen-
+tation with unrolled loops; Kernel Calls: implementations
+that divide the loops into different function calls; Blocked:
+tiled implementations; Goto: implementations of the Goto
+algorithm [29]; Strassen: implementations of the Strassen
+algorithm [55]; Intrinsics: implementations using Intel intrin-
+sics.
+In addition, we selected 50 non-GEMM projects to check
+whether any of the approaches gave false positives.
+Convolutions. We explored GitHub looking for C and
+C++ 4D convolution implementations. We analyzed around
+50 programs from which we a selected list of 15 programs
+based on the same methodology used for selecting GEMMs.
+The list of convolution programs is shown in Table 1. We
+have included codes from the most relevant convolution
+implementations: Direct: the direct convolution algorithm;
+im2col+gemm: an algorithm that casts the input as matrices
+(im2col) and later uses a GEMM, as in Caffe [32]; Winograd:
+the Winograd algorithm.
+7.2
+Methods
+We evaluate our approach against 4 well known schemes:
+IDL: Idioms are described using an idiom description lan-
+guage [28], which is translated into a set of constraints over
+LLVM IR.
+KernelFaRer: Uses different pattern matching to detect spe-
+cific code constructs, matching specific matrix-multiplication
+structures [20].
+Polly: Detects static control parts (SCoPs) in the code using
+the polyhedral model [30]. It does not replace the code with
+a call to an optimized library.
+FACC*: FACC uses neural embeddings and behavioral syn-
+thesis to detect candidates for acceleration [63]. It is limited
+to 1D arrays so we developed an extended version, FACC*,
+which supports multi-dimensional arrays.
+8
+Results
+8.1
+Detection
+Figure 9 shows the percentage of GEMM programs matched
+by each technique across each of 8 categories listed in Table 8.
+IDL. The constraint based scheme [28] only matches 6
+out of 50 cases. These programs are largely naive implemen-
+tations of GEMM, with a simple loop structure. It is able to
+manage 2 programs containing unrolled loops but fails on
+anything more complex. Matching more diverse cases would
+require writing a new IDL constraint description for each
+sub-class.
+KernelFaRer. This code matching approach [20] is more
+successful, matching 11 GEMMs due to a more robust pattern
+matcher. For straightforward sequential implementations, it
+is able to match all but one of the cases. However, any code
+variation, including loop unrolling, defeats it.
+Polly. Although it does not match and replace GEMMs,
+it can detect SCoPs which may be candidates for replace-
+ment with appropriate API calls. It is less successful than
+KernelFaRer in detecting naive implementations but is more
+robust across other more complex categories including one
+parallel and unrolled versions and 2 blocked cases. It slightly
+outperforms KernelFaRer, matching 13 vs. 11 out of 50 cases.
+FACC*. Unlike the other approaches, FACC* performed
+poorly on naive implementations, but better on others. Here,
+the size of the mapping search space is the limiting factor. It
+was able to find 10 cases in the available time (timeout ≤ 10
+mins). We examine the reasons for this in Section 8.3.
+ATC. Our approach is significantly more robust across all
+categories, matching 42 out of 50 cases. It is able to detect all
+naive implementations and the majority within each other
+category. It detects more naive parallel implementations,
+unrolled and blocked programs than Polly and is the only
+technique to detect GEMMs in codes containing kernel calls
+and intrinsic instructions.
+8.1.1
+Accuracy. Figure 10 provides a summary of ATC’s
+success and failure by type. In 8 cases ATC failed to detect
+that the program contained a GEMM. In one case, program
+23, this is due to there being too many candidate matches,
+280 which is above our timeout threshold of 100 candidates.
+The remaining cases are due to overly aggressive search
+7
+
+CC ’23, February 25–26, 2023, Montréal, QC, Canada
+P.A. Martínez, J. Woodruff, J. Armengol-Estapé, G. Bernabé, J.M. García, M.F.P. O’Boyle
+Algorithm
+Code
+LoC
+Layout
+Sizes
+Optimizations
+Naive
+1
+22
+Column-major
+Squared
+None
+2
+127
+Both
+Any
+None
+3
+18
+Row-major
+Any
+None
+4
+41
+Column-major
+Squared
+None
+5
+11
+Row-major
+Any
+None
+6
+11
+Row-major
+Any
+None
+7
+30
+Row-major
+Any
+None
+8
+18
+Column-major
+Any
+None
+9
+40
+Column-major
+Any
+None
+10
+39
+Column-major
+Any
+None
+11
+43
+Row-major
+Any
+None
+12
+11
+Row-major
+Squared
+None
+Naive
+parallel
+13
+39
+Row-major
+Squared
+OpenMP
+14
+28
+Column-major
+Squared
+OpenMP
+15
+164
+Row-major
+Any
+OpenMP
+16
+22
+Row-major
+Multiple of nthreads
+C++ threads
+17
+107
+Row-major
+Squared
+C++ threads
+Unrolled
+18
+57
+Row-major
+Any
+None
+19
+50
+Row-major
+Any
+None
+20
+63
+Row-major
+Squared
+OpenMP
+21
+38
+Row-major
+Squared, multiple of bs
+None
+Kernel Calls
+22
+46
+Column-major
+Any
+None
+23
+115
+Column-major
+Any
+OpenMP
+24
+61
+Column-major
+Any
+None
+25
+105
+Column-major
+Any
+Unrolled
+Algorithm
+Code
+LoC
+Layout
+Sizes
+Optimizations
+Kernel Calls
+26
+164
+Column-major
+Any
+Unrolled
+Blocked
+27
+104
+Row-major
+Any
+Block
+28
+30
+Row-major
+Squared
+OpenMP
+29
+52
+Column-major
+Any
+None
+30
+35
+Row-major
+Squared
+None
+31
+38
+Column-major
+Squared
+None
+32
+42
+Row-major
+Multiple of bs
+Unrolled
+33
+49
+Row-major
+Squared
+None
+34
+18
+Row-major
+Squared
+None
+35
+21
+Row-major
+Squared
+None
+Goto
+36
+247
+Column-major
+Squared
+Intrinsics (SSE)
+37
+89
+Row-major
+Squared
+None
+Strassen
+38
+210
+Row-major
+Squared
+None
+39
+315
+Row-major
+Squared, power of 2
+None
+40
+162
+Row-major
+Squared
+None
+Intrinsics
+41
+102
+Row-major
+Squared
+Intrinsics (AVX2)
+42
+91
+Row-major
+Multiple of 8
+Intrinsics (AVX2)
+43
+82
+Row-major
+Multiple of 8
+Intrinsics (AVX2)
+44
+58
+Row-major
+Any
+Intrinsics (SSE)
+45
+112
+Row-major
+Multiple of bs
+Intrinsics (AVX2)
+46
+136
+Row-major
+Multiple of bs
+Intrinsics (AVX2)
+47
+120
+Row-major
+Any
+Intrinsics (AVX2)
+48
+143
+Row-major
+Multiple of bs
+Intrinsics (AVX2)
+49
+57
+Row-major
+Multiple of bs
+Intrinsics (AVX2)
+50
+60
+Row-major
+Any
+Intrinsics (SSE)
+Figure 8. List of GEMM codes
+Naive
+Naive p.
+Unrrolled
+Kernels
+Blocked
+Goto
+Strassen
+Intrinsics
+All
+0
+20
+40
+60
+80
+100
+4
+9
+11
+1
+12
+0
+1
+0
+3
+4
+2
+1
+0
+1
+3
+0 0 0 0
+4
+0
+2
+0
+2
+6
+0 0 0
+1 1
+0 0 0
+2
+3
+0 0 0 0
+9
+6
+131110
+42
+% of matched codes
+IDL
+POLLY
+KFR
+FACC*
+ATC
+Figure 9. Percentage of matched GEMM codes by different techniques.
+0
+20
+40
+60
+80
+% of programs
+Matched
+Too many candidates
+Missed matches
+Figure 10. Percentage of matched GEMM codes by ATC
+divided by failure reason.
+pruning, missing a legal match. Improved search heuristics
+are likely to improve program coverage.
+False positives. None of the methods classified any of the
+50 non-GEMMs as a GEMM. Across all methods, there were
+no false positives.
+8.2
+Performance
+The performance of each approach is shown in Figure 11.
+Polly is not included here as although it can detect SCoPs, it
+does not explicitly identify them as GEMMs for API replace-
+ment. We show two bars for KernelFaRer, which correspond
+to the strategy of GEMM code with an optimized CPU im-
+plementation as described in [20] and KFR (XPU) which is
+our extension, replacing the CPU library with the optimized
+XPU implementation. IDL and FACC* directly target the ac-
+celerator, while ATC chooses the CPU or accelerator based
+on its SVM platform predictor. This runtime prediction cost
+is negligible ≤ 0.3𝑚𝑠𝑒𝑐 and included in Figure 11.
+What is immediately clear is that detecting more GEMMs
+leads to better overall speedup. In the Naive category, KFR
+and ATC are both able to achieve good performance, with
+a speedup of 726x and 1031x, respectively. The gap is nar-
+rowed when using KFR (XPU). However, KFR is unable to
+detect GEMMs in any other category leading to just a 6.2x
+8
+
+Matching linear algebra and tensor code to specialized hardware accelerators
+CC ’23, February 25–26, 2023, Montréal, QC, Canada
+Naive
+Naive p. Unrrolled Kernels
+Blocked
+Goto
+Strassen Intrinsics
+All
+1
+10
+100
+1000
+10000
+Speedup
+IDL
+KFR (CPU)
+KFR (XPU)
+FACC*
+ATC
+Figure 11. Geometric mean speedup obtained by IDL, KernelFaRer, FACC* and ATC in GEMM programs with 𝑛 = 8192.
+speedup overall while ATC achieves 344.0x. Unsurprisingly,
+there is more performance available on naive sequential im-
+plementations than in those cases where the programmer
+has spent effort in optimizing the program.
+8.3
+Candidate search complexity and compile time
+One of the key challenges in matching code to APIs is search-
+ing for program variables that map to API formal parameters.
+As the width of the API and complexity of the user program
+increase, this becomes combinatorially expensive. Figure 12
+evaluates FACC* naive matching of variables and our ap-
+proach based on the Levenshtein distance. Naive matching
+varies considerably from just 4 candidates to over 1 million.
+Our approach greatly reduces the number of candidates for
+the majority of the programs. There is one special case, code
+23, where we reduce the number of candidates, but it is still
+too high.
+Figure 13 shows the compilation time of ATC. The initial
+neural classifier has a negligible constant execution time of
+1.3 seconds, while the other phases’ compilation time grows
+with the number of candidates.
+As the number of candidates begins to increase compi-
+lation time becomes prohibitively expensive. Code 23 has
+280 candidates which would take 35 mins more to evaluate.
+We limit the number of candidates considered to 100 which
+corresponds to a timeout of ≤ 10 minutes.
+8.4
+Profitability accuracy
+To measure the accuracy of the SVM platform predictor, we
+built a model offline and tested it on unseen data values.
+Table 2 summarizes the SVM accuracy with different input
+sizes and shapes. The SVM achieves a global accuracy of
+99.7%, where the misprediction occurs between 𝑚 = 2000
+and 𝑚 = 8000 which is the “edge” between the CPU and the
+XPU. In all other intervals, the prediction is always correct.
+The best accuracy is achieved with non-squared matrices,
+while square matrices give slightly lower accuracy. Overall,
+this is a highly accurate predictor with a negligible runtime
+overhead of ‘ ≤ 0.3𝑚𝑠𝑒𝑐.
+Parameter
+Value
+(mnk)
+m
+Global
+Accuracy
+2000
+4000
+6000
+8000
+10000
+111
+100%
+100%
+100%
+70.0%
+100%
+93.8%
+123
+100%
+78.9%
+100%
+100%
+100%
+95.9%
+312
+100%
+84.3%
+100%
+100%
+100%
+96.9%
+136
+100%
+89.5%
+100%
+100%
+100%
+97.9%
+Table 2. SVM accuracy for different sizes. 111 means m = 1
+× m, n = 1 × m, k = 1 × m. 123 means m = 1 × m, n = 2 × m,
+k = 3 × m etc
+8.5
+Convolutions
+Our approach is generic and can be applied to other APIs
+other than GEMMs. As an example, we consider tensor con-
+volutions which are a significant component of DNN work-
+loads. While IDL, KernelFaRer, Polly and FACC* were unable
+to detect any of the convolutions, ATC detected 10 of the
+15 convolutions as shown in Figure 14; we were unable to
+match 5 due to the excessive number of candidates.
+Figure 15 shows the performance achieved by replacing
+with library code for each of the programs we are able to
+accelerate. Across all codes, the SVM predicts that the TPU
+accelerator outperforms the CPU, giving an average 17.8x
+performance improvement across the programs.
+9
+Related work
+Matching in Programs. Matching high-level program
+structure has been used to discover parallelism [23], het-
+erogenous offloading [6, 44] and many other core compiler
+tasks [27]. Constraint languages make these tasks easier [10,
+27, 28] but their constraints are very sensitive to code struc-
+ture [20].
+For matrix multiplications in particular, KernelFaRer [20]
+provides a more robust approach, detecting characteristics
+that define matrix multiplications. Polyhedral analyses can
+also be used to target matrix multiplication accelerators [9,
+58], but both these techniques fail to scale to the diversity
+9
+
+CC ’23, February 25–26, 2023, Montréal, QC, Canada
+P.A. Martínez, J. Woodruff, J. Armengol-Estapé, G. Bernabé, J.M. García, M.F.P. O’Boyle
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+1
+101
+102
+103
+104
+105
+106
+Code
+Candidates generated
+4s
+45s
+10m
+1h
+12h
+2d
+20d
+Approx. compilation time
+FACC*
+ATC
+Threshold
+Figure 12. Comparison of the number of candidates generated for matching GEMM codes: FACC* vs our approach.
+1
+3
+8
+48
+0
+20
+40
+60
+80
+100
+Code 2
+Code 21
+Code 7
+Code 1
+Number of candidates
+Time (s)
+IO Testing
+Tests
+Generation
+Candidates
+Generation
+Neural
+Embeddings
+280
+0
+500
+1,000
+1,500
+2,000
+Code 23
+Figure 13. Compilation time for different number of candi-
+dates.
+Direct
+i2mcol+gemm
+Winograd
+All
+0
+20
+40
+60
+80
+100
+6
+3
+3
+0
+1
+2
+10
+5
+% of matched codes
+Matched
+Not matched
+Figure 14. Matched convolution codes by ATC.
+1
+3
+4
+5
+6
+8
+10
+11
+13
+15
+All
+1
+10
+100
+1000
+Code
+Speedup
+Speedup (CPU)
+Speedup (TPU)
+ATC
+Figure 15. ATC speedup in convolution programs with ℎ =
+𝑤 = 224, 𝑘𝑤 = 𝑘ℎ = 11, 𝑐 = 3, 𝑘 = 96 and 𝑛 = 100.
+of real code. FACC [63] uses IO equivalence, which is ro-
+bust to program structure, but only addresses the challenges
+of FFTs and does not scale to longer function signatures
+used for GEMM. To support any accelerator type, the com-
+piler should support multi-dimensional arrays, while FACC
+only supports 1D arrays. Because in 1D arrays and FFTs the
+search space in matching the API parameters is small, FACC
+does not include anything to reduce it. With more complex
+programs and domains, this limitation makes compiling pro-
+grams intractable.
+Mask [51] uses symbolic execution to prove equivalence,
+which does not work well for floating-point problems. Fuzzy
+classification techniques based on code clone detection [40,
+57], domain-classification [59], pattern matching [15], code
+embeddings [2, 3, 21] and identifiers [36, 47] can be used
+to help compile to accelerators [63]. These classification
+strategies are able to classify diverse code structures, but do
+not provide a compilation strategy for using an accelerator
+on their own.
+A large class of techniques focus on migrating between
+APIs. These techniques often use program synthesis [16],
+NLP [46] and code embeddings [45, 48]. These techniques
+are unable to extract existing code into APIs.
+Compiling for GEMM Accelerators. Existing compila-
+tion strategies largely focus on lowering code from intrinsics
+to accelerators using rewrite rules [52, 53, 62] and synthesis
+techniques [17].
+Existing approaches to extracting matrix multiplications [20,
+28] are brittle. Synthesis-based techniques [1, 7, 41] and
+rewriting-based techniques [11, 54] have been developed
+to extract these DSLs that can then be lowered: but they
+largely require flexible DSLs, rather than APIs presented by
+hardware accelerators.
+Performance Prediction. Predicting code the performance
+of hardware accelerators is challenging, as the break-even
+point may depend on many different arguments within a
+function’s interface [4]. LogCA [4] introduces static perfor-
+mance comparison models for hardware accelerators and
+similar models have been applied in offloading tasks [64]. Ma-
+chine learning has often been applied in profitability settings,
+10
+
+Matching linear algebra and tensor code to specialized hardware accelerators
+CC ’23, February 25–26, 2023, Montréal, QC, Canada
+such as OpenCL Kernels [60, 61] and OpenMP [42]. Similar
+techniques have been applied to FPGAs, by estimating pow-
+er/performance [26] and tracking actual performance [50].
+10
+Conclusions
+This work presented ATC, a flexible domain-agnostic com-
+piler that matches legacy linear algebra code to accelerators.
+By using IO behavioral equivalence and smart search space
+reduction, we are able to match over 80% of challenging
+real-world programs to accelerator APIs, significantly out-
+performing all alternative approaches.
+Supporting new domains different from GEMM and convo-
+lution is easy because ATC focuses on behavior rather than
+code structure, which makes it very flexible and extensible.
+Furthermore, to support other accelerators in GEMM or con-
+volution, only the accelerator API is needed: ATC adapts to
+the new specification automatically.
+Future work will examine how to further reduce the search
+space using online learning and to expand the complexity
+of user code considered. Longer-term, we wish to automati-
+cally target a range of accelerators with diverse functionality,
+matching and transforming user code to maximize perfor-
+mance.
+Acknowledgments
+Grant TED2021-129221B-I00 funded by MCIN/AEI/10.13039/
+501100011033 and by the “European Union NextGenera-
+tionEU/PRTR”.
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+

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+MNRAS 000, 1–7 (2022)
+Preprint 6 January 2023
+Compiled using MNRAS LATEX style file v3.0
+Vortex weighing and dating of planets in protoplanetary discs
+Roman R. Rafikov1,2★, Nicolas P. Cimerman1
+1Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
+2Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+High-resolution sub-mm observations of some protoplanetary discs reveal non-asixymmetric features, which can often be
+interpreted as dust concentrations in vortices that form at the edges of gaps carved out by the embedded planets. We use recent
+results on the timescale for the planet-driven vortex development in low-viscosity discs to set constraints on the mass and age of
+a planet producing the vortex. Knowledge of the age of the central star in a vortex-bearing protoplanetary disc system allows one
+to set a lower limit on the planetary mass at the level of several tens of 𝑀⊕. Also, an independent upper limit on the planetary
+mass would constrain the planetary age, although given the current direct imaging detection limits this constraint is not yet very
+stringent (it is also sensitively dependent on the disc scale height). These results can be extended to account for the history of
+planetary mass accretion if it is known. We apply our calculations to several protoplanetary discs harbouring vortex-like features
+as revealed by ALMA and set limits of (30−50)𝑀⊕ (for disc aspect ratio of 0.1) on the minimum masses of putative planets that
+could be responsible for these vortices. Our vortex-based method provides an independent way of constraining the properties of
+embedded planets, complementary to other approaches.
+Key words: hydrodynamics – instabilities – shock waves – accretion discs – planets and satellites: formation – methods:
+numerical
+1 INTRODUCTION
+Observations of protoplanetary discs (PPDs) in dust continuum emis-
+sion with ALMA revealed a variety of substructures (Andrews 2020),
+including axisymmetric gaps and rings as well as non-axisymmetric
+clumps and arcs. An intriguing possibility is that these features could
+be produced by young embedded planets. In particular, gravitational
+coupling between a massive planet and the disc is known to result in
+formation of observable gaps around the planetary orbit (Papaloizou
+& Lin 1984; Rafikov 2002b). The evolution of vortensity (potential
+vorticity) at the edges of these gaps (Lin & Papaloizou 2010; Dong
+et al. 2011; Cimerman & Rafikov 2021) due to shock dissipation of
+the planet-driven density waves (Goodman & Rafikov 2001; Rafikov
+2002a) can trigger the Rossby Wave Instability (RWI, Lovelace et al.
+1999) resulting in the formation of fluid vortices at these locations.
+Dust accumulation inside the vortices (Barge & Sommeria 1995)
+naturally leads to observable non-axisymmetric arcs and lobes.
+Formation of these structures does not necessarily require massive
+(Jovian) planets. For example, it was shown (Dong et al. 2017; Bae
+et al. 2017; Miranda & Rafikov 2019, 2020a,b) that multiple visible
+gaps and rings in the dust distribution can result from nonlinear
+damping of multiple spirals triggered by a single sub-Jovian mass
+planet in a low viscosity disc. The mass of the planet 𝑀p can in fact be
+below the so-called thermal mass defined as 𝑀th = �𝐻p/𝑅p
+�3 𝑀★ =
+ℎ3p 𝑀★, where 𝐻p is the disc scale height at the planetary distance 𝑅p,
+ℎp = 𝐻p/𝑅p is the disc aspect ratio there, and 𝑀★ is the stellar mass.
+Emergence of vortices at the edges of planetary gaps also does not
+★ E-mail: rrr@damtp.cam.ac.uk (RRR)
+require massive planets if the disc is almost inviscid (Hammer et al.
+2021; Hallam & Paardekooper 2020), and sub-𝑀th mass planets can
+easily trigger them (Cimerman & Rafikov 2023, hereafter CR23).
+In this study we focus on non-axisymmetric disc features which
+can be interpreted as planet-induced vortices (other ways to produce
+vortices are mentioned in Section 2). Their development is not an
+instantaneous process as the evolution of vortensity near the planetary
+orbit towards the RWI takes a certain amount of time. As we show
+in this work, this fact can be exploited to set useful constraints on
+the mass and/or age of a putative planet responsible for production
+of the observed vortices in a PPD. This method relies on the recent
+calculation of CR23 who studied the development of vortices in
+inviscid PPDs. In that work, we showed that the time it takes for
+vortices to emerge at the edge of the gap carved out by a sub-𝑀th
+mass planet can be approximated as
+𝜏vrt ≈ 𝐴 𝑃p
+� 𝑀p
+𝑀th
+� 𝛼
+ℎ𝛽
+p ,
+where
+(1)
+𝐴 ≈ 1.6,
+𝛼 ≈ −2.7,
+𝛽 ≈ −0.86,
+(2)
+and 𝑃p is the orbital period at 𝑅p. This result assumes 𝑀p to be fixed
+in time. Interestingly, CR23 found 𝜏vrt to only weakly depend on the
+radial profile of surface density in the disc.
+We describe the general idea of our method in Section 2 and show
+how it can constrain the mass and age of the planet in Section 3 and
+Section 4, respectively. In Section 5 we extend our constraints to the
+case of a planet accreting its mass over an extended period of time.
+We apply our results to observed PPDs in Section 6 and discuss them
+in Section 7.
+© 2022 The Authors
+arXiv:2301.01789v1  [astro-ph.EP]  4 Jan 2023
+
+2
+R. R. Rafikov and N. P. Cimerman
+2 GENERAL IDEA OF THE METHOD
+Let us suppose that observations of dust continuum emission reveal
+a gap in a PPD, together with a non-axisymmetric lobe or arc at
+the gap edge, indicative of a vortex which traps dust grains at this
+location (e.g. van der Marel et al. 2016; Kraus et al. 2017; Dong
+et al. 2018; Pérez et al. 2018). We will interpret this observation by
+assuming that a planet (not necessarily directly visible) is located
+within the gap and is responsible for creating both the gap and the
+vortex (via the RWI). The spatial association of a vortex with an
+adjacent gap provides strong support to this interpretation and makes
+some other possibilities for triggering vortices, e.g. global baroclinic
+instability (Klahr & Bodenheimer 2003), convective overstability
+(Teed & Latter 2021), vertical shear instability (Richard et al. 2016)
+less attractive.
+We assume the disc viscosity to be low (essentially inviscid),
+consistent with many observations of PPDs (Pinte et al. 2016; Rafikov
+2017; Flaherty et al. 2020). For now we will also assume that 𝑀p has
+been constant ever since the planet appeared in the disc, a constraint
+that we will relax in Section 5. With these conditions fulfilled, the
+result (1)-(2) applies. We can then use the observation of the vortex
+to set a constraint on a particular combination of the planetary mass
+𝑀p and the planetary age 𝜏p — the time that has passed since the
+planet has reached its final mass 𝑀p.
+Indeed, the observation of a vortex at the gap edge implies that the
+RWI had enough time to fully develop into the non-linear stage in
+that region, i.e. that
+𝜏p > 𝜏vrt.
+(3)
+Together with equation (1) this leads to the following combined
+constraint on 𝜏p and 𝑀p:
+𝜏p𝑀−𝛼
+p
+> 𝐴𝑃p𝑀−𝛼
+th ℎ𝛽
+p .
+(4)
+With fit parameters (2) we can write this in physical units as
+𝜏p
+Myr
+�
+𝑀p
+102𝑀⊕
+�2.7
+> 0.11
+�
+𝑅p
+50AU
+�1.5 � 𝑀★
+𝑀⊙
+�2.2 � ℎp
+0.1
+�7.2
+.
+(5)
+This condition must be fulfilled whenever a vortex is observed at the
+gap edge. It is illustrated in Fig. 1 for several values of ℎp and 𝑅p.
+In a similar vein, the absence of vortex-like structures at the edges
+of a visible gap in a disc might be interpreted as meaning that 𝜏p <
+𝜏vrt, i.e. that planet-driven accumulation of vortensity has not yet led
+to RWI. If that were the case, the inequality in the constraint (4)-(5)
+would change its sign. However, this possibility has an important
+caveat as the absence of a vortex may also be interpreted differently:
+it could have formed at the gap edge earlier but then got destroyed
+through one of the processes that tend to destabilize vortices once
+they evolve into the nonlinear regime: the elliptical instability (Lesur
+& Papaloizou 2009), baroclinic effects (Rometsch et al. 2021; Fung &
+Ono 2021), dust feedback (Fu et al. 2014), etc. Also, vortex formation
+may have been delayed or suppressed altogether if disc viscosity
+is sufficiently high (Hammer et al. 2017; Hallam & Paardekooper
+2020). Thus, the lack of a vortex near a planetary gap cannot be
+unambiguously interpreted as meaning that the embedded planet did
+not get a chance to create it, i.e. that 𝜏p < 𝜏vrt. For that reason (and
+unlike Hallam & Paardekooper 2020) in the following we will not
+draw any conclusions from the absence of vortices at the edges of
+putative planetary gaps found in sub-mm observations.
+We will now show how equations (4) & (5) can be used to sepa-
+rately constrain 𝑀p or 𝜏p.
+10
+100
+1000
+Mp [M
+]
+0.01
+0.1
+1
+10
+p [Myr]
+no vortex
+hp = 0.07
+hp = 0.1
+hp = 0.15
+Rp = 100 AU
+Figure 1. Combined constraint (5) on the planetary mass 𝑀p and age 𝜏p,
+shown for different parameters of a system with 𝑀★ = 𝑀⊙. Solid lines are
+for 𝑅p = 50 AU and ℎp = 0.07 (fuchsia), ℎp = 0.1 (blue), ℎp = 0.15
+(green). Blue dashed line is for 𝑅p = 100 AU, ℎp = 0.1. Grey shaded region
+is excluded as no vortices should appear in this part of the parameter space
+(bounded by ℎp = 0.07 curve for illustration). Arrows indicate 𝑀th calculated
+using ℎp corresponding to the arrow color (same as in the legend). Constraint
+(5) — solid curves — is strictly valid only for 𝑀p ≲ 𝑀th.
+0.1
+1
+10
+sys [Myr]
+10
+100
+1000
+Mp [M
+]
+no vortex
+hp = 0.07
+hp = 0.1
+hp = 0.15
+Rp = 100 AU
+Figure 2. Mass constraint (6), (7) as a function of the system (stellar) age
+𝜏sys. Meaning of curves, arrows and shading are the same as in Fig. 1.
+3 VORTEX WEIGHING OF PLANETS
+Let us suppose that the age (time since formation) of the protostar-
+disc system 𝜏sys is known, e.g. from isochrone fitting of the charac-
+teristics of the central star. This is usually the case at some level of
+accuracy. Since, obviously, the planet is younger than its parent star,
+one must have 𝜏sys > 𝜏p. However, the presence of a vortex at the gap
+edge means that the inequality (3) is also fulfilled, which necessarily
+implies that 𝜏sys > 𝜏vrt. Using equation (4), this condition can be
+converted into a lower limit on 𝑀p:
+𝑀p > 𝑀vrt = 𝑀th
+�
+𝐴 ℎ𝛽
+p
+𝑃p
+𝜏sys
+�−1/𝛼
+.
+(6)
+In physical units,
+𝑀vrt ≈ 40𝑀⊕
+� 𝜏sys
+Myr
+�−0.37 �
+𝑅p
+50AU
+�0.56 � ℎp
+0.1
+�2.7 � 𝑀★
+𝑀⊙
+�0.81
+.
+(7)
+Note a strong dependence of 𝑀vrt on ℎp, but a rather weak scaling
+with 𝜏sys. This constraint is illustrated in Fig. 2.
+Note that for the mass constraint (6)-(7) to be valid, the timescale
+fit (1) should be justified in the first place. For this to be the case,
+the planetary mass must be in the sub-thermal mass regime. One can
+MNRAS 000, 1–7 (2022)
+
+Vortex weighing and dating
+3
+easily show that
+𝑀vrt
+𝑀th
+≈ 0.13
+� 𝜏sys
+Myr
+�−0.37 �
+𝑅p
+50AU
+�0.56 � ℎp
+0.1
+�−0.32 � 𝑀★
+𝑀⊙
+�−0.19
+,
+(8)
+i.e. the condition 𝑀p ≲ 𝑀th should be not difficult to satisfy in
+general (in Figs. 1,2 we illustrate the values of 𝑀th with arrows).
+Thus, we expect 𝑀vrt to provide a lower limit on 𝑀p quite generally.
+The constraint (6)-(7) can be improved (i.e. 𝑀vrt increased) if we
+had some independent way to set an upper limit on 𝜏p, which is
+lower than the system age 𝜏sys. In practice, however, such refined
+information on 𝜏p may be difficult to obtain.
+4 VORTEX DATING OF PLANETS
+One can also turn the argument around and assume that, in addition
+to observing a vortex adjacent to a gap, we also know the mass 𝑀p
+of the gap-opening planet — either via atmospheric modelling if the
+planet is visible, or through indirect dynamical measurements if it
+has not been imaged. We can then use the presence of the vortex to
+set a lower limit on the planetary age 𝜏p via equation (3), in which
+𝜏vrt ≈ 105yr
+�
+𝑀p
+102𝑀⊕
+�−2.7 �
+𝑅p
+50AU
+�1.5 � ℎp
+0.1
+�7.2 � 𝑀★
+𝑀⊙
+�2.2
+.
+(9)
+If only an upper limit 𝑀↓ on planetary mass is available to us,
+𝑀p < 𝑀↓ (e.g. from non-detection of the planet through near-IR
+imaging), then one should use 𝑀↓ instead of 𝑀p in (9). Solid and
+dashed lines in Fig. 1 give 𝜏vrt (as a function of 𝑀p or 𝑀↓) for
+different values of ℎp and 𝑅p.
+A constraint on 𝜏p would be extremely useful for understanding the
+timing of planet formation. It can also serve as a consistency check for
+calculations of planetary evolution post-formation, since the present
+day temperature and luminosity of the planet are themselves functions
+of its age 𝜏p (e.g. Linder et al. 2019), see Section 7. Unfortunately, the
+accuracy of the lower limit (3) & (9) may be somewhat compromised
+by the uncertainties in the determination of various parameters that
+enter it, e.g. 𝑀p and, especially, ℎp, given how steeply 𝜏vrt scales
+with them.
+5 ACCOUNTING FOR ACCRETION HISTORY OF A
+PLANET
+Our results (1) & (2) for 𝜏vrt have been obtained in CR23 for a
+constant 𝑀p (not varying in time). This implicitly assumes that planet
+has grown to its final 𝑀p very rapidly, having accreted its mass
+almost instantaneously; this accretion history is illustrated in panel
+(a) of Fig. 3. In panel (b) we also illustrate the corresponding growth
+of the characteristic amplitude1 𝐴𝜁 of the planet-induced vortensity
+perturbation 𝜁, which is the variable that eventually determines vortex
+generation (CR23): very crudely, one may expect the RWI to set in
+when 𝐴𝜁 reaches some threshold value (illustrated with red dotted
+line). The growth rate of 𝐴𝜁 in panel (b) is constant since it sensitively
+depends on 𝑀p and 𝑀p is fixed in this case.
+One may consider other representative histories of planetary mass
+evolution. For example, in Fig. 3c 𝑀p undergoes an initial period of
+accretion and then stays at its final value until the RWI sets in. As
+another example, in panel (e) the planetary mass increases steadily
+1 E.g. a maximum or minimum value of 𝜁 as a function of radius, see CR23.
+and the RWI gets triggered while 𝑀p is still growing. For these
+growth histories, the increase of 𝐴𝜁 is no longer purely linear, see
+panels (d) and (f), and using the final planetary mass2 in formula
+(1) we would underestimate the true age of the planet 𝜏p (illustrated
+in top panels), i.e. the time since its growth has started and until
+the present day when the vortex has emerged. Instead, application of
+equation (1) would give us some other time 𝜏0, which is illustrated
+by the orange lines (based on the growth rate of 𝐴𝜁 at the time when
+RWI sets in) in panels (d) & (f). Since growth of 𝜁 accelerates (quite
+steeply) for higher 𝑀p, the growth rate of 𝐴𝜁 can only increase in
+time, so that 𝜏p ≥ 𝜏0 always (with equality only for 𝑀p(𝑡) = const.,
+see Fig. 3a,b).
+Very importantly, this complication does not affect the validity
+of our time constraint, since 𝜏0 is given by our equation (1) and
+we just saw that 𝜏p ≥ 𝜏0. However, in some scenarios, e.g. in the
+continuous accretion case shown in panels (e),(f), 𝜏0 can be much
+shorter than 𝜏p, making our time constraint (3) too conservative.
+Thus, it is desirable to find ways to somehow account for the history
+of accretion (provided that it is known) to improve limits on 𝜏p.
+One way to do this has already been discussed in CR23 and
+amounts to replacing 𝜏p𝑀−𝛼
+p
+with
+∫ 𝜏p
+0
+�
+𝑀p(𝑡)
+�−𝛼 d𝑡 in equation
+(4); this modification allows us to account for the evolution of the
+vortensity (or 𝐴𝜁 ) growth rate, which is proportional to 𝑀−𝛼
+p
+, as
+𝑀p(𝑡) increases. Thus, we generalize the combined constraint on 𝜏p
+and 𝑀p in the case of an accreting planet to
+∫ 𝜏p
+0
+�
+𝑀p(𝑡)
+�−𝛼 d𝑡 > 𝐴𝑃p𝑀−𝛼
+th ℎ𝛽
+p .
+(10)
+Since this constraint must reduce to the inequality (4), we will assume
+all its parameters — 𝛼, 𝛽, 𝐴 — to be still given by the equation3 (2).
+Then in physical units equation (10) becomes
+(Myr)−1
+∫ 𝜏p
+0
+� 𝑀p(𝑡)
+102𝑀⊕
+�2.7
+d𝑡 > 0.11
+�
+𝑅p
+50AU
+�1.5
+×
+� 𝑀★
+𝑀⊙
+�2.2 � ℎp
+0.1
+�7.2
+.
+(11)
+For 𝑀p(𝑡) = const this inequality reduces to (5).
+We can apply this generalized criterion to the simulations of Hal-
+lam & Paardekooper (2020) who considered planetary accretion his-
+tory in the form 𝑀p(𝑡) = 𝑀f sin2 [(𝜋/2)(𝑡/𝑡G)] (where 𝑡G is the
+growth time) and determined the values of the final planet mass 𝑀f
+such that the RWI would marginally set in at 𝑡 = 𝑡G. In our notation
+this means setting 𝑡G = 𝜏vrt. We can use our results and determine the
+relation between such 𝑡G and 𝑀f by changing inequality to equality
+in equation (10) and setting 𝜏p = 𝑡G. We find, using the definition of
+𝑀th and introducing 𝑞f = 𝑀f/𝑀★,
+𝑡G = 𝐴 𝜅−1𝑃p 𝑞𝛼
+f ℎ𝛽−3𝛼
+p
+,
+(12)
+where, for a particular accretion history of Hallam & Paardekooper
+(2020), 𝜅 =
+∫ 1
+0 [sin(𝜋𝑥/2)]2𝛼𝑑𝑥 ≈ 0.33.
+As these authors also included the effects of viscosity, which is
+2 For simplicity we neglect the possible growth of 𝑀p after the vortex has
+appeared and until the present time.
+3 This assumption is only approximate since the non-trivial history of accre-
+tion may modify the radial profile of 𝜁 , which determines the RWI stability
+(Cimerman & Rafikov 2021). Also, the RWI threshold itself is not entirely
+universal (CR23). But this approximation should not be too bad as the RWI
+onset is mainly determined by the late-time behavior of 𝑀p(𝑡). Finally, note
+that theoretical arguments suggest 𝛼 = 2.6, but the numerical results are
+closer to 𝛼 ≈ 2.7 (CR23).
+MNRAS 000, 1–7 (2022)
+
+4
+R. R. Rafikov and N. P. Cimerman
+0
+Mp(t)
+p
+a
+Instant accretion
+p
+c
+Initial episode of accretion
+p
+e
+Continuous accretion
+t
+0
+A (t)
+0
+b
+t
+0
+d
+t
+0
+f
+Figure 3. Illustration of the different representative planetary accretion histories: (left) very rapid (instant) initial accretion to the final mass, (centre) extended
+initial interval of accretion, (right) continuous accretion. Top panels illustrate 𝑀p(𝑡) (blue) while the bottom panels show the corresponding growth of the
+characteristic amplitude 𝐴𝜁 (green) of the planet-driven vortensity perturbation (this calculations assumes d𝐴𝜁 /d𝑡 ∝ [𝑀p(𝑡)]2.7, see text). Arrows in the top
+panels indicate the planetary age 𝜏p (time since the start of its accretion), while in the bottom panels they show the "time to vortex formation" 𝜏0 calculated
+using equation (1) and assuming 𝑀p given by its final value. The red dotted line indicates the critical value of 𝐴𝜁 when the vortices are expected to appear. The
+key point illustrated here is that 𝜏0 ≤ 𝜏p always.
+known to delay the onset of RWI (Hammer et al. 2017), we cannot
+directly compare our results for 𝑡G with theirs. However, if we focus
+on their smallest 𝑞f = 1.5×10−4 (since in their setup this corresponds
+to the lowest value of viscosity, closer to our inviscid setup) and adopt
+their ℎp = 0.05 and the fit parameters (2), we find the age of the planet
+to satisfy 𝜏p ≳ 𝑡G ≈ 39𝑃p. This is comfortably below 𝑡G ≈ 200𝑃p
+that Hallam & Paardekooper (2020) find for the same 𝑞f, consistent
+with the viscosity-driven delay. Equally importantly, had we used the
+equation (4), that assumes 𝑀p = const, instead of (10), we would
+have found 𝜏p ≳ 13𝑃p (a factor of 𝜅 lower), far less constraining
+than the result that we obtained accounting for the (known) accretion
+history.
+Given the steep dependence of the integrand in (11) on 𝑀p (reflect-
+ing 𝑀p-dependence of the 𝜁 growth rate), we expect 𝐴𝜁 to increase
+the most when 𝑀p(𝑡) is close to its final value. This is indeed what
+we see in Fig. 3d, in which the initial accretion episode contributes
+only weakly to the total increase of 𝐴𝜁 , despite its duration being
+comparable to the time interval when 𝑀p stayed at its final value (our
+calculation in this plot assumed d𝐴𝜁 /d𝑡 ∝ 𝑀2.7
+p
+for compatibility
+with equation (11), see CR23). Thus, it is the history of 𝑀p accre-
+tion at late times that is most important for determining the age of a
+putative planet in an observed vortex-hosting system.
+6 APPLICATION TO OBSERVED DISCS
+We apply the constraints derived above to several protostellar
+systems observed by ALMA, for which the vortices have been
+invoked as a possible explanation of the observed non-axisymmetric
+features — arcs, clumps, etc. It is important to remember that
+the features detected in continuum emission by ALMA are due to
+thermal emission of dust grains, while our results on the emergence
+of vortices apply to the gaseous component of the disc. However,
+it has been shown by a number of authors (Barge & Sommeria
+1995; Godon & Livio 1999; Fu et al. 2014) that vortices are very
+efficient at trapping dust, providing support to our association of
+the dust asymmetries with the gas vortices in PPDs. Since our
+limits on 𝜏p and 𝑀p are highly sensitive to the disk aspect ratio ℎp,
+which is poorly known in most cases, we will retain the scaling with
+ℎ0.1 = ℎp/0.1 in our estimates.
+HD 135344B (SAO 206462)
+This 𝑀★ = 1.5𝑀⊙, 𝜏sys ≈ 9 Myr old (Asensio-Torres et al. 2021)
+Herbig F star harbours a transitional disc. ALMA dust continuum
+observations reveal an axisymmetric inner ring separated by a gap-
+like structure (centered around 70 AU) from an (outer) arc that can
+be interpreted as a vortex at the outer gap edge (van der Marel
+et al. 2016; Cazzoletti et al. 2018). The possibility of a planetary
+origin of these structures is supported by the near-IR scattered light
+observations of a two-armed spiral (Muto et al. 2012), although a
+unified model explaining all these features at once is lacking. We
+will nevertheless assume that the gap and the outer vortex are due to
+the (unseen) gap-opening planet at 𝑅p ≈ 70 AU, and the inner ring
+reflects dust trapping at the pressure maximum at the inner gap edge.
+These data and equations (6)-(7) allow us to constrain planetary mass
+as 𝑀p ≳ 32ℎ2.7
+0.1𝑀⊕.
+Direct imaging of HD 135344B with VLT/SPHERE sets an upper
+limit of 𝑀↓ ≈ 4𝑀J on the mass of a planetary object at ∼ 102AU
+scales (Asensio-Torres et al. 2021). Unfortunately, this 𝑀↓ is higher
+than the thermal mass 𝑀th = 1.5ℎ3
+0.1𝑀J, which makes the use of the
+timescale constraint (3),(9) unjustified (its blind application would
+give 𝜏vrt ≈ 500ℎ7.2
+0.1 yr, comparable to the orbital period at the gap
+location and not constraining 𝜏p effectively).
+HD 36112 (MWC 758)
+This 𝑀★ ≈ 1.8𝑀⊙, 𝜏sys ≈ 9 Myr old (Asensio-Torres et al. 2021)
+star harbours two clumps on top of the two rings separated by a gap
+in the outer disc (Dong et al. 2018). Neglecting the slight eccentricity
+of the disc and assuming the rings with clumps to correspond to the
+inner and outer edges of the gap carved by a planet, we will adopt
+𝑅p ≈ 70 AU for the planetary orbit. Then equations (6)-(7) allow us
+to set a mass constraint 𝑀p ≳ 37ℎ2.7
+0.1𝑀⊕.
+Analysis of the direct imaging observations of this system by
+Asensio-Torres et al. (2021) suggests that the upper limit on the
+possible point source inside the assumed gap is ∼ 8𝑀J, significantly
+higher than 𝑀th, precluding us from meaningfully constraining the
+MNRAS 000, 1–7 (2022)
+
+Vortex weighing and dating
+5
+age of the planet.
+HD 143006
+This G-type T Tauri star with 𝑀★ = 1.8𝑀⊙ and an estimated age
+of 𝜏sys ≈ 8 Myr harbours a disc rich in substructures (Pérez et al.
+2018). In addition to a misaligned inner disc, it features two outer
+rings separated by a gap centered around 52 AU, with an arc just
+outside the outermost ring. Interpreting these features as produced
+by an unseen planet inside the gap at 𝑅p = 52 AU, we get the mass
+constraint 𝑀p ≳ 33ℎ2.7
+0.1𝑀⊕ from equations (6)-(7).
+NaCo/VLT direct imaging does not provide a useful constraint on
+the mass of a putative planet, with 𝑀↓ at the level of several tens of
+𝑀J at the outer gap location (Jorquera et al. 2021). Thus, we cannot
+set a useful lower limit on the planetary age.
+V1247 Ori
+V1247 Ori is a 𝜏sys = 7.5 Myr old, 𝑀★ ≈ 1.9𝑀⊙ star (Willson et al.
+2019) harbouring a pre-transitional disc. ALMA dust continuum ob-
+servations (Kraus et al. 2017) reveal an inner disc (or ring) separated
+by a gap from the outer arc, which may be interpreted as a vortex at
+the outer gap edge. Assuming a planet to be in the gap, at 𝑅p ≈ 90
+AU, one finds that the planetary mass must satisfy 𝑀p ≳ 48ℎ2.7
+0.1𝑀⊕.
+While we could not find explicit limits on the mass of the putative
+planet in V1247 Ori system from direct imaging observations, Kraus
+et al. (2017) found that an 𝑀p = 3𝑀J planet can roughly match
+the shape of the spiral observed in scattered light using HiCIAO.
+Unfortunately, this 𝑀p is again above 𝑀th, not allowing the age of
+the planet to be meaningfully constrained.
+7 DISCUSSION
+7.1 Combination of multiple constraints
+Our limits on 𝑀p and 𝜏p based on the presence of vortices next
+to gaps in PPDs become even more powerful when combined with
+additional constraints on these key parameters. In particular, young
+planets passively lose thermal energy that they have been endowed
+with at formation, resulting in their luminosity decreasing with time.
+As more massive planets retain more heat at formation, it takes
+them longer to cool. Thus, if one can observationally constrain the
+luminosity of a planet 𝐿p to lie below a certain limit (or determine
+it in the case of direct detection), this would provide an additional
+constraint on 𝑀p and 𝜏p.
+We illustrate this approach in Fig. 4, where we show the vortex-
+based constraints from Fig. 1 together with the constraint 𝐿p <
+10−6𝐿⊙ (outside the pink shaded region to the right of the black
+dotted curve) based on the work4 of Linder et al. (2019). We also
+show the 𝐿p = 10−7𝐿⊙ curve (red dotted) which may be relevant
+for future direct imaging experiments. In addition, we impose a con-
+straint 𝜏p < 15 Myr (region below the orange dot-dashed line) since
+protoplanetary discs usually do not survive for that long (similar to
+the logic used in Section 3). There are other, complementary ways
+of constraining planetary properties, for example gap width/depth
+fitting (Dong & Fung 2017; Asensio-Torres et al. 2021) which can
+provide model-dependent information on 𝑀p for individual systems;
+we will not consider them here.
+4 We use the tracks for the bolometric luminosity 𝐿p of a fixed mass planet
+from Fig. 6 of Linder et al. (2019), which assume evolution with a cloud-free
+atmosphere of solar metallicity and use the petitCODE grid.
+10
+100
+1000
+Mp [M
+]
+0.01
+0.1
+1
+10
+p [Myr]
+no vortex
+hp = 0.07
+hp = 0.1
+hp = 0.15
+Rp = 100 AU
+Lp = 10
+6L
+Lp = 10
+7L
+p = 15 Myr
+Figure 4. Combination of the different constraints on the mass 𝑀p and age
+𝜏p of a planet in a vortex-hosting PPD. Grey shaded region is excluded
+(for ℎp = 0.07) as vortices have no time to develop in this part of the
+parameter space, analogous to Fig. 1 (solid and dashed lines are the same as
+in that figure). Pink shaded region is excluded as it corresponds to planetary
+(bolometric) cooling luminosity 𝐿p exceeding 𝐿p = 10−6𝐿⊙ (black dotted
+curve). We also show the curve 𝐿p = 10−7𝐿⊙ (red dotted curve); the 𝐿p
+curves are based on Linder et al. (2019). The orange shaded region above
+the orange dot-dashed curve excludes planetary ages above 15 Myr. Planets
+satisfying all three constraints reside in the unshaded part of the parameter
+space.
+A combination of the three constraints — based on planetary lu-
+minosity, age and presence of vortices — limits planetary 𝑀p and
+𝜏p to lie within the unshaded region. This region shrinks for hotter
+discs with larger ℎp (compare fuchsia and green solid curves), as
+well as for larger 𝑅p (compare blue solid and dashed lines). Thus, the
+vortex-based limits are more stringent in hotter discs and for more
+distant planets. Also, the allowed region would shrink even further as
+the upper limit on 𝐿p gets lowered in the future. It should also be re-
+membered that the luminosity-based (dotted) curves assume that the
+(possible ongoing) gas accretion provides insignificant contribution
+to 𝐿p. If the planetary accretion luminosity is non-negligible, this
+would additionally shift the dotted curves to the left, constraining
+𝑀p and 𝜏p even further.
+7.2 Utility of the vortex-based constraint
+The sub-Jovian value of 𝑀vrt implied by the equation (7) and our
+estimates in Section 6 is very relevant in light of the recent results
+(Dong et al. 2017; Bae et al. 2017; Miranda & Rafikov 2019, 2020a,b)
+showing that in a low viscosity disc a single sub-𝑀th planet can give
+rise to a series of several prominent gaps and rings in the radial dust
+distribution. For example, for the AS 209 system (𝑀★ = 0.83𝑀⊙,
+𝜏sys ≈ 1 Myr, Andrews et al. 2018) imaged with ALMA Zhang et al.
+(2018) have shown that a single planet with 𝑀p as low as 25𝑀⊕ orbit-
+ing within the outer (primary) gap at 𝑅p ≈ 100AU can be responsible
+for creating all five gaps observed in this disc. This possibility makes
+the typical values of 𝑀p implied by the constraint (6)-(7) very inter-
+esting for understanding the architecture of the underlying planetary
+system.
+Given the upper limits on 𝑀p based on direct imaging in several
+systems covered in Section 6, we found our age constraint (3) & (9)
+to be not very useful at present. However, things will improve as
+𝑀↓ decreases in the future. Once 𝑀↓ is below 𝑀th, our constraint
+(3) & (9) becomes valid and may provide useful information on the
+MNRAS 000, 1–7 (2022)
+
+6
+R. R. Rafikov and N. P. Cimerman
+planetary age. The decrease of 𝑀↓ may not necessarily come from
+improved direct imaging capabilities. In particular, one may use the
+technique of multiple gap fitting used by Zhang et al. (2018) for AS
+209 to get a much better measurement of 𝑀p or 𝑀↓.
+Just for illustration, let us imagine that AS 209 did possess a vortex
+at the edge of its outermost gap (just outside 𝑅p = 100AU). Then
+using 𝑀p ≈ 25𝑀⊕ (based on Zhang et al. 2018) equation (9) would
+predict 𝜏p ≳ 8ℎ7.2
+0.1 Myr. This 𝜏p is much longer (for ℎp = 0.1) than
+the age of the system 𝜏sys ≈ 1 Myr, and could have implied that either
+the planetary mass is underestimated (by a factor of ∼ 2), or that the
+stellar age is underestimated (by almost an order of magnitude), or
+that the disc is somewhat colder — using ℎp = 0.075 (consistent
+with Paneque-Carreño et al. 2022) in (9) would reconcile 𝜏p with its
+estimated 𝜏sys.
+The latter possibility represents a simple way to resolve the age
+discrepancy for this imaginary AS 209-like system. It also highlights
+the importance of good knowledge of the thermal state of the disc
+near the planet, which sets ℎp. Indeed, 𝜏vrt depends very sensitively
+on ℎp and a mis-estimate of ℎp by a factor of 2 would result in a
+factor of ≈ 150 error in the determination of 𝜏vrt and the planetary
+age. The situation is somewhat improved for the mass constraint
+(6)-(7), in which variation of ℎp by a factor of 2 results in 𝑀vrt
+changing by a factor of ≈ 6.5. In any case, good understanding
+of disc thermodynamics is clearly needed when applying the age
+constraint (3) & (9). Recent ALMA measurements of emission heights
+of different molecular lines in PPDs (Law et al. 2021, 2022; Paneque-
+Carreño et al. 2022) provide a (model-dependent) way to determine
+disc aspect ratio at different radii, generally finding values in the
+range ℎp ∼ (0.07 − 0.1) for 𝑅p ∼ (50 − 100) AU.
+On the other hand, our constraints (5),(7) & (9) should be rather
+insensitive to the radial profile of the disc surface density near the
+planet. Indeed, CR23 showed that the parameters of the fit (1),(2)
+show little variation when changing the slope of the surface density
+profile near the planet. Also, the dependence of the vortex-based con-
+straints on 𝑅p and 𝑀★ is not as steep as for ℎp, and the characteristic
+accuracy with which these parameters can be measured is (10−20)%
+or better.
+7.3 Additional processes and further extensions
+Since the constraints (3)-(6) are lower limits on 𝜏p and 𝑀p, respec-
+tively, they do not change if the vortices we observe in discs now are
+not the first generation vortices. It is possible that the vortices that
+formed early on have then dissolved and what we are seeing now are
+the second (or multiple) generation vortices (Hammer et al. 2021).
+Nevertheless, even in this case the condition 𝜏p > 𝜏vrt would still
+need to be fulfilled, definitely for the first generation of vortices, as
+well as for the following generations, confirming the validity of the
+constraints (3)-(6).
+Similarly, dust trapped in vortices can maintain observable non-
+axisymmetric distribution even after the vortices in the gaseous com-
+ponent dissolve (Fu et al. 2014). Thus, when we see an asymmetry in
+dust continuum observations, the original vortex that has led to it may
+have already been gone. However, this would again not invalidate the
+constraints obtained in Sections 3 & 4.
+The fit (1),(2) for 𝜏vrt was derived by CR23 for discs which are
+inviscid or have low viscosity, an assumption which is consistent
+with observations of many systems (see Section 2). We can roughly
+estimate the upper limit on the viscosity 𝜈 below which the inviscid
+assumption should be valid by demanding the timescale on which the
+vortensity structures produced by the planet get viscously diffused
+away to be longer that the age of the system 𝜏sys. For the characteristic
+radial scale of the vortensity structures 𝐿 ∼ 𝐻p(𝑀p/𝑀th)−0.4 (Dong
+et al. 2011; CR23) this timescale is ∼ 𝐿2/𝜈 ∼ 𝑃p𝛼−1(𝑀p/𝑀th)−0.8,
+where we adopted the 𝛼-ansatz for the viscosity 𝜈 = 𝛼Ωp𝐻2p (and
+Ωp = 2𝜋𝑃−1
+p ). For this to exceed 𝜏sys for a sub-thermal mas planet
+we require that roughly 𝛼 ≲ 𝑃p/𝜏sys ∼ 10−4, given the long orbital
+periods at 𝑅p = 50 − 100 AU. A more refined estimate of the critical
+𝛼 can be found in CR23.
+However, even if the disc were sufficiently viscous (i.e. for
+𝛼 ≳ 10−4), the RWI development would get only delayed (Hallam &
+Paardekooper 2020) or the instability may be suppressed altogether,
+see Hammer et al. (2017), CR23. Because of that, our inviscid esti-
+mate for 𝜏vrt continues to provide a lower limit on 𝜏p in the presence
+of a vortex, i.e. the equation (3) and all other constraints remain valid
+(see Section 5 for application of this logic).
+On the other hand, some other effects may accelerate vortex pro-
+duction compared to the results of CR23. For example, this could
+happen as a result of baroclinicity of the disc near the planet since
+the RWI is sensitive to entropy gradients (Lovelace et al. 1999).
+Under certain circumstances dust feedback can also promote vor-
+tex production (Lin & Youdin 2017). These processes, if they are
+important, may somewhat weaken our constraints on 𝑀p and 𝜏p.
+We demonstrated in Section 5 how our constraints can be modified
+to account for the evolution of planetary mass 𝑀p. Other relevant
+parameters might change as well, for example 𝑅p can vary as a result
+of planet migration, or ℎp can change as the disc evolves in time.
+CR23 outlined ways in which one can account for these processes
+to derive a new estimate for 𝜏vrt instead of (1),(2), thus providing a
+pathway to modifying our constraints on 𝑀p and 𝜏p.
+Of the four systems considered in Section 6, three show vortex-like
+non-axisymmetries only at the outer edge of the putative planetary
+gap, and only one, MWC 758, has them on both sides of the gap.
+This is somewhat surprising, since the simulations of CR23 not only
+show the emergence of vortices on both sides of the gap, but also
+demonstrate that the time interval separating their production by
+RWI is typically smaller than 𝜏vrt (see Table 1 in that work). Thus,
+one would expect to see vortices on both sides of the gap more
+often. It is not clear why this expectation fails. It could be that the
+dust concentration is more efficient in the outer vortices5 or that it
+tends to survive there considerably longer than in the inner ones.
+Or that some physical processes neglected in our study suppress the
+formation of the inner vortices. Expanding the sample of observed
+discs with vortex-like asymmetries would help in resolving this issue
+in the future.
+8 SUMMARY
+In this work we used the results of CR23 on the time it takes visible
+gas vortices to appear next to a gap carved by a low-mass planet in a
+low-viscosity PPD to set constraints on the masses 𝑀p and ages 𝜏p of
+planets in PPDs with observed vortex-like structures. We found that
+the presence of a vortex sets a lower limit on a particular combination
+of 𝑀p and 𝜏p, with separate constraints on these variables possible
+if some additional information (such as the system age 𝜏sys or the
+upper limit on the planetary mass 𝑀↓) is available. These considera-
+tions allowed us to constrain the masses of putative planets in several
+vortex-bearing PPDs to be above several tens of 𝑀⊕. The limits
+5 Outer vortices should form first in inviscid discs with radially decreasing
+surface density (CR23).
+MNRAS 000, 1–7 (2022)
+
+Vortex weighing and dating
+7
+on the planetary age are not very constraining at the moment, but
+they will improve as future observations lower 𝑀↓. Our constraints
+can be extended to account for the non-trivial history of planetary
+mass accretion, and we provide a recipe for doing that in Section
+5. Finally, we showed the robustness of our constraints in light of
+additional complications (e.g. non-zero disc viscosity, multiple gen-
+eration of vortices, etc.) and demonstrated their useful synergy with
+other types of constraints on 𝑀p and 𝜏p, e.g. based on the upper lim-
+its on the planetary cooling luminosity coming from direct imaging
+observations.
+ACKNOWLEDGEMENTS
+Software: Matplotlib (Hunter 2007). Authors are grateful to Ewine
+van Dishoeck for illuminating discussions and to an anonymous ref-
+eree for useful suggestions. R.R.R. acknowledges financial support
+through the Science and Technology Facilities Council (STFC) grant
+ST/T00049X/1 and Ambrose Monell Foundation. N.P.C. is funded
+by a STFC and Isaac Newton studentship.
+DATA AVAILABILITY
+The data underlying this article will be shared on reasonable request
+to the corresponding author.
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+

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+1 
+ 
+A note on the constant characteristic time of failure incubation processes under various high-
+rate loads 
+Ivan Smirnov 
+Saint Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg, 199034, Russia 
+Corresponding author: ivansmirnov.sci@gmail.com 
+ 
+Abstract. The research reveals the existence of a constant characteristic time of preparatory micro-
+structural processes before the onset of macro-failure at various high loading rates of brittle and 
+quasi-brittle materials. The presence of this characteristic is analysed based on available data in the 
+literature from dynamic tests for uniaxial compression and splitting. It is shown that the 
+characteristic time can be determined experimentally and used to calculate the strain rate 
+dependencies of either critical failure stresses or time to failure, at least in the case of linearly 
+growing loads. In addition, it is discussed that the presence of this constant parameter opens up a 
+prospective opportunity for research and development of new methods for assessing the structural-
+temporal and scale characteristics of the strength and failure of materials under dynamic loads. 
+ 
+Keywords: quasi-brittle material, high strain rate, dynamic strength, time to fracture, characteristic 
+time, failure criteria. 
+ 
+1. Introduction 
+The problems of deformation and failure of materials at high-rate intensive loads have been 
+relevant for a significant period of time. Today, these studies continue to both certify the behaviour 
+of materials at a given range of strain rates [1–3], and solve fundamental issues in order to develop 
+approaches for qualitative forecasting the behaviour of materials, regardless of loading history 
+[1,4,5]. However, there are still no universally accepted test methods or parameters that could allow 
+us to evaluate and model the dynamic strength of materials based on simple engineering principles. 
+
+2 
+ 
+At sufficiently slow loads, the tensile strength and yield strength can, in fact, be assumed to be 
+constant and considered as characteristic strength parameters of a material. This forms the basis of 
+systems for state and industry standards to determine the strength characteristics of brittle and 
+plastic materials. When the strain or loading rates exceed the standardized quasi-static range, these 
+strength characteristics can become unstable and often strongly depend on the load history [5,6]. 
+Therefore, under dynamic loads, they can no longer be considered as material parameters. To date, 
+there is no agreement on which characteristics can serve as the primary indicators of the properties 
+of materials under high-rate and shock loads. There is an urgent need to identify the characteristics, 
+which do not depend on the history and method (set-up) of load application. 
+This short communication presents an important, previously unknown effect of the 
+connection between failure incubation processes in the material structure and a constant time 
+characteristic, which are independent of strain/loading rate. It is shown that this characteristic time 
+can be determined experimentally and applied to criteria for evaluating the strength of a material 
+within a wide range of loading rates. 
+ 
+2. The characteristic time of failure incubation processes 
+Let us consider the case of uniaxial loading for which failure begins at the stage of load 
+growth without the pronounced and irreversible deformation of a specimen. This situation is typical 
+when testing brittle and quasi-brittle materials, using the scheme of uniaxial compression or 
+splitting tensile tests, for example. Let the beginning of a decrease in stresses in the stress-time or 
+stress-strain diagram be the factor indicating the beginning of failure. The strength of a material 
+then corresponds to the maximum value of stress. This is a common procedure for determining the 
+strength of such materials with quasi-static tests. However, with an increase in the load or strain rate 
+above the quasi-static range, the maximum failure stresses increase. Further in this paper, the 
+maximum failure stresses are assumed as critical stresses. 
+Fig. 1 demonstrates the case described above. Since the loading area up until the moment of 
+failure can be considered linear, the next expression is valid for the critical stress σcr: 
+
+3 
+ 
+𝜎𝑐𝑟 = 𝐸𝜀̇𝑡𝑐𝑟,    𝑡𝑐𝑟 > 0,                                                                 (1) 
+where E is the modulus of elasticity, 𝜀̇ is the strain rate and tcr is the time before the start of failure. 
+When the critical stress σcr is greater than the quasi-static strength σst, the time to failure and 
+expression (1) can be written as the sum of two terms: 
+𝑡𝑐𝑟 = 𝑡𝑠𝑡 + 𝑡𝑑 = 𝜎𝑠𝑡
+𝐸𝜀̇ + 𝑡𝑑,                                                            (2) 
+𝜎𝑐𝑟 = 𝜎𝑠𝑡 + 𝐸𝜀̇𝑡𝑑,
+𝑡𝑠𝑡 + 𝑡𝑑 > 0,                                          (3) 
+where td is the time from time tst when the quasi-static strength of a material is reached to the time 
+of the actual critical stress tcr. As a rule, experimental equipment allows for recording time diagrams 
+of load and strain. The strain or loading rate is specified by the input test conditions. For the case 
+under consideration, the stress rate is 𝜎̇ = 𝐸𝜀̇. Thus, all components of expression (3) can be 
+determined from the experiment. 
+ 
+Fig. 1. Schematic representation of loading at different rates. Designations are disclosed in the text. 
+Then we can estimate the time td that characterises the dynamics of micro-processes in the 
+structure of the material after reaching the quasi-static strength of the material σst until the onset of 
+‘macro’ failure tcr: 
+𝑡𝑑 = 𝜎𝑐𝑟 − 𝜎𝑠𝑡
+𝐸𝜀̇
+.                                                                     (4) 
+Let us consider such an estimate regarding various experimental data. Dynamic strength is 
+typically evaluated by the dependence of critical stress on the strain or stress rate. An example of 
+(3
+cr,t3
+cr)
+(1
+cr,t1
+cr)
+ 
+ 
+st
+st
+st
+3td
+4td
+2td
+Stress
+Time
+td
+st
+(2
+cr,t2
+cr)
+
+4 
+ 
+such dependence for the case of dynamic uniaxial compression of concrete is presented in Fig. 2. 
+Substituting the experimental points in expression (4), we derive the dependence of time td on the 
+strain rate. Fig. 3 presents such calculations for the uniaxial compression and splitting test of 
+various materials. Usually experimental points have a spread, so it is convenient to use the equation 
+of the slope of a regression line to determine the time td. The results show that, regardless of the 
+material and loading scheme, the time td for each experiment can be considered constant. 
+ 
+Fig. 2. Strain rate dependencies of critical stresses and time to failure for dynamic uniaxial 
+compression testing of concrete. Experimental data were obtained by [7]. The solid curves ware 
+calculated using expressions (2) and (3). The dashed line is a linear approximation. 
+Note that we use the same notation for the case of compression and splitting. This is not 
+only to avoid unnecessary notation, but also to show the commonality of the effect for various tests. 
+A caveat, however, is that when considering a specific test method, these load and material 
+parameters only apply to this method of testing. 
+Up to this point, it has been assumed that the time to failure tcr is greater than the time td. 
+However, the question arises: up to what value does the time to failure decrease, and would it be 
+less than time td? In order to answer this question, it is necessary to consider the stress impulse 
+before the start of failure. 
+0
+100
+200
+300
+400
+40
+60
+80
+100
+120
+ 
+ - cr
+- tcr
+- td
+Strain rate (1/s)
+cr (MPa)
+0
+20
+40
+60
+80
+100
+tcr, td (s)
+
+5 
+ 
+ 
+ 
+Fig. 3. The characteristic times of incubation micro-processes from the moment of reaching quasi-
+static strength to the moment of the onset of macro-failure of various materials under high-rate 
+loading. The calculations were carried out by Eq. (4) for the experimental data: a) compression tests 
+of concrete 1 [7], 2 [8], 3 [9], 4 [10], and granite 5 [11] and 6 [12]; b) splitting test of granite 7 [13], 
+ceramics 8 [14] and 9 [15] and glass 10 [16]. 
+A stress analysis shows that for critical stress σst < σcr < 2σst, the stress impulse applied to a 
+specimen is constant over a time from tcr – 2td to tcr (at the condition that td is constant). These 
+segments of the stress pulses Jcr are indicated by the shaded areas in Fig. 1. It is evident that the 
+stress impulse Jcr is equal to 
+𝐽𝑐𝑟 = 2𝜎𝑠𝑡𝑡𝑑.                                                                       (5) 
+101
+102
+103
+104
+0.1
+1
+10
+100
+td (s)
+ 
+ 
+Concrete 4
+Concrete 3
+Concrete 1
+Strain rate (1/s)
+Concrete 2
+Granite 5,6
+a)
+102
+103
+104
+105
+0.1
+1
+10
+100
+ 
+ 
+td (s)
+Stress rate (GPa/s)
+Glass 10
+Macor 9
+Al2O3 8
+Granite 7
+b)
+
+6 
+ 
+Suppose this impulse is the minimum value of the stress impulse that must be applied to the 
+specimen in order to initiate and prepare its macro failure. Then, if the time to failure is tcr ≤ 2td, the 
+failure stress impulse must also be at least 2σsttd. Since for tcr ≤ 2td  2σsttd =0.5σcrtcr, then the 
+expressions for the strain rate dependence of the critical stress and time to failure, taking (1) into 
+account, can be written in the following form: 
+𝜎𝑐𝑟 = √4𝐸𝜎𝑠𝑡𝑡𝑑𝜀̇,    𝑡𝑐𝑟 ≤ 2𝑡𝑑,                                                   (6) 
+𝑡𝑐𝑟 = √4𝜎𝑠𝑡𝑡𝑑
+𝐸𝜀̇
+,    𝑡𝑐𝑟 ≤ 2𝑡𝑑.                                                        (7) 
+In addition, the condition for time to failure in expression (3) should now be specified as 
+tcr ≥ 2td. 
+The characteristic time td can be easily obtained from expression (6). Fig. 4 and 5 present 
+these calculations for the uniaxial compression and splitting test of various materials. Thereby, the 
+time td was determined by formula (4) for the case σcr <2σst and from formula (6) for σcr ≥ 2σst. This 
+example also demonstrates that the time td can be considered constant for various strain and loading 
+rates. 
+ 
+Fig. 4. Strain rate dependencies of critical stresses and time to failure for dynamic uniaxial 
+compression tests of concrete in the cases of tcr ≥ 2td and tcr ≤ 2td. Experimental data was obtained 
+by [17]. The solid curves were calculated using conditions (2), (3), (6) and (7). The dashed line is a 
+linear approximation. 
+0.0
+2.0x104
+4.0x104
+6.0x104
+8.0x104
+0
+200
+400
+600
+800
+−cr 
+ − tcr
+− td
+Stress rate (GPa/s)
+cr (MPa)
+15
+30
+45
+60
+75
+tcr, td (s)
+
+7 
+ 
+ 
+ 
+ 
+Fig. 5. The characteristic times calculated for different conditions of σcr < 2σst and σcr ≥ 2σst. The 
+calculations were carried out according to Eq. (4) and (6) for the experimental data: a) compression 
+tests of concrete 11 [18] and 12 [19], glass 10 [16] and 13 [20], brick 14 [21], and mortar 15 [22]; b) 
+splitting test of rocks 6 [12], 16 [23], and 17 [24], concrete 11 [18] and 12 [19], and 18 [25], mortar 
+19 [25], and ceramics 20 [26]. 
+ 
+3. Discussion and future work 
+Thus, the experimental data provides reason to introduce the characteristic time of 
+incubation processes of failure td, which can be considered a constant value independent of a strain 
+rate above the quasi-static range, at least for cases of continuous linear increase in load during the 
+101
+102
+103
+104
+10-1
+100
+101
+102
+103
+Mortar 15
+Glass 13
+ 
+ 
+Glass 10
+td (s)
+Strain rate (1/s)
+a)
+Concrete 11, 12
+Brick 14
+100
+101
+102
+103
+104
+105
+106
+107
+10-1
+100
+101
+102
+103
+Concrete 11
+Mortar 19
+Concrete 12
+ 
+ 
+td (s)
+Stress rate (GPa/s)
+b)
+Ceramics 20
+Argillite 17
+Concrete 18
+Granite 6,16
+
+8 
+ 
+uniaxial compression or splitting tests. This time parameter characterises the individual response of 
+a material to dynamic loads. Therefore, this characteristic time can be used in criteria conditions of 
+structural-temporal approaches to failure analysis and strength prediction. For example, similar 
+expressions for (3) and (6) were obtained theoretically using the incubation time criterion [5,27,28]. 
+The criterion assumes that the following condition must be implemented for failure to occur: 
+∫ 𝜎(𝑡)𝑑𝑡 ≥ 𝜎𝑠𝑡𝜏
+𝑡
+𝑡−𝜏
+,                                                                   (8) 
+where σ(t) is the stress profile at the failure place, σst is the quasi-static strength, and τ is the 
+incubation time of failure. The parameters σst and τ are strength parameters of a material for a 
+particular test scheme, for example, compression or tension. The incubation time was introduced as 
+a hypothetical characteristic period of time responsible for the incubation period of macro-failure. 
+This was necessary to ensure the possibility of a smooth transition from pulsed loads to quasi-static 
+loads using the Nikiforovsky-Shemyakin integral criterion for spall fracture (full integral of the 
+stress over time at the spall section should reach a critical value) [5,29]. Therefore, according to (8), 
+failure will occur in the case that the stress impulse in the region of failure is not less than σstτ for 
+the time t ≤ τ (for t < 0, σ(t) = 0). A similar assumption is made in the discussion of (5). 
+Substituting (1) for σ(t) in (8), we can obtain simple relationships for calculating the strain 
+rate dependencies of critical stresses σcr and the time to failure tcr: 
+{  
+  𝜎𝑐𝑟 = 𝜎𝑠𝑡 + 0.5𝐸𝜀̇𝜏,             𝑡𝑐𝑟 = 𝜏
+2 + 𝜎𝑠𝑡
+𝐸𝜀̇ ,      𝑡𝑐𝑡 > 𝜏,
+𝜎𝑐𝑟 = √2𝐸𝜎𝑠𝑡𝜏𝜀̇,
+       𝑡𝑐𝑟 = √4𝜎𝑠𝑡𝑡𝑑
+𝐸𝜀̇
+,       𝑡𝑐𝑟 ≤ 𝜏.  
+                           (9) 
+Substituting τ = 2td, we derive the expressions in (9), which correspond exactly to 
+expressions (2), (3), (6) and (7). 
+The incubation time approach allows us to successfully solve a number of dynamic 
+problems related to fracture mechanics [5,27–30]. However, the question of experimental 
+determination of the incubation time of failure still remains open-ended. The obtained result shows 
+
+9 
+ 
+that the incubation time of failure, at least for the case of a controlled linear increase in fast loading, 
+can be determined experimentally by estimating td. 
+The demonstrated effect opens up new possibilities for solving important problems related 
+to the mechanics of dynamic deformation and fracture of materials. One of these tasks relates to the 
+possibility of determining the parameters of the dynamic strength of materials using basic tests that 
+are both publicly available and generally accepted. Furthermore, it is necessary to study appropriate 
+methods and basic rules to determine the characteristic time td. Experimental conditions affecting 
+time td should also be established. 
+Another problem relates to the determination of the strain rate dependence of the yield 
+strength. The presented reasoning can be applied to consider similar characteristic time of 
+incubation processes involved in the plastic deformation [30]. 
+The found characteristic time of failure incubation processes can provide a new approach to 
+the determination of scale levels of failure. Since we have a constant time characteristic for the 
+implementation of preparatory failure processes, we can assume that there is also a constant 
+characteristic structural scale at which these processes are realised. For example, according to the 
+structural-temporal approach [29], this characteristic scale can relate to the propagation distance of 
+the elastic wave during time τ (d = τC, in which C is the speed of sound); as well, the scale can be 
+introduced as 𝑑 = (2𝐾𝐼𝐶
+2)/(𝜋𝜎𝑠𝑡
+2 ) (KIC as the stress intensity factor). However, the accuracy of 
+these expressions for the elementary scale of failure is still being studied. Thus, the relationship of 
+time td, scale factors and the dynamics of the preparatory microstructural processes of macro-
+fracture should be studied. 
+ 
+4. Conclusions 
+The presented study shows the existence of a characteristic time for preparatory micro-
+processes of macro-failure of brittle and quasi-brittle materials. This time does not depend on the 
+strain or loading rate, at least in terms of the compression and splitting test methods under 
+consideration. The value of the characteristic time can be determined directly from the experiment. 
+
+10 
+ 
+The presented results show that this integral time characteristic of the dynamic failure process can 
+be used for the development of experimental and theoretical foundations to determine and predict 
+the strength characteristics of constructional materials across a wide range of loading rates. 
+ 
+Acknowledgements 
+This work was supported by the Russian Science Foundation, grant № 18-79-00193. 
+ 
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+

VNE_T4oBgHgl3EQfxhzT/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf,len=399
+page_content='1  A note on the constant characteristic time of failure incubation processes under various high- rate loads Ivan Smirnov Saint Petersburg State University, Universitetskaya nab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' 7/9, St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Petersburg, 199034, Russia Corresponding author: ivansmirnov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content='sci@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content='com  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' The research reveals the existence of a constant characteristic time of preparatory micro- structural processes before the onset of macro-failure at various high loading rates of brittle and quasi-brittle materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' The presence of this characteristic is analysed based on available data in the literature from dynamic tests for uniaxial compression and splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' It is shown that the characteristic time can be determined experimentally and used to calculate the strain rate dependencies of either critical failure stresses or time to failure, at least in the case of linearly growing loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' In addition, it is discussed that the presence of this constant parameter opens up a prospective opportunity for research and development of new methods for assessing the structural- temporal and scale characteristics of the strength and failure of materials under dynamic loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content='  Keywords: quasi-brittle material, high strain rate, dynamic strength, time to fracture, characteristic time, failure criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Introduction The problems of deformation and failure of materials at high-rate intensive loads have been relevant for a significant period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Today, these studies continue to both certify the behaviour of materials at a given range of strain rates [1–3], and solve fundamental issues in order to develop approaches for qualitative forecasting the behaviour of materials, regardless of loading history [1,4,5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' However, there are still no universally accepted test methods or parameters that could allow us to evaluate and model the dynamic strength of materials based on simple engineering principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content='  2  At sufficiently slow loads, the tensile strength and yield strength can, in fact, be assumed to be constant and considered as characteristic strength parameters of a material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' This forms the basis of systems for state and industry standards to determine the strength characteristics of brittle and plastic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' When the strain or loading rates exceed the standardized quasi-static range, these strength characteristics can become unstable and often strongly depend on the load history [5,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Therefore, under dynamic loads, they can no longer be considered as material parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' To date, there is no agreement on which characteristics can serve as the primary indicators of the properties of materials under high-rate and shock loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' There is an urgent need to identify the characteristics, which do not depend on the history and method (set-up) of load application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' This short communication presents an important, previously unknown effect of the connection between failure incubation processes in the material structure and a constant time characteristic, which are independent of strain/loading rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' It is shown that this characteristic time can be determined experimentally and applied to criteria for evaluating the strength of a material within a wide range of loading rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' The characteristic time of failure incubation processes Let us consider the case of uniaxial loading for which failure begins at the stage of load growth without the pronounced and irreversible deformation of a specimen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' This situation is typical when testing brittle and quasi-brittle materials, using the scheme of uniaxial compression or splitting tensile tests, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Let the beginning of a decrease in stresses in the stress-time or stress-strain diagram be the factor indicating the beginning of failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' The strength of a material then corresponds to the maximum value of stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' This is a common procedure for determining the strength of such materials with quasi-static tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' However, with an increase in the load or strain rate above the quasi-static range, the maximum failure stresses increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Further in this paper, the maximum failure stresses are assumed as critical stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' 1 demonstrates the case described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Since the loading area up until the moment of failure can be considered linear, the next expression is valid for the critical stress σcr:  3  𝜎𝑐𝑟 = 𝐸𝜀̇𝑡𝑐𝑟,    𝑡𝑐𝑟 > 0,                                                                 (1) where E is the modulus of elasticity, 𝜀̇ is the strain rate and tcr is the time before the start of failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' When the critical stress σcr is greater than the quasi-static strength σst, the time to failure and expression (1) can be written as the sum of two terms: 𝑡𝑐𝑟 = 𝑡𝑠𝑡 + 𝑡𝑑 = 𝜎𝑠𝑡 𝐸𝜀̇ + 𝑡𝑑,                                                            (2) 𝜎𝑐𝑟 = 𝜎𝑠𝑡 + 𝐸𝜀̇𝑡𝑑, 𝑡𝑠𝑡 + 𝑡𝑑 > 0,                                          (3) where td is the time from time tst when the quasi-static strength of a material is reached to the time of the actual critical stress tcr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' As a rule, experimental equipment allows for recording time diagrams of load and strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' The strain or loading rate is specified by the input test conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' For the case under consideration, the stress rate is 𝜎̇ = 𝐸𝜀̇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Thus, all components of expression (3) can be determined from the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Schematic representation of loading at different rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Designations are disclosed in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Then we can estimate the time td that characterises the dynamics of micro-processes in the structure of the material after reaching the quasi-static strength of the material σst until the onset of ‘macro’ failure tcr: 𝑡𝑑 = 𝜎𝑐𝑟 − 𝜎𝑠𝑡 𝐸𝜀̇ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' (4) Let us consider such an estimate regarding various experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Dynamic strength is typically evaluated by the dependence of critical stress on the strain or stress rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' An example of (\uf0733 cr,t3 cr) (\uf0731 cr,t1 cr)  \uf034\uf073st  \uf033\uf073st  \uf032\uf073st  3td  4td  2td  Stress  Time  td  \uf073st  (\uf0732  cr,t2  cr)  4  such dependence for the case of dynamic uniaxial compression of concrete is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Substituting the experimental points in expression (4), we derive the dependence of time td on the strain rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' 3 presents such calculations for the uniaxial compression and splitting test of various materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Usually experimental points have a spread, so it is convenient to use the equation of the slope of a regression line to determine the time td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' The results show that, regardless of the material and loading scheme, the time td for each experiment can be considered constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Strain rate dependencies of critical stresses and time to failure for dynamic uniaxial compression testing of concrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Experimental data were obtained by [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' The solid curves ware calculated using expressions (2) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' The dashed line is a linear approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Note that we use the same notation for the case of compression and splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' This is not only to avoid unnecessary notation, but also to show the commonality of the effect for various tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' A caveat, however, is that when considering a specific test method, these load and material parameters only apply to this method of testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Up to this point, it has been assumed that the time to failure tcr is greater than the time td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' However, the question arises: up to what value does the time to failure decrease, and would it be less than time td?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' In order to answer this question, it is necessary to consider the stress impulse before the start of failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' 0 100 200 300 400 40 60 80 100 120  \uf073cr  tcr  td Strain rate (1/s) \uf073cr (MPa) 0 20 40 60 80 100 tcr, td (\uf06ds)  5  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' The characteristic times of incubation micro-processes from the moment of reaching quasi- static strength to the moment of the onset of macro-failure of various materials under high-rate loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' The calculations were carried out by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' (4) for the experimental data: a) compression tests of concrete 1 [7], 2 [8], 3 [9], 4 [10], and granite 5 [11] and 6 [12];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' b) splitting test of granite 7 [13], ceramics 8 [14] and 9 [15] and glass 10 [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' A stress analysis shows that for critical stress σst < σcr < 2σst, the stress impulse applied to a specimen is constant over a time from tcr – 2td to tcr (at the condition that td is constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' These segments of the stress pulses Jcr are indicated by the shaded areas in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' It is evident that the stress impulse Jcr is equal to 𝐽𝑐𝑟 = 2𝜎𝑠𝑡𝑡𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' (5) 101 102 103 104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content='1 1 10 100 td (\uf06ds)  Concrete 4  Concrete 3  Concrete 1  Strain rate (1/s)  Concrete 2  Granite 5,6  a)  102  103  104  105  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content='1  1  10  100  td (\uf06ds)  Stress rate (GPa/s)  Glass 10  Macor 9  Al2O3 8  Granite 7  b)  6  Suppose this impulse is the minimum value of the stress impulse that must be applied to the specimen in order to initiate and prepare its macro failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Then, if the time to failure is tcr ≤ 2td, the failure stress impulse must also be at least 2σsttd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Since for tcr ≤ 2td  2σsttd =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content='5σcrtcr, then the expressions for the strain rate dependence of the critical stress and time to failure, taking (1) into account, can be written in the following form: 𝜎𝑐𝑟 = √4𝐸𝜎𝑠𝑡𝑡𝑑𝜀̇,    𝑡𝑐𝑟 ≤ 2𝑡𝑑,                                                   (6) 𝑡𝑐𝑟 = √4𝜎𝑠𝑡𝑡𝑑 𝐸𝜀̇ ,    𝑡𝑐𝑟 ≤ 2𝑡𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' (7) In addition, the condition for time to failure in expression (3) should now be specified as tcr ≥ 2td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' The characteristic time td can be easily obtained from expression (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' 4 and 5 present these calculations for the uniaxial compression and splitting test of various materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Thereby, the time td was determined by formula (4) for the case σcr <2σst and from formula (6) for σcr ≥ 2σst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' This example also demonstrates that the time td can be considered constant for various strain and loading rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Strain rate dependencies of critical stresses and time to failure for dynamic uniaxial compression tests of concrete in the cases of tcr ≥ 2td and tcr ≤ 2td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Experimental data was obtained by [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' The solid curves were calculated using conditions (2), (3), (6) and (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' The dashed line is a linear approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content='0x104 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content='0x104 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content='0x104 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content='0x104 0 200 400 600 800 −\uf073cr − tcr − td Stress rate (GPa/s) \uf073cr (MPa) 15 30 45 60 75 tcr, td (\uf06ds)  7  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' The characteristic times calculated for different conditions of σcr < 2σst and σcr ≥ 2σst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' The calculations were carried out according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' (4) and (6) for the experimental data: a) compression tests of concrete 11 [18] and 12 [19], glass 10 [16] and 13 [20], brick 14 [21], and mortar 15 [22];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' b) splitting test of rocks 6 [12], 16 [23], and 17 [24], concrete 11 [18] and 12 [19], and 18 [25], mortar 19 [25], and ceramics 20 [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Discussion and future work Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' the experimental data provides reason to introduce the characteristic time of incubation processes of failure td,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' which can be considered a constant value independent of a strain rate above the quasi-static range,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' at least for cases of continuous linear increase in load during the 101 102 103 104 10-1 100 101 102 103 Mortar 15 Glass 13  Glass 10  td (\uf06ds)  Strain rate (1/s)  a)  Concrete 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' 12  Brick 14  100  101  102  103  104  105  106  107  10 1  100  101  102  103  Concrete 11  Mortar 19  Concrete 12  td (\uf06ds)  Stress rate (GPa/s)  b)  Ceramics 20  Argillite 17  Concrete 18  Granite 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content='16  8  uniaxial compression or splitting tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' This time parameter characterises the individual response of a material to dynamic loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Therefore, this characteristic time can be used in criteria conditions of structural-temporal approaches to failure analysis and strength prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' For example, similar expressions for (3) and (6) were obtained theoretically using the incubation time criterion [5,27,28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' The criterion assumes that the following condition must be implemented for failure to occur: ∫ 𝜎(𝑡)𝑑𝑡 ≥ 𝜎𝑠𝑡𝜏 𝑡 𝑡−𝜏 ,                                                                   (8) where σ(t) is the stress profile at the failure place, σst is the quasi-static strength, and τ is the incubation time of failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' The parameters σst and τ are strength parameters of a material for a particular test scheme, for example, compression or tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' The incubation time was introduced as a hypothetical characteristic period of time responsible for the incubation period of macro-failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' This was necessary to ensure the possibility of a smooth transition from pulsed loads to quasi-static loads using the Nikiforovsky-Shemyakin integral criterion for spall fracture (full integral of the stress over time at the spall section should reach a critical value) [5,29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Therefore, according to (8), failure will occur in the case that the stress impulse in the region of failure is not less than σstτ for the time t ≤ τ (for t < 0, σ(t) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' A similar assumption is made in the discussion of (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content='  Substituting (1) for σ(t) in (8), we can obtain simple relationships for calculating the strain rate dependencies of critical stresses σcr and the time to failure tcr: { 𝜎𝑐𝑟 = 𝜎𝑠𝑡 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content='5𝐸𝜀̇𝜏,             𝑡𝑐𝑟 = 𝜏 2 + 𝜎𝑠𝑡 𝐸𝜀̇ ,      𝑡𝑐𝑡 > 𝜏, 𝜎𝑐𝑟 = √2𝐸𝜎𝑠𝑡𝜏𝜀̇, 𝑡𝑐𝑟 = √4𝜎𝑠𝑡𝑡𝑑 𝐸𝜀̇ ,       𝑡𝑐𝑟 ≤ 𝜏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' (9) Substituting τ = 2td, we derive the expressions in (9), which correspond exactly to expressions (2), (3), (6) and (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' The incubation time approach allows us to successfully solve a number of dynamic problems related to fracture mechanics [5,27–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' However, the question of experimental determination of the incubation time of failure still remains open-ended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' The obtained result shows  9  that the incubation time of failure, at least for the case of a controlled linear increase in fast loading, can be determined experimentally by estimating td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' The demonstrated effect opens up new possibilities for solving important problems related to the mechanics of dynamic deformation and fracture of materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' One of these tasks relates to the possibility of determining the parameters of the dynamic strength of materials using basic tests that are both publicly available and generally accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Furthermore, it is necessary to study appropriate methods and basic rules to determine the characteristic time td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Experimental conditions affecting time td should also be established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Another problem relates to the determination of the strain rate dependence of the yield strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' The presented reasoning can be applied to consider similar characteristic time of incubation processes involved in the plastic deformation [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' The found characteristic time of failure incubation processes can provide a new approach to the determination of scale levels of failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Since we have a constant time characteristic for the implementation of preparatory failure processes, we can assume that there is also a constant characteristic structural scale at which these processes are realised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content='  For example, according to the structural-temporal approach [29], this characteristic scale can relate to the propagation distance of the elastic wave during time τ (d = τC, in which C is the speed of sound);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' as well, the scale can be introduced as 𝑑 = (2𝐾𝐼𝐶 2)/(𝜋𝜎𝑠𝑡 2 ) (KIC as the stress intensity factor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' However, the accuracy of these expressions for the elementary scale of failure is still being studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Thus, the relationship of time td, scale factors and the dynamics of the preparatory microstructural processes of macro- fracture should be studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' Conclusions The presented study shows the existence of a characteristic time for preparatory micro- processes of macro-failure of brittle and quasi-brittle materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' This time does not depend on the strain or loading rate, at least in terms of the compression and splitting test methods under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content=' The value of the characteristic time can be determined directly from the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content='  10  The presented results show that this integral time characteristic of the dynamic failure process can be used for the development of experimental and theoretical foundations to determine and predict the strength characteristics of constructional materials across a wide range of loading rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
+page_content='  Acknowledgements This work was supported by the Russian Science Foundation, grant № 18-79-00193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'}
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WdE_T4oBgHgl3EQfyhxs/content/tmp_files/2301.08318v1.pdf.txt

@@ -0,0 +1,1758 @@
+arXiv:2301.08318v1  [gr-qc]  18 Jan 2023
+Gravitational-wave equation in
+effective one-body background for spinless binary
+Ya Guo1, Hiroaki Nakajima2 and Wenbin Lin1, 2, ∗
+1 School of Physical Science and Technology, Southwest Jiaotong University,
+Chengdu, 610031, China
+2 School of Mathematics and Physics, University of South China,
+Hengyang, 421001, China
+∗ Email: lwb@usc.edu.cn
+Abstract
+We construct the gravitational-wave equation in the background of the effec-
+tive one-body system for the spinless binary, which is in general available with the
+spherically symmetric background as well. The gauge conditions are given in terms
+of the metric perturbation.
+
+1
+Introduction
+The direct observation of gravitational waves by LIGO and Virgo [1] has opened the
+new era of cosmology. The binary system such as two black holes is a good object to
+observe the gravitational waves, where the analytical calculation is also possible. One of
+the calculation method of the gravitational waves radiated from the binary system is the
+theory of post-Newtonian approximations [2]. On the other hand, one can also use the
+black hole perturbation theory [3], which is useful for the case of the extreme mass-ratio
+inspiral (EMRI). The advantage of this method is that there is no divergence and one can
+calculate the observable quantity at very high post-Newtonian order [4].
+The reason why the black-hole perturbation theory is available for EMRI is because
+the field of one object is regarded as the background and the other body is regarded as
+the test particle. If one can map the two-body dynamics into the dynamics of the test
+particle in some appropriate background exactly, then the calculation beyond the EMRI
+approximation would be possible in terms of the black-hole perturbation theory. This
+approach is called the effective one-body (EOB) dynamics [5, 6]. In Newtonian limit, it
+is well-known that the effect of the two-body dynamics can exactly be included by just
+replacing the light (heavy) mass in the EMRI approximation with the reduced (total)
+mass. However when the correction of general relativity is included, the correspondence
+between the two-body dynamics and the dynamics of the test particle in some background
+becomes complicated. It turns out that the Hamiltonians of the two dynamics are related
+by a very nontrivial way [5, 6].
+The gravitational-wave equation in the EOB background has been studied in [7] using
+the Newman-Penrose formalism [8], which is a natural extension of the method to obtain
+the Teukolsky equation [9] in the Schwarzschild spacetime. It is obtained for some special
+cases of the background and is restricted to the even-parity mode. Later the same group
+obtained the equation both for the even- and odd-parity modes using the different choices
+of the gauge conditions [10]. The wave equation from the metric perturbation in the
+generally spherically symmetric background has also been studied in [11, 12].
+Inspired by the work of Jing et al. [7], in this paper, we show that the gravitational-
+wave equation for both the even- and odd-parity modes can be obtained using the gauge
+conditions taken in their work. Moreover our formalism can be applicable for more general
+spherically symmetric backgrounds.
+The reminder of this work is organized as follows: in section 2, we briefly review the
+EOB dynamics.
+In section 3, we consider the wave equation for the perturbed Weyl
+scalars. In section 4, we discuss the gauge condition. In section 5, we derive the explicit
+gravitational-wave equation. Section 6 is devoted to summary and discussion.
+2
+EOB dynamics
+The EOB system for the spinless binary is first introduced in [5] in the post-Newtonian
+formalism. The effective background metric geff
+µν is taken as the spherically symmetric
+1
+
+form:
+ds2
+eff = geff
+µνdxµdxν = A(r)dt2 − B(r)dr2 − C(r)r2(dθ2 + sin2 θdϕ2),
+(2.1)
+where the function C(r) can be freely chosen by the coordinate transformation for the
+radial coordinate r, and we here choose the Schwarzschild coordinate corresponding to
+C(r) ≡ 1. The explicit forms of A(r) and B(r) can be determined by the comparison
+with the two-body dynamics. For the Newman-Penrose formalism [8], we will take the
+corresponding null tetrad basis
+lA
+µ dxµ = dt − D(r)
+A(r)dr,
+nA
+µdxµ = A(r)
+2
+dt + D(r)
+2
+dr,
+mA
+µdxµ = − r
+√
+2(dθ + i sin θdϕ),
+¯mA
+µdxµ = − r
+√
+2(dθ − i sin θdϕ),
+(2.2)
+where D(r) =
+�
+A(r)B(r). The suffix A denotes the background quantities. The null
+tetrads satisfy the orthonormal condition as
+lA
+µ nµ
+A = 1,
+mA
+µ ¯mµ
+A = −1,
+(2.3)
+and the other inner products vanish. From the tetrad basis, one can compute the spin
+coefficients, the components of the Ricci tensor and the Weyl scalars as
+κA = νA = σA = λA = πA = τ A = ǫA = 0,
+(2.4)
+ρA = − 1
+rD,
+µA = − A
+2rD,
+γA = A′
+4D,
+αA = −βA = − cot θ
+2
+√
+2r,
+(2.5)
+ΦA
+01 = ΦA
+10 = ΦA
+02 = ΦA
+20 = ΦA
+12 = ΦA
+21 = 0,
+(2.6)
+ΦA
+00 = �� D′
+rD3,
+ΦA
+22 = −A2D′
+4rD3,
+(2.7)
+ΦA
+11 = −
+1
+8r2D3
+�
+2D3 − 2AD − r2(A′D′ − A′′D)
+�
+,
+(2.8)
+ΛA =
+1
+24r2D3
+�
+−2D3 − r2A′D′ + 2A(D − 2rD′) + rD(4A′ + rA′′)
+�
+,
+(2.9)
+ΨA
+0 = ΨA
+1 = ΨA
+3 = ΨA
+4 = 0,
+(2.10)
+ΨA
+2 =
+1
+12r2D3
+�
+2(AD + rAD′ − D3) − rA′(2D + rD′) + r2A′′D
+�
+,
+(2.11)
+where the prime denotes the ordinary derivative with respect to r. One can find that the
+background belongs the petrov type D from (2.10), but is not in the vacuum since there
+are nonvanishing components of the Ricci tensor. This type D property is important to
+derive the gravitational-wave equation in the next section. Note that for the special case
+D(r)=1, which was taken in [7, 10], we have
+ΦA
+00 = ΦA
+22 = 0,
+(2.12)
+2
+
+and the nonvanishing quantities in the above becomes simplified as
+ρA = −1
+r,
+µA = − A
+2r,
+γA = A′
+4 ,
+αA = −βA = − cot θ
+2
+√
+2r,
+(2.13)
+ΦA
+11 = − 1
+8r2(2 − 2A + r2A′′),
+(2.14)
+ΛA =
+1
+24r2(−2 + 2A + 4rA′ + r2A′′),
+(2.15)
+ΨA
+2 =
+1
+12r2(−2 + 2A − 2rA′ + r2A′′),
+(2.16)
+Note that when we choose A = 1 − 2M/r and D = 1, the background (2.1) is reduced to
+the Schwarzschild spacetime.
+The EOB Hamiltonian Heff can be determined from the geodesic motion under the
+background (2.1). The action S satisfies the Hamilton-Jacobi equation
+gµν
+eff PµPν − m2
+0 = 0,
+(2.17)
+where Pµ = ∂S/∂xµ is the momentum. m0 is the mass of the test particle and is matched
+as the reduced mass in the two-body dynamics. From (2.17), Heff is computed as
+Heff = m0
+�
+A
+�
+1 + AP 2r
+m2
+0D2 +
+P 2ϕ
+m2
+0r2
+�
+,
+(2.18)
+where the motion plane is fixed on θ = π/2 due to the spherical symmetry. The effective
+Hamiltonian Heff and the real two-body Hamiltonian Hreal are compared by matching the
+masses and the action variables. It turns out that the two Hamiltonians are related by a
+rather nontrivial way as [5, 6]
+Hreal = M0
+�
+1 + 2m0
+M0
+�Heff
+m0
+− 1
+�
+,
+(2.19)
+where M0 is the total mass in the two-body dynamics. The functions A(r) and D(r) are
+obtained as [5]
+A(r) = 1 − 2M0
+r
++ 2m0
+M0
+�M0
+r
+�3
++ · · · ,
+D(r) = 1 − 3m0
+M0
+�M0
+r
+�2
++ · · · .
+(2.20)
+However hereafter we leave A(r) and D(r) arbitrarily. Because of that, the metric (2.1)
+takes the most general form of the spherically symmetric background, which can also be
+applied to other kinds of the background. Moreover we will see later that when D(r)=1,
+the gravitational-wave equation and the gauge condition becomes simplified drastically.
+3
+
+3
+Wave equation for perturbed Weyl scalars
+It has been shown that the background (2.1) is classified as the nonvacuum Petrov type
+D background, which is useful to derive the gravitational-wave equation for the perturbed
+Weyl scalars using the Newman-Penrose formalism, as in the Teukolsky equation [9] in
+the Schwarzschild and the Kerr background.
+We begin with the following equations in Newman-Penrose formalism:
+(δ + 4β − τ)Ψ4 − (∆ + 4µ + 2γ)Ψ3 + 3νΨ2
+= (¯δ − ¯τ + 2¯β + 2α)Φ22 − (∆ + 2γ + 2¯µ)Φ21 − 2λΦ12 + 2νΦ11 + ¯νΦ20,
+(3.1)
+(D + 4ǫ − ρ)Ψ4 − (¯δ + 4π + 2α)Ψ3 + 3λΨ2
+= (¯δ − 2¯τ + 2α)Φ21 − (∆ + 2γ − 2¯γ + ¯µ)Φ20 + ¯σΦ22 − 2λΦ11 + 2νΦ10,
+(3.2)
+(∆ + µ + ¯µ + 3γ − ¯γ)λ − (¯δ + π − ¯τ + ¯β + 3α)ν + Ψ4 = 0.
+(3.3)
+We split all the quantities in the above into the background part (A) and the perturbation
+part (B), e. g. Ψ4 = ΨA
+4 + ΨB
+4 , etc. Now we have to take into account the case where ΦA
+22
+is nonvanishing, then the background part of the equation becomes nontrivial, i. e.
+(¯δ − ¯τ + 2¯β + 2α)AΦA
+22 = 0,
+(3.4)
+has to be satisfied1, and one can confirm that it is indeed the case.
+The part of the
+first-order perturbation in (3.1)–(3.3) becomes2
+(δ + 4β − τ)AΨB
+4 − (∆ + 4µ + 2γ)AΨB
+3 + 3νBΨA
+2
+= (¯δ − ¯τ + 2¯β + 2α)BΦA
+22 + (¯δ − ¯τ + 2¯β + 2α)AΦB
+22
+− (∆ + 2γ + 2¯µ)AΦB
+21 + 2νBΦA
+11,
+(3.5)
+(D + 4ǫ − ρ)AΨB
+4 − (¯δ + 4π + 2α)AΨB
+3 + 3λBΨA
+2
+= (¯δ − 2¯τ + 2α)AΦB
+21 − (∆ + 2γ − 2¯γ + ¯µ)AΦB
+20 + ¯σBΦA
+22 − 2λBΦA
+11,
+(3.6)
+(∆ + µ + ¯µ + 3γ − ¯γ)AλB − (¯δ + π − ¯τ + ¯β + 3α)AνB + ΨB
+4 = 0.
+(3.7)
+Again, when ΦA
+22 is nonvanishing, the first term in the right hand side in (3.5) appears,
+and more perturbed quantities ¯τ B, ¯βB and αB contribute to the equation, compared with
+the vacuum case. In order to reduce the number of those quantities, we require that this
+term should vanish by the gauge condition;
+Ξ22 ≡ (¯δ − ¯τ + 2¯β + 2α)BΦA
+22 = 0,
+(3.8)
+1Here the superscript A (B) on the parentheses denotes that all the quantities and the operators inside
+the parentheses are in the background (perturbation).
+2Here we will not use πA = τA = ǫA = 0 from the beginning and keep them for a while, which would
+be useful to extend the result into the spinning case.
+4
+
+Now we will obtain the wave equation for ΨB
+4 in a similar way with the method used
+to derive the Teukolsky equation. First we show the following commutation relation of
+the differential operators:
+[∆ + (p + 1)γ − ¯γ − qµ + ¯µ]A (¯δ + pα − qπ)A
+−
+�¯δ + (p + 1)α + ¯β − ¯τ − qπ
+�A (∆ + pγ − qµ)A
+= νADA − λAδA − p [(β + τ)λ − (ρ + ǫ)ν + Ψ3]A
++ q [−Dν + δλ + (¯π + τ + 3β − ¯α)λ − (3ǫ + ¯ǫ + ρ − ¯ρ)ν + 2Ψ3]A
+= 0,
+(3.9)
+where p and q are arbitrary constants and we have used νA = λA = ΨA
+3 = 0. We operate
+(∆ + 3γ − ¯γ + 4µ + ¯µ)A to (3.6) and (¯δ + 3α + ¯β − ¯τ + 4π)A to (3.5), and then subtract
+one equation from the other. The terms with ΨB
+3 cancel by (3.9) with p = 2, q = −4 and
+the remaining becomes
+�
+(∆ + 3γ − ¯γ + 4µ + ¯µ)(D + 4ǫ − ρ) − (¯δ + 3α + ¯β − ¯τ + 4π)(δ + 4β − τ)
+�A ΨB
+4
++ 3ΨA
+2
+�
+(∆ + 3γ − ¯γ + 4µ + ¯µ)AλB − (¯δ + 3α + ¯β − ¯τ + 4π)AνB�
++ 3λB∆AΨA
+2 − 3νB¯δAΨA
+2
+= T4 + (∆ + 3γ − ¯γ + 4µ + ¯µ)A(¯σBΦA
+22) − 2λB∆AΦA
+11 − 2νB¯δAΦA
+11
+− 2ΦA
+11
+�
+(∆ + 3γ − ¯γ + 4µ + ¯µ)AλB + (¯δ + 3α + ¯β − ¯τ + 4π)AνB�
+,
+(3.10)
+where T4 is defined by
+T4 = (∆ + 3γ − ¯γ + 4µ + ¯µ)A �
+(¯δ − 2¯τ + 2α)AΦB
+21 − (∆ + 2γ − 2¯γ + ¯µ)AΦB
+20
+�
+− (¯δ + 3α + ¯β − ¯τ + 4π)A �
+(¯δ − ¯τ + 2¯β + 2α)AΦB
+22 − (∆ + 2γ + 2¯µ)AΦB
+21
+�
+.
+(3.11)
+For the third line in (3.10), we have
+∆AΨA
+2 = −3µAΨA
+2 − 2µΦA
+11 − (D − ¯ρ + 2ǫ + 2¯ǫ)AΦA
+22 − 2∆AΛA,
+(3.12)
+¯δAΨA
+2 = −3πAΨA
+2 + 2πAΦA
+11 − 2¯δAΛA,
+(3.13)
+and then substituting the above into (3.10), we get
+�
+(∆+3γ−¯γ+4µ + ¯µ)(D+4ǫ−ρ)−(¯δ + 3α + ¯β − ¯τ + 4π)(δ + 4β − τ)
+�A ΨB
+4
++ 3ΨA
+2
+�
+(∆ + 3γ − ¯γ + µ + ¯µ)AλB − (¯δ + 3α + ¯β − ¯τ + π)AνB�
+= T4 + (∆ + 3γ − ¯γ + 4µ + ¯µ)A(¯σBΦA
+22) − 2λB∆AΦA
+11 − 2νB¯δAΦA
+11
+− 2ΦA
+11
+�
+(∆ + 3γ − ¯γ + µ + ¯µ)AλB + (¯δ + 3α + ¯β − ¯τ + π)AνB�
++ 3λB(D − ¯ρ + 2ǫ + 2¯ǫ)AΦA
+22 + 6λB∆AΛA − 6νB¯δAΛA.
+(3.14)
+5
+
+The second line in (3.14) becomes −3ΨA
+2 ΨB
+4 using (3.7), and there is also similar terms in
+the fourth line but the relative sign is positive. We now require more gauge conditions as
+λB = σB = 0.
+(3.15)
+Then the fourth line in (3.14) becomes −2ΦA
+11ΨB
+4 and the terms with ΦA
+22 disappear.
+Thus under the gauge conditions (3.8) and (3.15), the decoupled wave equation for ΨB
+4 is
+obtained as
+�
+(∆ + 3γ − ¯γ + 4µ + ¯µ)(D + 4ǫ − ρ)
+− (¯δ + 3α + ¯β − ¯τ + 4π)(δ + 4β − τ) − 3Ψ2 + 2Φ11
+�AΨB
+4 = T4,
+(3.16)
+One can also consider the wave equation for ΨB
+0 , which can be obtained in a similar way.
+The resultant equation becomes
+�
+(D − 3ǫ + ¯ǫ − 4ρ − ¯ρ)(∆ − 4γ + µ)
+− (δ − 3β − ¯α + ¯π − 4τ)(¯δ − 4α + π) − 3Ψ2 + 2Φ11
+�AΨB
+0 = T0.
+(3.17)
+The gauge conditions are (3.15) and
+Ξ00 ≡ (δ + ¯π − 2¯α − 2β)BΦA
+00 = 0.
+(3.18)
+Note that for the case D=1 we have (2.12), then (3.8) and (3.18) are obviously satisfied.
+The other conditions (3.15) are also relaxed as just λB = 0 for (3.16) and just σB = 0 for
+(3.17).
+4
+Gauge conditions
+Here we will study the consistency of the gauge conditions (3.8), (3.18) and (3.15) (or
+just (3.15) for D = 1). In Newman-Penrose formalism, there are ten gauge degrees of
+freedom. Six of them are the tetrad rotation (the local Lorentz transformation), which
+can be decomposed as the following three kinds [13]:
+lµ → lµ,
+mµ → mµ + alµ,
+¯mµ → ¯mµ + ¯alµ,
+nµ → nµ + ¯amµ + a ¯mµ + a¯alµ,
+(4.1)
+nµ → nµ,
+mµ → mµ + bnµ,
+¯mµ → ¯mµ + ¯bnµ,
+lµ → lµ + ¯bmµ + b ¯mµ + b¯bnµ,
+(4.2)
+lµ → e−clµ,
+nµ → ecnµ,
+mµ → eiϑmµ,
+¯mµ → e−iϑ ¯mµ.
+(4.3)
+Here a and b are complex functions, and c and ϑ are real functions. The transformations of
+the quantities in Newman-Penrose equations (the spin coefficients, the Weyl scalars, etc.)
+under the above are shown in [14], for example. The other four are the general coordinate
+transformation x′µ = xµ+ξµ(x). Under the transformation, the null tetrads behave as the
+6
+
+one-forms or the vector fields, and the quantities in Newman-Penrose equations behave
+as the scalar fields. Because of this, the form of the transformation for the latter becomes
+X → X − ξµ∂µX,
+(4.4)
+where X represents either one of the spin coefficients, Ψ’s, Φ’s or Λ. Since we want to
+keep the background (2.4)–(2.11) (or (2.13)-(2.16) for D=1) intact, the functions a, b, c,
+ϑ and ξµ have to be at the order of the perturbed quantities, and in particular the second
+and the higher orders of them are neglected.
+The gauge conditions (3.8), (3.15) and (3.18) are transformed under the the gauge
+transformations (4.1), (4.2), (4.3) and (4.4) as3
+Ξ00 → Ξ00 + (δ + ¯π − 2¯α − 2β)A(ξµ∂µΦA
+00) + DA(aΦA
+00) − 2aρAΦA
+00
++ b(∆A + ¯µA − 2µA − 2¯γA − 2γA)ΦA
+00 + 2ΦA
+00δc,
+(4.5)
+Ξ22 → Ξ22 + (¯δ − ¯τ + 2α + 2¯β)A(ξµ∂µΦA
+22) + ¯a(DA − ¯ρA + 2ρA)ΦA
+22
++ ∆(¯bΦA
+22) + 2¯b(µA + γA + ¯γA)ΦA
+22 − 2ΦA
+22¯δc,
+(4.6)
+λB → λB + 2αA¯a + ¯δA¯a,
+(4.7)
+σB → σB + 2βAb − δAb.
+(4.8)
+One can find that there are nine real degrees (four ξ’s, a, b and c) of freedom for eight real
+conditions (Ξ00, Ξ22, λB and σB). In general, it is enough to satisfy all the conditions.
+In order to see the the condition explicitly, here we will use the form of the metric
+perturbation, following the A-K parametrization [15, 12]. First, the metric perturbation
+is given as
+gµν = gA
+µν + hBE
+µν + hBO
+µν ,
+(4.9)
+where gA
+µν = geff
+µν is the background metric (2.1). hBE
+µν and hBO
+µν are respectively the even-
+and the odd-parity part of the perturbation, which are parametrized as
+hBE
+µν =
+
+
+
+
+
+AS
+−DS
+−rB∂θS
+−rB∂ϕS
+KS
+rH∂θS
+rH∂ϕS
+r2E+
+F
+r2FPS1
+r2 sin2 θ E−
+F
+
+
+
+
+ ,
+(4.10)
+hBO
+µν = 1
+D
+
+
+
+
+
+0
+0
+rC csc θ∂ϕS
+−rC sin θ∂θS
+0
+−rJ csc θ∂ϕS
+rJ sin θ∂θS
+−r2G csc θPS1
+−1
+2r2GPS2
+r2GPS3
+
+
+
+
+ .
+(4.11)
+3Our gauge conditions are invariant under the transformation generated by ϑ.
+7
+
+Here the lower-left blanks have to be filled as hBE
+µν and hBO
+µν to be symmetric. A, B, C,
+D, E, F, G, H, J and K are the functions4 of t and r, and S is the function of θ and ϕ,
+which should in turn be identified as the spherical harmonics Ylm(θ, ϕ). The quantities
+E±
+F , PS1, PS2 and PS3 are defined by
+E±
+F =
+�
+E ± F
+�
+∂2
+θ + 1
+2l(l + 1)
+��
+S,
+(4.12)
+PS1 = (∂θ∂ϕ − cot θ∂ϕ)S,
+(4.13)
+PS2 = (csc θ∂2
+ϕ + cos θ∂θ − sin θ∂2
+θ)S,
+(4.14)
+PS3 = (sin θ∂θ∂ϕ − cos θ∂ϕ)S,
+(4.15)
+where l is one of the label in Ylm(θ, ϕ). Other perturbed quantities are also decomposed
+into the even- and the odd-parity modes, such as lB
+µ = lBE
+µ
++ lBO
+µ , etc.
+Now we have to find the null tetrads corresponding to the metric. Even though the
+metric (2.1) is fixed, the tetrads are not unique due to the tetrad rotation (4.1), (4.2) and
+(4.3). In other words, once a set of null tetrads is found, the others are obtained from
+those tetrads by the tetrad rotation. We will fix the reference tetrads as
+lE
+µ dxµ ≡ (lA
+µ + lBE
+µ )dxµ
+= dt − D
+A
+�
+1 − 1
+2A
+�
+A − 2A
+D D + A2
+D2K
+�
+S
+�
+dr −
+r
+AD(DB − AH)dS,
+(4.16)
+nE
+µ dxµ ≡ (nA
+µ + nBE
+µ )dxµ
+= A
+2
+�
+1+ AS
+A
+�
+dt+ D
+2
+�
+1+ 1
+2A
+�
+A−2A
+D D− A2
+D2K
+�
+S
+�
+dr− r
+2D(DB+AH)dS,
+(4.17)
+mE
+µ dxµ ≡ (mA
+µ + mBE
+µ )dxµ
+= − r
+√
+2
+��
+1 − E+
+F
+2
+�
+dθ + i
+�
+1 − E−
+F
+2 + i csc θFPS1
+�
+sin θdϕ
+�
+,
+(4.18)
+¯mE
+µ dxµ ≡ ( ¯mA
+µ + ¯mBE
+µ )dxµ
+= − r
+√
+2
+��
+1 − E+
+F
+2
+�
+dθ − i
+�
+1 − E−
+F
+2 − i csc θFPS1
+�
+sin θdϕ
+�
+,
+(4.19)
+for the even-parity modes and
+lO
+µ dxµ ≡ (lA
+µ + lBO
+µ )dxµ
+= dt − D
+Adr +
+r
+AD
+�
+C − A
+DJ
+�
+(csc θ∂ϕS dθ − sin θ∂θS dϕ),
+(4.20)
+nO
+µ dxµ ≡ (nA
+µ + nBO
+µ )dxµ
+4We hope the readers may not confuse the function D here and the differential operator D in Newman-
+Penrose formalism.
+8
+
+= A
+2 dt + D
+2 dr + r
+2D
+�
+C + A
+DJ
+�
+(csc θ∂ϕS dθ − sin θ∂θS dϕ),
+(4.21)
+mO
+µ dxµ ≡ (mA
+µ + mBO
+µ )dxµ
+= − r
+√
+2
+��
+1+ csc θ
+2D GPS1
+�
+dθ+i sin θ
+�
+1 − 1
+2D csc θG(PS1 + iPS2)
+�
+dϕ
+�
+, (4.22)
+¯mO
+µ dxµ ≡ ( ¯mA
+µ + ¯mBO
+µ )dxµ
+= − r
+√
+2
+��
+1+ csc θ
+2D GPS1
+�
+dθ−i sin θ
+�
+1 − 1
+2D csc θG(PS1 − iPS2)
+�
+dϕ
+�
+, (4.23)
+for the odd-parity modes. Here dS = (∂θS)dθ + (∂ϕS)dϕ. Note that the above choice
+of the reference tetrads is different from the one taken in [7] for both the even- and the
+odd-parity modes even under the Regge-Wheeler gauge B = F = H = G = 0 [16, 17, 18]
+or the EZ gauge B = E = F = G = 0 [15, 19]. This is again due to the different choice of
+the reference and they are equivalent. The advantage of our choice is that the expressions
+of λB and σB become relatively simple as
+λBE = −A
+4
+��
+∂2
+θ + 1
+2l(l + 1)
+�
+S − i csc θPS1
+� � 1
+A∂tF − 1
+D∂rF
+�
+,
+(4.24)
+σBE = 1
+2
+��
+∂2
+θ + 1
+2l(l + 1)
+�
+S + i csc θPS1
+� � 1
+A∂tF + 1
+D∂rF
+�
+,
+(4.25)
+for the even-parity modes and
+λBO = A
+8 csc θ (2PS1 − iPS2)
+� 1
+A∂t
+�G
+D
+�
+− 1
+D∂r
+�G
+D
+��
+,
+(4.26)
+σBO = −1
+4 csc θ (2PS1 + iPS2)
+� 1
+A∂t
+�G
+D
+�
++ 1
+D∂r
+�G
+D
+��
+,
+(4.27)
+for the odd-parity modes. In particular, one can easily find that they vanish under the
+Regge-Wheeler or the EZ gauge5, which also means that the above reference tetrads under
+those gauges can be used as the standard form of the perturbation.
+Next we will consider the gauge invariants, are obtained in [12, 15] as
+α = J − r
+2D∂r
+�G
+D
+�
+,
+(4.28)
+β = −C − r
+2∂tG,
+(4.29)
+χ = H − D2
+2AE − l(l + 1)D2
+4A
+F − r
+2∂rF,
+(4.30)
+ψ = 1
+2K − r
+4
+�D2
+A
+�′
+E − D2
+2AE − rD2
+2A ∂rE
+5For the even-parity modes, the choice taken in [7] can also lead to λBE = σBE = 0 under F = 0, but
+not for the odd-parity mode under G = 0.
+9
+
+− r
+8l(l + 1)
+�D2
+A
+�′
+F − D2
+4Al(l + 1)F − rD2
+4A l(l + 1)∂rF
+(4.31)
+δ = D + rD2
+2A ∂tE +
+�rA′
+A − 1
+�
+B − r∂rB
+− r2
+2 ∂t∂rF −
+�
+r − r2A′
+2A − rD2
+4A l(l + 1)
+�
+∂tF,
+(4.32)
+ǫ = −1
+2A − rA′
+4 E − r∂tB − rA′
+8 l(l + 1)F − r2
+2 ∂2
+t F.
+(4.33)
+The perturbed Weyl scalars ΨB
+4 and ΨB
+0 can be written as the gauge-invariant combina-
+tions of the perturbation as
+ΨBE
+4
+= csc θ
+8r2D3(PS2 + 2iPS1)
+�
+−A2Dψ − AD2δ + D3ǫ
+−rAD2∂tχ + rA2D∂rχ + A2(D − rD′)χ
+�
+,
+(4.34)
+ΨBO
+4
+= −i csc θ
+8rD (PS2 + 2iPS1)
+�
+−A
+D∂tα + A2
+D2∂rα + A2(D − 2rD′)
+rD3
+α
++ ∂tβ − A
+D∂rβ +
+�AD′
+D2 − A − rA′
+rD
+�
+β
+�
+,
+(4.35)
+ΨBE
+0
+=
+csc θ
+2r2A2D3(PS2 − 2iPS1)
+�
+−A2Dψ + AD2δ + D3ǫ
++rAD2∂tχ + rA2D∂rχ + A2(D − rD′)χ
+�
+,
+(4.36)
+ΨBO
+0
+= i csc θ
+2rA2D(PS2 − 2iPS1)
+�
+A
+D∂tα + A2
+D2∂rα + A2(D − 2rD′)
+rD3
+α
++ ∂tβ + A
+D∂rβ +
+�
+−AD′
+D2 + A − rA′
+rD
+�
+β
+�
+,
+(4.37)
+The gauge conditions (3.8), (3.18) and (3.15) can be expressed as the gauge invariants
+and the set of the variables (B, F, H, G) or (B, E, F, G). The explicit form for the latter
+is shown in Appendix.
+5
+Explicit PDE and ODE for radial coordinate
+In section 3, we have obtained the wave equation for the perturbed Weyl scalars ΨB
+4
+and ΨB
+0 as (3.16) and (3.17), respectively. By substituting the background (2.4)–(2.11)
+(or (2.13)-(2.16) for D = 1), the partial differential equations (the master equations) are
+obtained, Then by the separation of variables, the ordinary differential equation for the
+radial coordinate r and that for the angular coordinates θ and ϕ are also obtained.
+10
+
+We define ψ(−2) as
+ψ(−2) ≡ (ρA)−4ΨB
+4 = (rD)4ΨB
+4 .
+(5.1)
+From (3.16), ψ(−2) satisfies the following partial differential equation:
+r2
+A
+∂2ψ(−2)
+∂t2
+− (r2A)2D7 ∂
+∂r
+�
+(r2A)−1D−9∂ψ(−2)
+∂r
+�
++
+�2r2A′
+AD − 4r
+D
+� ∂ψ(−2)
+∂t
++
+�
+−24r2A(D′)2
+D4
++ 4r2AD′′
+D3
+− 3r2A′D′
+D3
+− 12rAD′
+D3
+− r2A′′
+D2
+− 2rA′
+D2
+�
+ψ(−2)
+−
+1
+sin θ
+∂
+∂θ
+�
+sin θ∂ψ(−2)
+∂θ
+�
+−
+1
+sin2 θ
+∂2ψ(−2)
+∂ϕ2
++ 4i cot θ
+sin θ
+∂ψ(−2)
+∂ϕ
++(4 cot2 θ+2)ψ(−2)
+= 2r6D4T4.
+(5.2)
+For homogeneous case (T4 = 0), by assuming the product form ψ(−2) = e−iωteimϕR(r)S(θ),
+one can obtain the separated equations as
+(r2A)2D7 d
+dr
+�
+(r2A)−1D−9dR(r)
+dr
+�
++
+�r2ω2
+A
++ iω
+�2r2A′
+AD − 4r
+D
+�
++ 24r2A(D′)2
+D4
+−4r2AD′′
+D3
++ 3r2A′D′
+D3
++ 12rAD′
+D3
++ r2A′′
+D2
++ 2rA′
+D2 − λ
+�
+R(r) = 0,
+(5.3)
+1
+sin θ
+d
+dθ
+�
+sin θdS(θ)
+dθ
+�
+−
+� m2
+sin2 θ − 4m cot θ
+sin θ
++ 4 cot2 θ + 2 − λ
+�
+S(θ) = 0,
+(5.4)
+where ω gives the frequency of the gravitational wave and λ is the separation constant.
+From (5.4) one can find that S(θ)eimϕ coincides with the s = −2 spin-weighted spherical
+harmonics, denoted as −2Ylm(θ, ϕ), and then the separation constant λ has to be
+λ = (l − 1)(l + 2),
+(5.5)
+where l and m take the integer value with l ≥ 2 and −l ≤ m ≤ l, respectively. For
+inhomogeneous case (T4 ̸= 0), one can expand ψ(−2) and T4 as
+ψ(−2) =
+�
+dω
+�
+l,m
+R(−2)lω(r)−2Ylm(θ, ϕ)e−iωt,
+(5.6)
+2r6D4T4 = −
+�
+dω
+�
+l,m
+G(−2)lω(r)−2Ylm(θ, ϕ)e−iωt.
+(5.7)
+Then the ordinary differential equation for the radial coordinate r is
+(r2A)2D7 d
+dr
+�
+(r2A)−1D−9dR(−2)lω(r)
+dr
+�
++
+�r2ω2
+A
++ iω
+�2r2A′
+AD − 4r
+D
+�
++24r2A(D′)2
+D4
+− 4r2AD′′
+D3
++ 3r2A′D′
+D3
++ 12rAD′
+D3
++ r2A′′
+D2
++ 2rA′
+D2
+− (l − 1)(l + 2)
+�
+R(−2)lω(r) = G(−2)lω(r).
+(5.8)
+11
+
+In the case of D=1, (5.8) is reduced to
+(r2A)2 d
+dr
+�
+(r2A)−1dR(−2)lω(r)
+dr
+�
++
+�r2ω2
+A
++ iω
+�2r2A′
+A
+− 4r
+�
++ r2A′′ + 2rA′ − (l − 1)(l + 2)
+�
+R(−2)lω(r)
+= G(−2)lω(r),
+(5.9)
+Note that for the case of Schwarzschild background, i. e. A = 1 − 2M/r, the differential
+equation (5.9) is reduced to the Teukolsky equation [9] with the spin weight s = −2.
+The equation for ψ(2) ≡ ΨB
+0 can be obtained in a similar way. The master equation
+becomes
+r2
+A
+∂2ψ(2)
+∂t2
+− (r2A)−2D−1 ∂
+∂r
+�
+(r2A)3D−1∂ψ(2)
+∂r
+�
+−
+�2r2A′
+AD − 4r
+D
+� ∂ψ(2)
+∂t
++
+�3r2A′D′
+D3
+− 3r2A′′
+D2
+− 10rA′
+D2
+− 4A
+D2
+�
+ψ(2)
+−
+1
+sin θ
+∂
+∂θ
+�
+sin θ∂ψ(2)
+∂θ
+�
+−
+1
+sin2 θ
+∂2ψ(2)
+∂ϕ2 −4i cot θ
+sin θ
+∂ψ(2)
+∂ϕ +(4 cot2 θ + 2)ψ(2)
+= 2r2T0.
+(5.10)
+After the separation of the variables, for the angular part, we have the s = 2 spin-weighted
+spherical harmonics 2Ylm(θ, ϕ). For the radial part, we have the following equation:
+(r2A)−2D−1 d
+dr
+�
+(r2A)3D−1dR(2)lω(r)
+dr
+�
++
+�r2ω2
+A
+− iω
+�2r2A′
+AD − 4r
+D
+�
+− 3r2A′D′
+D3
++ 3r2A′′
+D2
++ 10rA′
+D2
++ 4A
+D2
+− (l − 2)(l + 3)
+�
+R(2)lω(r) = G(2)lω(r),
+(5.11)
+which is reduced to the Teukolsky equation with s = ±2 for A = 1 − 2M/r and D ≡ 1.
+6
+Summary and discussion
+In this paper, we have studied the wave equations for the perturbed Weyl scalars ΨB
+4 and
+ΨB
+0 in the background of the EOB dynamics for the spinless binary. We also obtained
+the Teukolsky-like equations for the function of the radial coordinate r. In the previous
+work [7], the special case D=1 is studied and the (decoupled) wave equation is obtained
+only for the even-parity mode. On the other hand, here we have studied the odd-parity
+mode also, and have obtained the same form of the wave equation. Moreover, we have
+12
+
+considered a more general case including D ̸=1. The different form of the wave equation
+is proposed in [10] by the use of the different gauge, although it is again restricted to the
+case of D=1. It would be interesting to find the relation between our equations and their
+ones.
+One can easily find that the background is assumed to be just spherically symmetric
+and general enough. Even in the case D = 1, it contains many kinds of the black holes.
+A similar wave equations in the spherically symmetric background are obtained using
+the metric perturbation in [12], which are resemble to the Regge-Wheeler and the Zerilli
+equations [16, 17], and are different from the ones obtained here. This is because we
+have used the different gauge and the different master variables. However in the case of
+Schwarzschild background, there is a transformation originated from the connection of
+the different gauges, which is called the Chandrasekhar transformation [20]. Moreover
+the relation between the R(−2)lω(r) in (5.8) and R(2)lω(r) in (5.11) can also be regarded
+as the special case of the Chandrasekhar transformation [9, 21]. It would be interesting
+to find a similar transformation in the present case.
+Another possible generalization would be to include the effect of the spin, where the
+background becomes axially symmetric and contains the Kerr black hole as an example.
+In this case the method of the metric perturbation is difficult to perform, because it
+relies on the spherical symmetry and the expansion by the spherical harmonics. However
+the Newman-Penrose formalism would still be powerful enough, and it can be expected
+that the gravitational-wave equation could be obtained. Studying the equations for the
+electromagnetic and the scalar waves is also interesting.
+Acknowledgements
+This work was supported in part by the National Natural Science Foundation of China
+(Grant No. 11973025).
+A
+Gauge conditions with gauge invariants
+The gauge conditions (3.8), (3.18) and (3.15) can be written in terms of the gauge invari-
+ants and the variables (B, E, F, G). Each condition consists of the even-parity part (E),
+odd-parity part (O) and the transformation part (T).
+Ξ22 = ΞE
+22 + ΞO
+22 + ΞT
+22,
+(A.1)
+ΞE
+22 = A2(∂θS − i csc θ∂ϕS)
+4
+√
+2r
+�
+−AD′′
+D5
+�
+χ + r
+2∂rF + l(l + 1)
+4
+D2
+A F + D2
+2AE
+�
+− D′
+D3
+�
+− 1
+D∂tχ+ A
+2D2∂rχ+
+�
+− A
+2rD2 + 3A′
+2D2 −7AD′
+2D3
+�
+χ+ 3
+2rAǫ−
+A
+2rD2ψ
++ 1
+rDδ + 2
+A∂tB − D
+A∂tE +
+�A′
+A − 3D′
+2D − 1
+2r
+�
+E + 3r
+4A∂2
+t F + rA
+4D2∂2
+rF
+13
+
++
+� 1
+D − l(l + 1)
+2
+D
+A − rA′
+2AD
+�
+∂tF +
+�3rA′
+4D2 − 7rAD′
+4D3
+�
+∂rF
+−
+�1
+r + 3D′
+D − 2A′
+A
+� l(l + 1)
+4
+F
+��
+,
+(A.2)
+ΞO
+22 = −iA2(∂θS − i csc θ∂ϕS)
+4
+√
+2r
+�
+−AD′′
+D6
+�
+α + rD
+2 ∂r ˜G
+�
++ D′
+D3
+� 1
+D2∂tα − A
+2D3∂rα
++
+� A
+2rD3 − 3A′
+2D3 + 4AD′
+D4
+�
+α+
+1
+2AD∂tβ− 1
+D2∂rβ+
+� A′
+AD2 − 1
+rD2 + D′
+D3
+�
+β
++ r
+4A∂2
+t ˜G +
+� rA′
+2AD − 1
+D
+�
+∂t ˜G − rA
+4D2∂2
+r ˜G +
+�
+−3rA′
+4D2 + 7rAD′
+4D3
+�
+∂r ˜G
+��
+, (A.3)
+ΞT
+22 =
+A
+4rD4
+�AD′
+r
+− A′D′ + 3A(D′)2
+D
+− AD′′
+� �
+¯a − A
+2
+¯b
+�
+−A2D′
+4rD3
+�
+A′
+AD¯a− 1
+rD ¯a− A
+rD
+¯b+ 1
+2∂t¯b − A
+2D∂r¯b −
+√
+2
+r ∂θc +
+√
+2i csc θ
+r
+∂ϕc
+���
+, (A.4)
+Ξ00 = ΞE
+00 + ΞO
+00 + ΞT
+00,
+(A.5)
+ΞE
+00 = ∂θS + i csc θ∂ϕS
+√
+2r
+�
+−AD′′
+D5
+�
+χ + r
+2∂rF + l(l + 1)
+4
+D2
+A F + D2
+2AE
+�
+− D′
+D3
+� 1
+D∂tχ + A
+2D2∂rχ+
+� 3A′
+2D2 −
+A
+2rD2 − 7AD′
+2D3
+�
+χ− 5
+2rAǫ−
+A
+2rD2ψ
+− 1
+rDδ − 2
+A∂tB + D
+A∂tE −
+�3D′
+2D + 1
+2r
+�
+E − 5r
+4A∂2
+t F + rA
+4D2∂2
+rF
++
+�
+− 1
+D + l(l + 1)
+2
+D
+A + rA′
+2AD
+�
+∂tF +
+�3rA′
+4D2 − 7rAD′
+4D3
+�
+∂rF
+−
+�1
+r + 3D′
+D
+� l(l + 1)
+4
+F
+��
+,
+(A.6)
+ΞO
+00 = i(∂θS + i csc θ∂ϕS)
+√
+2r
+�
+−AD′′
+D6
+�
+α + rD
+2 ∂r ˜G
+�
+− D′
+D3
+� 1
+D2∂tα + A
+2D3∂rα
++
+� 3A′
+2D3 −
+A
+2rD3 −4AD′
+D4
+�
+α−
+1
+2AD∂tβ− 1
+D2∂rβ+
+� A′
+AD2 − 1
+rD2 + D′
+D3
+�
+β
+− r
+4A∂2
+t ˜G +
+� rA′
+2AD − 1
+D
+�
+∂t ˜G + rA
+4D2∂2
+r ˜G +
+�3rA′
+4D2 − 7rAD′
+4D3
+�
+∂r ˜G
+��
+, (A.7)
+ΞT
+00 =
+1
+rD4
+�D′
+r + 3(D′)2
+D
+− D′′
+� �
+a − A
+2 b
+�
+− D′
+rD3
+�
+2
+rDa+ 1
+A∂ta + 1
+D∂ra +
+A
+2rDb − A′
+D b +
+√
+2
+r ∂θc +
+√
+2i csc θ
+r
+∂ϕc
+�
+, (A.8)
+λB = λBE + λBO + λBT ,
+(A.9)
+14
+
+λBE = −A
+4
+��
+∂2
+θ + 1
+2l(l + 1)
+�
+S − i csc θPS1
+� � 1
+A∂tF − 1
+D∂rF
+�
+,
+(A.10)
+λBO = A
+8 csc θ (2PS1 − iPS2)
+� 1
+A∂t ˜G − 1
+D∂r ˜G
+�
+,
+(A.11)
+λBT =
+1
+√
+2r
+(∂θ¯a − i csc θ∂ϕ¯a − ¯a cot θ),
+(A.12)
+σB = σBE + σBO + σBT ,
+(A.13)
+σBE = 1
+2
+��
+∂2
+θ + 1
+2l(l + 1)
+�
+S + i csc θPS1
+� � 1
+A∂tF + 1
+D∂rF
+�
+,
+(A.14)
+σBO = −1
+4 csc θ (2PS1 + iPS2)
+� 1
+A∂t ˜G + 1
+D∂r ˜G
+�
+,
+(A.15)
+σBT = − 1
+√
+2r
+(∂θb + i csc θ∂ϕb − b cot θ).
+(A.16)
+Here ˜G = G/D and ξ’s are omitted since these are converted to fix the variables (B, E, F, G).
+Note that in the case of D =1, the conditions Ξ22 = Ξ00 = 0 becomes trivial from (2.12)
+and λB = σB = 0 can be achieved by setting F = G = a = b = 0.
+References
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+

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+1 
+Developing and deploying deep learning models in brain MRI:  
+a review   
+Kunal Aggarwal1.2, Marina Manso Jimeno3,4, Keerthi Sravan Ravi3,4, Gilberto Gonzalez5, Sairam 
+Geethanath1 
+ 
+1 Accessible MR Laboratory, Biomedical Engineering, and Imaging Institute, Dept. of Diagnostic, 
+Molecular and Interventional Radiology, Mount Sinai Hospital, New York City, New York 
+2 Department of Electrical and Computer Engineering, Technical University Munich, Munich, 
+Germany 
+3 Department of Biomedical Engineering, Columbia University in the City of New York, New York 
+City, New York, USA 
+4 Columbia University Magnetic Resonance Research Center, Columbia University in the City of 
+New York, New York City, New York, USA 
+5 Division of Neuroradiology, Department of Radiology, Massachusetts General Hospital, 
+Boston, Massachusetts 
+ 
+ 
+Word Count: 5952 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+2 
+Abstract 
+Magnetic Resonance Imaging (MRI) of the brain has benefited from deep learning (DL) to alleviate 
+the burden on radiologists and MR technologists, and improve throughput. The easy accessibility 
+of DL tools have resulted in the rapid increase of DL models and subsequent peer-reviewed 
+publications. However, the rate of deployment in clinical settings is low. Therefore, this review 
+attempts to bring together the ideas from data collection to deployment into the clinic  building on 
+the guidelines and principles that accreditation agencies have espoused. We introduce the need 
+for and the role of DL to deliver accessible MRI. This is followed by a brief review of DL examples 
+in the context of neuropathologies. Based on these studies and others, we collate the 
+prerequisites to develop and deploy DL models for brain MRI. We then delve into the guiding 
+principles to practice good machine learning practices in the context of neuroimaging with a focus 
+on explainability. A checklist based on the FDA's good machine learning practices is provided as 
+a summary of these guidelines. Finally, we review the current challenges and future opportunities 
+in DL for brain MRI. 
+ 
+Keywords: Accessible MRI, Neuroimaging, GMLPs, Explainable AI, FDA, Deep Learning, 
+Deployment, Brain 
+ 
+ 
+ 
+
+3 
+Abbreviations:  
+CAD: Computer Aided Detection 
+CAM: Class Activation Mapping 
+CNN: Convolutional Neural Network 
+DAGMNet: Dual attention gate network 
+DICOM: Digital Imaging and Communications in Medicine 
+DSC: DICE Similarity Coefficient 
+FL: Federated Learning 
+GDPR: General Data Protection Regulation 
+GMLPs: Good Machine Learning Practices 
+Grad-CAM: Gradient-weighted Class Activation Mapping 
+HR: High-Resolution 
+IRB: Institutional Review Board 
+LR: Low-Resolution 
+MCI: Mild Cognitive Impairment 
+OECD: Organisation for Economic Co-operation and Development 
+OOD: Out-of-Distribution 
+PPML: Privacy Protecting Machine Learning 
+PSNR: Peak Signal-to-Noise Ratio 
+RF: Radio Frequency 
+SSIM: Structural Similarity Index Measure 
+XAI: Explainable Artificial Intelligence 
+ 
+ 
+
+4 
+Introduction 
+As of 20181-2, the density of MR scanners was least in geographies with the highest populations 
+such as in sub-Saharan Africa and the Indian subcontinent. The severe lack of neurosurgeons 
+and skilled human resources required to operate, use, and interpret data from MR scanners is a 
+major barrier to accessing this life-saving technology2. In contrast, contemporary radiology 
+departments in the Organisation for Economic Co-operation and Development (OECD) countries 
+require a close interplay of a team with diverse expertise: radiologists specializing in different 
+anatomical sites, MR technologists, medical physicists, and radiologic nurses supported by the 
+vendor’s service and application engineers. This MR inaccessibility and the resulting disparity 
+necessitates the development and deployment of automated methods to augment existing local 
+expertise.  Recently, there has been the development of autonomous MRI methods to automate 
+protocolling3-4, identify artifacts5-6, and reconstruct images from accelerated scans7. These 
+advances are expected to assist local MR technicians, accelerate acquisitions, improve 
+throughput by assisting or replacing manual data processing steps, and reduce waiting times for 
+radiology reporting. In addition to positively impacting MR accessibility, these automation methods 
+facilitate MRI-based big data studies such as the Human Connectome Project8, UK Biobank9, and 
+Rhineland study10, among others. This is due to the high volume and velocity associated with 
+such studies. 
+ 
+These automation methods leverage the recent resurgence of artificial intelligence techniques in 
+general and supervised learning methods in particular. A deep cascade of neural networks - 
+termed deep learning (DL) models -  is trained with a set of input features on one side and the 
+resulting outcomes (labels) on the other side, using existing apriori annotated data11. These 
+trained DL models are then validated against an unseen but similar data set to fine-tune 
+“hyperparameters” of the DL model. Subsequently, the tuned DL model is used to perform 
+
+5 
+inference on a test dataset to evaluate its performance by comparing it with a human-evaluated 
+outcome. Finally, once the classification-tasked model performs satisfactorily with respect to 
+false-positive (FP), false-negative (FN), true-positive (TP), and true-negative (TN) prediction 
+metrics (captured in a “confusion matrix”), then it is considered for a deployment study and down-
+stream accreditation processes11. 
+ 
+In this review, we discuss studies that have developed and deployed deep learning models tasked 
+with multi-task classification. We then present an analysis of the related literature to provide 
+critical steps involved in data collection and curation required to set up deep learning studies. We 
+then highlight the importance of good machine learning practices and the explainability of the DL 
+results by illustrating examples and tools. A suggestive set of steps to enable mounting a 
+successful development and deployment of MRI DL models will be then discussed based on 
+recent literature. Finally, a brief overview of the challenges and opportunities related to MRI-based 
+DL studies is presented.  
+ 
+MRI DL studies of the brain 
+Easy and direct availability of vast amounts of MRI data from publicly available repositories such 
+as HCP8, UK Biobank9, among others, as well as accessible tools to build12 and optimize DL 
+models13 have significantly accelerated the application of DL methods to address challenges in 
+MRI. These studies have resulted in a substantial body of peer-reviewed literature (Figure 1), with 
+most of them sharing an open-source implementation of their models with sample data. This 
+review focuses on MRI DL studies that meet the following criteria: (i) published in the last five 
+years; (ii) Methods mostly focusing on classification and not regression tasks; (iii) studies 
+incorporating explainable AI components; (iv) works demonstrating deployment of the DL models 
+and preferably across multiple vendors and sites. These criteria were used to focus our review on 
+
+6 
+specific developments. Notably, the generic tools developed by mathematicians and computer 
+scientists in the DL community for explainable AI such as GradCAM14 and other methods15-16 have 
+focused more on classification tasks compared to regression tasks (Figure 1). The quantification 
+of the outcomes of a classifier network is relatively straightforward compared to a regression 
+model. The outcomes are directly compared to the “true annotated labels'' and hence result in a 
+binary or a multi-class decision. These  can easily be binned into TP, FP, TN, and FN and 
+evaluated for sensitivity and specificity. In contrast, regression models accomplishing tasks such 
+as DL denoising and synthesizing images are quantified using peak signal-to-noise ratio (PSNR) 
+and structural similarity index measure (SSIM). These metrics have a continuous value and are 
+hard to set thresholds for acceptance. In addition, regression models are typically explained using 
+activation maps that are subsequently interpreted by the authors rather than community-wide 
+tools independent of the developed methods. The interested reader is referred to 7 for a detailed 
+review of machine learning-based image reconstruction methods that leverage regression 
+models.  
+ 
+Example DL MRI solutions for neuroimaging 
+Brain tumors: Nalepa et al. have demonstrated a fully automated pipeline for DCE-MRI analysis 
+of brain tumors called Sens.AI DCE17. In particular, they have substituted manual segmentation 
+of brain tumors in T2 FLAIR images with deep-learning-based methods to demonstrate improved 
+reproducibility. They complement this with a real-time image processing algorithm to determine 
+the vascular input region. Finally, they include a new cubic model of the vascular function for PK 
+modeling. They have validated their package with the BraTs dataset for DICE coefficient and area 
+under the curve as well as by twelve readers from two institutions. These results show good to 
+excellent agreement between the gold standard BraTS dataset and Sens.AI DCE with a total 
+execution time of approximately 3 minutes. Another study focused on the automated identification 
+and classification of brain tumor MRI data classified into glioma, meningioma, and pituitary tumors 
+
+7 
+with an accuracy of 93.7% and a DICE similarity coefficient (DSC) of 95.8%18. A subsequent task 
+of classifying gliomas into high or low grade had an accuracy of 96.5% and a DSC of 94.3%. 
+These models were based on the GoogleNet variant architectures to efficiently combine local 
+features. The identification and classification tasks were accomplished in less than 3 minutes. 
+The method compared well with other state-of-the-art methods with respect to accuracy. Although 
+the study did not explicitly focus on explainable AI methods such as Grad-CAM for interpretation, 
+the authors performed an ablation study to demonstrate the effect of the locally chosen features. 
+Brain extraction is a key step in multiple neuroimaging pre-processing pipelines and in complying 
+with privacy laws. This task becomes more challenging in the presence of pathology such as 
+diffuse gliomas as most analytical and deep learning methods focus on healthy brain extraction. 
+Thakur et al have addressed this gap by developing and testing their models for brain extraction 
+in the presence of diffuse glioma, in a multi-institutional manner19. The authors considered 
+multiparametric MRI data from private and public repositories acquired with different acquisition 
+protocols to train a “modality-agnostic” tool that does not require retraining. The work 
+demonstrated a similar or better accuracy compared to other brain extraction models that worked 
+only on healthy brain extractions. The interested reader is pointed to these reviews for further 
+reading on deep learning methods for brain tumor imaging20, classification21, and segmentation22. 
+ 
+Stroke: The need for automated segmentation and classification of images, especially in 
+emergency room settings in this time-critical pathology is well understood. In this direction, Liu et 
+al developed a deep learning model that detected and segmented abnormalities in acute ischemic 
+stroke23. The work included several steps of pre-processing the data, such as skull stripping and 
+DWI intensity normalization, among others. The study compared T-score and the modified c-fuzzy 
+methods for lesion segmentation. In addition, the authors implemented the 3D Dual attention gate 
+network (DAGMNet) as a supervised learning method to delineate the lesions. The developed 
+model performs better than the unsupervised generic tools and is faster, publicly available, and 
+
+8 
+easy to deploy. They tested their algorithm on a hold-out data set of 280 MRIs and quantified the 
+improved performance using DICE scores, precision, sensitivity, subject detection rate, DICE 
+scores for the lesion volumes and lesion DWI contrast. Another study on stroke detection 
+performed by Zhang et al demonstrated an accuracy of 89.77% over 300 ischemic stroke 
+patients24. The authors evaluated three network architectures with labels drawn by experienced 
+radiologists from two hospitals. Their models predicted a bounding box covering the lesions. 
+Statistical analysis performed on the location, size, and shape correlated well with the radiologists’ 
+labeling. This implementation aids in rapid localization and preliminary characterization of the 
+lesion. The authors have committed to making the data available once a thousand patient data 
+are collected. An important task in imaging stroke is to grade the severity of the ischemia. A multi-
+class classification solution for detecting the severity has been developed by Acharya et al.25 The 
+authors extracted higher-order features from MR images, such as bispectrum entropy and its 
+phase, followed by support vector machines to classify the severity of the stroke into LACS, PACS 
+and TACS. The algorithm demonstrated high levels of accuracy without the need for any manual 
+intervention to augment the neuroradiologist. 
+ 
+Alzheimer’s disease: Supervised learning models have demonstrated the utility of automation 
+in the imaging of Dementia. A specific challenge in this area is the classification and staging of 
+the progression in mild cognitive impairment (MCI) patients. Kwak et al. have developed a deep 
+learning model based on brain atrophy patterns and associated these changes with differences 
+in amyloid burden, cognition, and metabolism26. This model is used to classify AD patients from 
+cognitively normal subjects. A secondary classification helps identify the trajectory of cognitive 
+decline in individuals with MCI. These results were validated with cognitive tests, fluid biomarkers, 
+and PET uptake data with good agreement. The approach is expected to benefit from integrating 
+cognitive and neurobiological features to capture the heterogeneity of MCI. Another approach 
+involved the development and testing of whole-brain 3D convolutional neural networks to detect 
+
+9 
+AD27. This model did not include any patient-specific information to allow the generalization of the 
+algorithm. The implementation included four steps of brain extraction, normalization, 3D CNN 
+followed by domain adaptation. A key feature of this study is the authors' focus on accomplishing 
+accountability. The method outperformed other state-of-the-art methods in the CAD Dementia 
+challenge test, along with explainable AI visualizations to aid the interpretation of classification 
+results. Ahmed et al. also used a 3D approach but collected an ensemble of 2D patches in three 
+orientations to train an ROI-based neural network to stage AD using MR images as per NIA 
+labels28. This approach was demonstrated on the GARD and ADNI datasets. This method was 
+compared to other state-of-the-art methods, and the performance was similar or better. The 
+landmarks delineated by the algorithm to indicate AD correlated well with known neuroanatomical 
+areas. However, the model could not accurately predict asymptomatic AD based on the high rates 
+of false positives.  
+ 
+Prerequisites for DL-based neuro-MRI 
+Data from routine MRI studies result in high volume, velocity, and variety: characteristics of big 
+data29–35. For training DL models, high volume and velocity are favorable factors: more data is 
+better than lesser; high velocity of data requires automated processing methods. However, the 
+variety in MRI data due to a large number of available acquisition parameters, reconstruction 
+methods, receive coil configurations, post-processing steps requires attention to fine details 
+before collating data for annotation and subsequent training. In line with the classification 
+suggested by Wald36, the sources of these variabilities need to be binned as emanating from the 
+MR system characteristics, subject-induced variations, and pathology-specific factors. Typically, 
+the goal of the DL models discussed in this work is to glean the subtle changes in 
+pathophysiological states based on the MR images. Therefore, standardization of parameters 
+getting affected by the system and subject priors is critical to ensure that the DL models focus 
+
+10 
+their attention on the pathology being investigated. The removal or reduction of the confounding 
+variables is therefore essential in building these DL models.  This exercise is facilitated by the use 
+of explainable AI tools. Therefore, two critical requirements to understand an MRI-based DL study 
+are standardization of procedures and methods  and explainable AI tools. A detailed discussion 
+on these prerequisites is performed below. 
+DL models require large amounts of data to achieve high accuracy levels37–39. Unlike models 
+trained on natural images, large datasets are challenging to achieve for medical imaging 
+applications37–39. Training a DL model of medical images entails data collection, curation, and 
+annotation. Additionally, data augmentation might be necessary in cases when the data are 
+insufficient or to strengthen the generalization ability of the model39–41. The steps of data collection 
+and annotation are the most expensive and time-consuming, and patient privacy policies often 
+restrict the use and sharing of images37,39,42,43. These factors significantly impact the models’ 
+performance in clinical settings, limiting their ultimate deployment40,42. The purpose of this section 
+is to review strategies and recommendations (Figure 2) at the data level for maximizing the 
+likelihood of successful deployment after training based on previously published work.  
+Data collection, including patient selection, imaging protocol, sequences, and scan parameters, 
+is determined based on the intended use of the model and its targeted application. Cohorts can 
+be prospective or retrospective, depending on whether the data are acquired for the study or 
+retrieved from a public or private repository44. An initial step in the process is  the approval by an 
+Institutional Review Board (IRB) or a similar board. Additionally, participants provide informed 
+consent about the use of their personal data, which is typically de-identified during data curation. 
+Privacy Protecting Machine Learning (PPML) is a niche area of research that aims at maximizing 
+the confidentiality of patient data while optimizing its use on data-driven models45. In this field, 
+Federated Learning (FL) alleviates the shortage of data problems by allowing training models on  
+large-scale, multi-center data without data sharing.  FL feasibility in MRI has been explored by 
+
+11 
+Sarma et al. and Sheller et al. for brain tumor and prostate segmentation tasks46,47. When dealing 
+with multi-institutional data, protocol harmonization can avoid model bias that may arise from 
+differences in image contrast, intensity, or noise distribution. In MRI, these differences may stem 
+from multiple sources, including variations in system manufacturer, field strength, Radio 
+Frequency (RF) coils, patient positioning, acquisition sequence and scan time, and even pre-
+processing and reconstruction pipelines.  
+Publicly available databases are the results of extensive research projects and contain large 
+amounts of data that can be leveraged for training. These datasets are typically acquired using 
+the same protocol and scanner or using highly harmonized protocols. Patient inclusion and 
+exclusion criteria in these research cohorts are strict, and the datasets are well-curated and 
+usually undergo multi-step post-processing pipelines and standardization operations. While the 
+reproducibility of models trained on publicly available datasets is easier to assess, the data  lack 
+the heterogeneity characteristic of clinical data observed during deployment. Martensson et al.43 
+systematically studied the performance variability of a DL model trained on different combinations 
+of training sets, including publicly available datasets and more heterogeneous Out-of-Distribution 
+(OOD) datasets. They observed that performance drops when models trained on homogeneous 
+data are applied to clinical data. However, inference on clinical data showed a better agreement 
+level with a radiologist reading if clinical cohorts were present in the training data.  
+A benefit of local data acquisition for training is having access to raw data. Most public or private 
+imaging repositories contain data in the Digital Imaging and Communications in Medicine 
+(DICOM) format. The raw data undergo several pre-processing steps, including coil-combination, 
+filtering, artifact correction, and phase removal before storage, stripping the images of features 
+that DL models in the process could recognize. Raw data might be favored for certain DL tasks, 
+data augmentation techniques, or for data synthesis via forward modeling. This is exemplified by 
+the increasing usage of the fastMRI dataset48, the only publicly-available dataset of raw MR knee 
+
+12 
+and brain data. It has become a benchmark for the validation and reproducibility assessment of 
+DL-based image reconstruction algorithms. Additionally, models for the detection or correction of 
+k-space-occurring artifacts such as motion49–51 and Gibbs ringing5,52 typically leverage raw data 
+for the simulation of artifact-corrupted images. 
+Data collection is followed by data curation. This step is performed to standardize and improve 
+dataset quality for subsequent deep neural network training42. The data used for the model 
+development entails a trade-off between distribution heterogeneity and representation bias. It 
+should represent varying patient populations and anatomy disparities while avoiding biasing 
+network representation. Successfully deployed models are typically developed with data acquired 
+using the same imaging protocol and the same system as the site targeted for deployment53,54. 
+Failure mode analysis of the prospective evaluation of an automatic kidney segmentation model 
+after deployment revealed segmentation errors arising from common clinical scenarios such as a 
+fluid-filled stomach and a distended bladder54. These cases are typically excluded during cohort 
+building or data curation and reduce the model’s tolerance to data variations. Poor generalizability 
+to unseen domains is one of the major challenges to successfully deploying DL models in the 
+clinic55. 
+For fully and semi-supervised learning tasks, data labeling or annotation is typically the most time-
+consuming step of an AI project. It may require localizing, delineating, or segmenting lesions or 
+organs of interest or labeling or annotating characteristics of the data. This step is performed 
+manually by an experienced reader via visual inspection. When human observations or clinicians' 
+expertise is required to annotate the data, multiple readers and ideally with variable levels of 
+experience, are preferred to estimate inter-reader variability and compare it to the model’s 
+performance.  
+
+13 
+Finally, data augmentation is the process of generating additional versions of the initial data to 
+enlarge the training set and improve the model's robustness, thereby avoiding overfitting56. 
+Typical techniques include translation, flipping, rotation, and cropping. Depending on the 
+application, other approaches may be useful, such as random k-space oversampling for model-
+based reconstruction techniques. Data augmentation can also be leveraged to reduce the gap 
+between the training data and prospective clinical data, for example, by simulating noise and 
+motion artifacts in the images. A recent study57 for accelerated MR reconstruction demonstrated 
+that with the introduction of these commonly-occurring artifacts using MR physics-driven data 
+augmentation techniques, model performance on both in-distribution and OOD data increases 
+compared to state-of-the-art image-based data augmentation methods. 
+ 
+Good Machine Learning Practices for MRI 
+AI's novelty and current regulatory paradigms are not well adapted to strike a balance between 
+patient safety and promoting the expansion of this new industry58. For assessing commercially 
+accessible algorithms to guarantee their dependability and safety, defining best practices is an 
+area of active research59,60, significant regulatory problems need to be resolved to move clinical 
+AI toward becoming safe and robust61.  Wu et. al. reviewed the FDA database for parameters that 
+were used to evaluate the AI algorithms of products. The parameters they found were - (i) number 
+of patients and sites used in the evaluation, (ii) prospective and retrospective collection of data, 
+and (iii) whether the performance was stratified by disease subtypes or not. Based on the FDA 
+summary, the study revealed that 126 out of 130 AI devices conducted solely retrospective 
+investigations at their submission. The influence of the AI decision tool on clinical practice must 
+be fully characterized, though, and this is crucial since human-computer interaction might differ 
+significantly from a model's intended purpose. For instance, most computer-aided detection 
+
+14 
+(CAD) diagnostic tools are meant to serve as decision-support aids rather than primary diagnostic 
+instruments61. Instead of independently diagnosing, staging, or triaging pathology, CAD is meant 
+to identify, mark, highlight, or otherwise draw attention to imaging characteristics62. The FDA 
+suggests regulating AI software based on function rather than technical components or intended 
+use, which is different from the case for most pharmaceutical items, gadgets, and foods58. 
+Therefore, FDA advocates ten guiding principles for medical device development known as 10 
+Good Machine Learning Practices (GMLPs) that take into consideration the prerequisites 
+discussed above. According to the first and second principle, all expertise related to the product 
+development should work together from the development phase until integration into the clinical 
+workflow. This includes neuroradiologists, neuroimaging scientists, MR technicians, and data 
+scientists implementing good software engineering and security practices. The third principle 
+states the importance of metadata in developing DL models and connects to the concept of data 
+security from the second principle. The fourth and fifth principles mention the importance of 
+datasets. The training and testing datasets should be independent and the reference datasets 
+should have the same characteristics as of the patients in the former datasets. According to the 
+sixth principle, the intended use of a model must be clearly defined along with its risks and 
+performance limitations on different datasets. This relates to principle number three in a sense 
+that metadata defines the scope of the model being used on the specific patients. Principle seven 
+states the involvement of humans and the fact that human intervention cannot be avoided at any 
+stage of development or deployment. This principle focuses more on AI in the loop rather than 
+human in the loop. Eighth and ninth principle centers on the user and states that the model should 
+be easy to understand for the end user and must list all the possible precautions in order to avoid 
+harm to the patient. Principle ten mentions that updates in the model are a mandatory part of the 
+DL deployment and must be considered frequently59. A checklist based on these GMLPs is 
+provided as a summary specifically designed for experts working in brain MRI (Table 1). 
+
+15 
+Explainable AI 
+Although deep learning techniques produce outcomes, they do not explain how those results were 
+obtained. One cannot just analyze the deep neural network to understand how that choice was 
+made. As a result, deep learning models are sometimes referred to as "Black Boxes"63. Medical 
+professionals believe these "black boxes" may be prejudiced in some way, which might have 
+negative effects when used in practical applications63. Additionally, laws like the General Data 
+Protection Regulation (GDPR, Article 15) of the European Union specify that patients have the 
+right to request an explanation for how a given diagnosis was reached if the standard deep 
+learning models cannot64. Therefore, Explainable Artificial Intelligence (XAI) techniques recently 
+developed with the primary objective of visualizing and interpreting the results of machine learning 
+(ML) and deep learning (DL) networks represent a potential remedy to close this gap between 
+high performance and deep-level understanding65 (Figure 3). They have been utilized in a variety 
+of applications, including the categorization of ECGs66 and the visualization of feature maps at 
+various Convolutional Neural Network (CNN) layers67. 
+Velden et al. categorized XAI approaches into three groups based on three criteria: (i) model-
+based vs post-hoc; (ii) model-specific against model-agnostic; and (iii) global versus local. These 
+categories are visual, textual, and example based. The most prevalent type of XAI in medical 
+imaging, out of these three categories, is the visual explanation. These approaches, sometimes 
+referred to as saliency mapping, employ a backpropagation methodology to highlight the key 
+elements of a picture for a certain model’s decision by emphasizing the pixels that had the 
+greatest influence on the results of the investigation64. Class activation mapping (CAM), a 
+technique used in the backpropagation methodology, was introduced by Zhou et al. in 2016. They 
+used global average pooling on the last convolutional feature maps to substitute the fully 
+connected layers at the conclusion of a CNN. It is a weighted linear sum of the visual patterns 
+that were observed and recorded by the filters at various spatial positions68. 
+
+16 
+Gradient-weighted class activation mapping is a generic strategy that includes CAM as one of its 
+specialized methods (Grad-CAM). Grad-CAM can operate with any CNN, but CAM needs global 
+average pooling in particular64. Grad-CAM delivers the ROI on an input image that has the 
+greatest influence on class prediction. Grad-CAM allows us to track the spatial attention changes 
+that occur between network layers, or more precisely, what each network layer focuses on in each 
+input image. To do this, the output gradient with respect to each neuron in the network is 
+calculated to ascertain its relative significance69. 
+Grad-CAM has been widely used to describe deep learning models. Jimeno et. al. used it to 
+identify and classify wrap-around and Gibbs ringing artifacts5. It was used by Windisch et al. in 
+2020 to identify brain MRI regions that caused the classifier to determine the existence of a 
+malignancy70. A model’s prediction of the fetus's brain age may also be explained using Grad-
+CAM, according to a 2020 publication by Liao et al., which will help avoid congenital 
+malformations71. It was also employed by Natekar et al. in 2020 to describe the brain tumor 
+segmentation network69. 
+Occlusion Sensitivity technique is another XAI tool64. The input MR image is disturbed by a small 
+perturbation, and the categorization choice is changed and examined. In order to quantify the 
+variation in the output prediction, it covers a piece of the input picture with a black patch. After 
+moving the patch across the whole picture, it is simple to determine which parts of the brain are 
+responsible for the categorization choice in question by looking at this variation65. This approach 
+was utilized by Bordin et al. to identify relationships between White matter hyperintensities and 
+the anatomical areas that are most important for the categorization of Alzheimer's disease. In 
+conclusion, these XAI approaches present a potentially important addition that may eventually 
+boost radiologist's confidence in the usage of AI models. 
+ 
+
+17 
+Challenges and opportunities  
+DL research has recently witnessed accelerating adoption in the field of MRI (Figure 1) impacting  
+image acquisition, reconstruction, processing, and radiological reporting tasks.  
+Image acquisition: Currently, DL for MRI acquisition can be classified into two broad categories: 
+(i) automatically generating MR pulse sequences for a target contrast or signal-to-noise ratio. In 
+this approach, once a vendor hardware of interest has been identified, imposing appropriate 
+constraints on the cost function (for example, slew rate) will facilitate easy implementation of the 
+optimized pulse sequence on the chosen hardware4. Second is the acceleration of existing 
+vendor-defined protocols, potentially relying on post-acquisition methods to recover SNR72–74. 
+This approach is inherently limited to a particular protocol and vendor. The emergence of physics-
+informed DL methods will allow researchers to develop models that are privy to the underlying 
+physical phenomena, potentially resulting in improved interpretability since the outputs can be 
+evaluated using existing task-specific knowledge75–79. Performing automated and intelligent slice 
+planning for localizers is also an active area of research80,81. 
+Image reconstruction and processing: Based on the work by Chaudhari et al.40, applications 
+of DL to image reconstruction and processing are classified into model-free image synthesis, 
+model-based image reconstruction, and classification and segmentation. Model-free image 
+synthesis pertains to the mapping of input images to output images. Examples are image super-
+resolution, denoising, artifact reduction or removal, and synthesis of missing contrasts. Image 
+super-resolution enables the acquisition of multiple low-resolution images, which can be upscaled 
+using DL models82–87. Compiling a training dataset for this task is not straightforward since it is 
+not trivial to acquire paired low-resolution (LR) and high-resolution (HR) data. Apart from logistical 
+challenges, image registration is a primary concern. It is therefore convenient to acquire HR 
+images and subsequently perform retrospective downsampling to generate LR images. However, 
+this does not faithfully replicate MRI encoding, and hence does not accurately represent real-
+
+18 
+world LR data. In some other cases, HR data is not readily available. One workaround is to 
+leverage a self-supervised learning framework to synthesise low-resolution images from high-
+resolution data, thereby mitigating the requirement of image registration88. Image denoising 
+models improve SNR post-acquisition72,74. Two common approaches to achieve image denoising 
+are to either directly synthesise the denoised image, or to synthesise the residual from which the 
+final denoised image can be obtained. In the first approach, the models are trained on pairs of 
+noisy/clean images to optimize for image quality whilst avoiding blurring artifacts and retaining 
+the anatomical structures present in the original image89–92. An alternative method is to obtain the 
+final denoised image from the difference of the original input image and the predicted residual93,94. 
+Next, artifact reduction or removal models improve image quality by partially or completely 
+correcting MR image artifacts that might otherwise interfere with diagnosis or reduce image 
+quality52,95–97. Finally, contrast-synthesis models enable performing a limited MR exam whilst still 
+obtaining the same diagnostic information as from a comprehensive MR exam, by generating the 
+missing contrasts98,99. They can also enable performing contrast-enhanced MR examinations with 
+reduced dosages of the exogenous contrast agents100,101. Model-based image reconstruction 
+involves transforming undersampled data into fully-sampled reconstructed images102–105. One 
+primary challenge associated with image synthesis and reconstruction is hallucination40. This 
+relates to the addition of features that are not present in the input image. Since the model’s 
+representations are learned implicitly, hallucinations typically tend to reflect the characteristics of 
+the training dataset. The challenge of distinguishing true image signals from hallucinated signals 
+is exacerbated in the task of contrast-synthesis. Existing explainable AI approaches applicable to 
+other tasks are not amenable to image synthesis tasks. Consequently, mitigation strategies to 
+avoid hallucinations are an active area of research in the broader DL community. For image 
+reconstruction, embedding data-consistency steps into the reconstruction process is a viable 
+strategy to mitigate hallucinations.  
+
+19 
+Radiological reporting: The typical workflow of a radiologist involves identifying, localizing, and 
+characterizing the pathology of interest. This is labour-intensive, and recent DL implementations 
+have attempted to alleviate this burden on the radiologist106. Examples range from predicting 
+diagnosis from input images, to generating a text-based radiological report from input images. 
+The superior performance of DL methods on identification and classification tasks lends itself to 
+the automated detection of findings from acquired images. Furthermore, several works have also 
+demonstrated a potential for automated interpretation of findings107. Finally, assisting clinical 
+decision support systems could improve quality of care108,109. However, the attribution of clinical 
+decisions that were assisted by DL systems is an unresolved problem. Along with other ethical 
+and legal challenges such as those related to data sharing and bias (refer to sections on 
+prerequisites and GMLP), these bottlenecks need to be addressed prior to a potential deployment 
+in a real-world scenario. Wang et al. discuss the entire workflow of medical imaging: from 
+tomographic raw data/features to reconstructed images and then extracted diagnostic 
+features/readings7. 
+Figure 4 briefly captures the broader challenges and opportunities associated with employing DL 
+in medical imaging. In general, DL methods present other challenges and opportunities apart from 
+the application-specific ones discussed above. First, the current state-of-the-art  DL qualifies as 
+narrow intelligence since it lacks global context108. This results in severe performance degradation  
+when tackling out of distribution data (OOD). Furthermore, it is not trivial to identify whether 
+unseen data is OOD110. This problem is exacerbated in diagnostic healthcare imaging because 
+the generated data is heterogeneous, noisy, and incomplete. This can be attributed to the 
+differences in vendor hardware and software, and the plethora of component configurations111. 
+Second, the lack of interpretability of DL models does not allow clinical users to develop trust in 
+the models’ predictions, resulting in stymied adoption and deployment in healthcare. Third, 
+training DL models to achieve the level of robustness necessary to handle this variety requires an 
+ImageNet-like breakthrough in the medical imaging community at large112. Recent works such as 
+
+20 
+RadImageNet112 are encouraging, and can potentially facilitate such advancements. However, 
+with ever-growing scales of data collection, the closely coupled and critical task of data curation 
+grows in complexity, at least for supervised learning frameworks. This relates to the fifth 
+challenge, which involves ensuring bias-free data curation. The performance of any DL model is 
+directly dependent on the quality of the data it was trained on. To avoid any biases in the output 
+which could potentially compound in downstream analyses, the training dataset has to be free of 
+all confounding factors. Training DL models on large-scale datasets requires prohibitively 
+expensive hardware setups to provide the required compute, coupled with extremely long training 
+durations. Consequently, this time-, cost-, and resource-intensive workflow raises the 
+development barrier thereby mostly limiting research efforts to well-funded organizations and 
+institutions. However, the recently increasing availability of commercial cloud solutions by 
+Amazon, Microsoft, Google, etc., unlocks cost-effective compute that is globally accessible. In 
+addition to the pre-existing heterogeneity of the data, acquisition methods are constantly evolving, 
+introducing another dimension of variability to the data. This will require deployed DL models to 
+be capable of online training to avoid incorrect or irrelevant predictions, or potential misdiagnoses 
+in downstream analyses when encountering OOD data. Any development workflow lag, 
+regardless of the duration, will result in incorrect treatment planning until updated models are 
+deployed. On the other hand, disengaging the models until newer versions are available will result 
+in workflow interruptions and throughput degradation. Lastly, the ethical and legal uncertainties 
+involved critically need to be resolved prior to any potential deployments. Different countries 
+enforce different medical data custody laws, necessitating region-specific modifications to the DL 
+tool and the data pipeline to ensure compliance. Most importantly, the ownership of a DL-assisted 
+clinical decision is an open question. Despite these challenges, the ability to automate tasks such 
+as image interpretation and diagnosis will alleviate the immense burden on healthcare providers, 
+allowing them to focus on other important tasks whilst improving the quality of their work lives. 
+Providing more accurate and timely diagnosis, reduced costs, increased efficiency, and tailored 
+
+21 
+treatments to individual patients based on their specific characteristics and needs all result in 
+improved patient outcomes. These are strong motivators to strategize immediate or near-future 
+adoption of existing DL methods. Directing research efforts to explore opportunities and 
+simultaneously addressing existing issues will aid in the wider adoption and improved realization 
+of DL’s potential. Along with addressing weaknesses and leveraging strengths, incorporating the 
+GMLP principles (Section 4) across the development lifecycle of DL-assisted medical applications 
+will aid in maximizing safety, efficiency, and quality during clinical deployment.  
+Conclusion 
+Our literature review indicates an increase in DL models for brain MRI tasks related to the 
+acquisition, reconstruction, image analysis, and reporting in the last five years across 
+neuropathologies such as tumors, stroke, and Alzheimer’s disease. These studies were 
+summarized as a suggestive DL pipeline for brain MRI studies.  Importantly, the proportion of 
+studies that adhere to GMLP principles and contain XAI components are significantly low. This 
+DL neuro-MRI GMLP checklist in this review is motivated by this gap and emanates from the ten-
+point guidelines espoused by the accreditation agencies for these principles tailored to brain MRI.  
+Finally, our assessment of the opportunities and challenges in DL studies on brain MRI indicates 
+that the inclusion of the GMLPs significantly reduces the challenges associated with cost, and 
+lack of interpretability, bias in the training data among others (Figure 4). Overcoming these 
+challenges will unlock the potential to improve multiple aspects of neuroimaging using MRI 
+through the successful deployment of accreditation agency-approved DL models. 
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+[110] Ren J, Liu PJ, Fertig E, Snoek J, Poplin R, DePristo MA, et al. Likelihood ratios for out-
+of-distribution detection. In: Proceedings of the 33rd International Conference on Neural 
+Information Processing Systems. Curran Associates Inc.; 2019:14707-14718. 
+
+30 
+[111] Yan W, Huang L, Xia L, Gu S, Yan F, Wang Y, et al. MRI Manufacturer Shift and 
+Adaptation: Increasing the Generalizability of Deep Learning Segmentation for MR Images 
+Acquired with Different Scanners. Radiol Artif Intell. 2020;2(4):e190195. 
+[112] Mei X, Liu Z, Robson PM, Marinelli B, Huang M, Doshi A, et al. RadImageNet: An Open 
+Radiologic Deep Learning Research Dataset for Effective Transfer Learning. Radiol Artif 
+Intell. 2022;4(5):e210315. 
+ 
+ 
+
+31 
+Figures and Tables: 
+ 
+ 
+ 
+Figure 1: Publication trend in the past five years for deep learning in MRI. We used the PubMed 
+database with the following keywords - regression MRI deep learning, classification MRI deep 
+learning, and MRI deep learning. 
+ 
+ 
+ 
+ 
+Figure 2: Steps for creating a dataset for the development of a deep learning model.  
+ 
+
+Studydesign
+DataQA
+Data
+Database
+andIRB
+and
+privacyand
+creation
+curation
+security
+and sharing
+ApprovalData
+Data
+Data
+acquisition
+annotation
+augmentation
+orretrievallearningMRl
+publications
+2018
+2019
+2020
+2021
+2022
+Year of Publication Publications
+Abstracts
+containing
+1000-
+'classification"
+Deep
+#
+500-Deep learning MRl publications
+2000
+Abstracts
+containing32 
+ 
+ 
+Figure 3: Mountain represents the number of publications getting reduced as we focus our 
+scope. At the base of the mountain, we have a number of publications on AI in MRI. As we 
+move up the mountain, we specialize more into accreditation agency approved XAI models in 
+MRI following GMLPs. We can see that a small fraction of all publications actually make it to the 
+top of the mountain, i.e., follow all the requirements of the accreditation agency. Many of the 
+publications follow black box approach which doesn’t explain the decision making methodology 
+of their models and thus end up nowhere. 
+ 
+ 
+
+CAM
+Artificial Intelligence in MRI
+(6486publications)imaging-392
+Guided Backpropagation
+LRP
+Black BoxApproach-Doesn'tanswer
+Occlusion Sensitivity
+why, what and how
+GradCAMAccreditation agency approved XAl models in MRI following Good
+MachineLearningPractices(GMLPs)-35
+Accreditation agency approved XAl models in MRl-166
+Accreditation agency approved XAl models in medical33 
+ 
+ 
+Figure 4: Challenges and opportunities of employing deep learning (DL) in neuroimaging. 
+Developing DL methods poses several challenges, such as (clockwise from top) ensuring bias-
+free data curation and annotation, difficulty in assembling datasets reflective of real-world 
+heterogeneity, black-box models stymieing development of trust in predictions, region-specific 
+ethical and legal requirements, and managing the massive compute requirements to train and 
+deploy models. However, unrealized opportunities such as (clockwise from top) more timely and 
+accurate clinical decisions, augmenting available human expertise to alleviate the burden on 
+skilled personnel, increased throughput and the associated reduction in operating costs are 
+strong motivators to work toward a successful deployment. 
+ 
+ 
+
+andradiologists
+Improved
+Lack oftrust inmodels'predictions
+throughput
+dueto lack of interpretabilitydecisions
+$$$
+$$$
+Ethical& legal
+Growing compute
+Lackof
+Reduced costs
+Augment expertise
+requirements
+requirementsto
+robustness due
+from increased
+to alleviate burden
+train and deploy
+tovariegate data
+efficiency
+on MRtechniciansCHALLENGES
+OPPORTUNITIES
+Bias-free data
+curation and
+Improved
+annotation
+clinical34 
+ 
+Checklist of GMLPs for brain MRI  
+1. 
+Are neuroradiologists, neuroimaging scientists, MR technician and data scientist 
+working together throughout the whole life cycle of the product? 
+ 
+2. 
+Is the patient's personal information anonymous in the brain MR images? 
+ 
+3. 
+Is the metadata being filled for each patient scan with proper details of all 
+parameters? 
+ 
+4. 
+Does training and testing MR datasets contain different scans? There shouldn’t be 
+any common scan in both datasets.  
+ 
+5. 
+Does reference MR dataset for validation of model have completely unique scans 
+with same parameters as training and testing dataset? 
+ 
+6. 
+Are you using the model for segmenting brain structures from the specific contrast 
+for which it has been trained for? If so, don’t use it for other contrasts. 
+ 
+7. 
+Is the output of the model accepted and readable by the neuroradiologist? 
+ 
+8. 
+Has the model been tested in the neuroradiology department under the supervision 
+of an expert neuroradiologist before deployment? 
+ 
+9. 
+Are the precautions and potential dangers of using the model explicitly 
+mentioned? 
+ 
+10. 
+Is the model being updated frequently for incorporating the changes in the new 
+scans that may occur naturally? 
+ 
+ 
+Table 1: This checklist represents the 10 guiding principles framed by the FDA for medical 
+device development known as Good Machine Learning Practices (GMLPs). This is a simpler 
+version of those principles rephrased specifically for experts working in brain MRI, such as 
+neuroradiologists, neuroimaging scientists and associated data scientists. In order to get 
+approved by the FDA, the AI model developed for MRI must fulfill all these questions.  
+ 
+

_dAzT4oBgHgl3EQfhPzZ/content/tmp_files/2301.01483v1.pdf.txt

@@ -0,0 +1,640 @@
+Proceedings of the 9th International and 49th National Conference on Fluid Mechanics and Fluid Power (FMFP)
+December 14-16, 2022, IIT Roorkee, Roorkee-247667, Uttarakhand, India
+FMFP2022-594
+Performance enhancement of a 100 watts class Tesla turbine
+Arindam Mandal1, Rajosik Adak1, and Sandeep Saha1
+1Department of Aerospace Engineering, IIT Kharagpur, Kharagpur 721302, India
+ABSTRACT Tesla turbines are an attractive but less
+explored
+area
+in
+low-power
+applications.
+This
+article
+presents an experimental investigation of a centimeter
+scale
+Tesla
+turbine
+in
+bi
+and
+uni-directional
+outlet
+configuration with compressed air at 6 bar. The turbine’s
+performance is enhanced by ≈ 38% for a uni-directional
+outlet configuration. Furthermore, we also investigate the
+electrical power of the turbine in a bi-directional outlet
+configuration by coupling the turbine with a generator.
+Despite achieving higher performance in uni-directional
+outlet configuration, we observe substantial losses at the
+inlet we use for the experiment. To illustrate and improve
+the losses, we numerically investigate the turbine inlet
+at a total pressure and temperature difference of 2 bar
+and
+50◦C,
+respectively.
+Subsequently,
+we
+design
+two
+more nozzles and compare their performance with the
+nozzle we used in our experiment. Our findings suggest
+that nozzle 3 performs the best in delivering the highest
+Mach no and uniformity across the slits. This observation
+would help optimize the nozzle suitable for the Tesla turbine.
+Keywords: Tesla turbine, Friction turbine, Low power
+application, Slit nozzle,
+I.
+INTRODUCTION
+Small scale low microturbines are crucial since there is a
+growing need for them in numerous applications. Examples
+of notable applications include waste heat recovery, pico-
+hydro power, organic Rankine cycle technologies, biomass,
+waste heat recovery, micro GT, and many more. The increas-
+ing need for energy harvesting at these scales poses a variety
+of challenges to the performance and manufacturability of
+the turbine due to their compact sizes, high rotational speeds,
+and increased viscous losses. An alternative expansion de-
+vice addressing these challenges can be highly beneficial
+regarding its techno-economic feasibility. The tesla turbine
+is one of its kind, which has a uniquely simple design and
+a unique mechanism of momentum transfer. The turbine
+rotor consists of several co-axial discs closely packed with
+each other. Each of the rotor discs has outlet ports near the
+center. A casing covering the rotors helps guide the flow
+from the inlet. The turbine shaft is attached to the rotor-
+casing configuration with the help of two bearings. Figure 1
+shows the different components of the turbine. After being
+injected through the inlet system of the turbine, the fluid
+flows spirally inward and exits through the ports located
+near the shaft in the axial direction. The momentum transfer
+from the fluid to the rotor takes place using the adhesion
+and viscosity of the fluid. This mechanism makes the turbine
+attractive at small scales where the dominance of the viscous
+force becomes significant.
+This turbine was conceptualized by Nicola Tesla [39]
+in the year 1907. Initially, the device was unable to attract
+market attention because of the no potential requirement of
+low-power harvesting technologies. After that, till the twen-
+tieth century, a considerable amount of thrust was given to
+the possible design modifications [27], [28], [12], improved
+seal designs [2] and loss analysis [6], [30] of the turbine
+and its ancillary components for enhancing performance. In
+addition, a few simplified theoretical models were developed
+[1], [21] to get an insight into the influence of the parameters
+associated to the turbine.
+Figure 1: Schematic of the exploded view of the turbine.
+a: Casing, b: Rotors, c: Inlet configurations (i,ii and iii),
+d: Shaft, e. Bearing. (f) shows the fluid flow between two
+consecutive corotating discs
+More recently, there has been a surge in interest in
+exploring this technology due to its several advantages
+over conventional energy harvesting technologies. Due
+to its simple design and flow mechanics, the turbine can
+handle particle-laden fluids and two-phase expansion with
+minimal damage. Numerical simulations of the gas flow
+and mass transfer between two coaxially co-rotating discs
+were performed by Sandilya et al. [32] where they found a
+satisfactory match between the numerical and experimental
+value
+of
+mass
+transfer
+coefficient.
+Using
+numerical
+simulations, Ladino [18] presented the load coefficient
+curve, efficiency, and degree of reaction variation after
+maintaining a constant rotational speed. Upon developing
+an analytical model, Deam et al. [7] scaled down the turbine
+in millimeter size to achieve better efficiency. Lemma et al.
+[22] investigated the Parasitic, viscous, and other dissipative
+losses in the bearings and end walls to mitigate their effect
+on decaying the performance.
+An explicit description of a flexible test rig was de-
+1
+arXiv:2301.01483v1  [physics.flu-dyn]  4 Jan 2023
+
+(i)
+(ii)
+(ii)
+(c)
+(p)
+(b)
+(a)
+f
+(e)veloped by Hoya et al. [14] where they calculated the
+low torque at high RPM using the angular acceleration
+method. A comparative study in the efficiency offered by
+Tesla and small bladed microturbine in a micro power plant
+was addressed by Lampart et al. [20], [19] to establish
+the competitiveness of a Tesla turbine. To understand the
+transport phenomena inside the turbine rotor, a number
+of numerical and analytical models solving the Navier-
+Stokes equations using different approaches are present in
+the literature repository [15], [9], [36], [10], [31], [35], [34],
+[5]. Aside from that, flow diagnostics using particle tracking
+velocimetry by Schosser et al. [33] provides an insight into
+the component-wise velocity profiles inside the rotor gaps at
+different radial locations. In recent years, A wide range of
+application based studies of Tesla turbine related to Micro-
+air vehicles [23], Organic Rankine cycle [37], [38], [8],
+[25], [24], Combined heat plant [3], Pico-hydro applications
+[13], [4], [16], [17], Ammonia synthesys [11] have been
+investigated or under investigation [26], [29].
+The recent resurgence in interest indicates the impor-
+tance of investigating the turbine further to mitigate the
+losses due to the nonuniformity and disturbances associated
+with the inflow to rotor and rotor to outflow interaction.
+This article presents the experimental investigation of a
+centimeter-scaled Tesla turbine in uni and bi-directional
+outlet configuration with compressed air. We compare the
+performance in terms of Mechanical power output for the
+two configurations. Furthermore, we investigate the losses in
+the inlet due to the sharp divergent and the slit configuration
+at the inlet rotor junction. Finally, we compare the nozzles’
+performance by looking at the peak discharge Mach number
+and channel-wise disparity in flow injection to the rotor. Our
+numerical results can be beneficial in coming up with better
+inlet configurations for extracting maximum energy from the
+fluid, which leaves a further scope for further investigation.
+II.
+METHODOLOGY AND EXPERIMENT
+A lab scale turbine prototype (in fig. 2a) is fabricated
+for experimentation in the Aerospace Engineering depart-
+ment, IIT Kharagpur. The turbine consists of 10 consecutive
+corotating discs having an outer diameter of 10 cm and a
+thickness of 2 mm. The gap between the consecutive rotating
+discs is 2 mm, and the distance between two extreme discs to
+the adjacent casing wall is 1 mm. The turbine has four outlet
+holes near the shaft, having a center distance of 2 cm from
+the shaft center. The shaft and the exhaust ports’ diameters
+are 1.5 cm and 1 cm, respectively. The distance between the
+shroud and the disc’s edge is 1 mm. The inlet system of the
+turbine is designed to connect the compressor outlet port of
+6mm dia to the turbine having an inlet of 4 cm thickness.
+The area of the inlet is 0.4 cm2, which is segregated using
+slit configurations to guide air to each gap with minimum
+interaction with the peripheral walls of the rotor.
+The turbine inlet is connected to the compressor by
+Polyeurathane pneumatic pipes with an installed pres-
+sure gauge. The RPM of the turbine is measured using
+an Arduino-based tachometer. The angular acceleration-
+deceleration approach computes the net accelerating and de-
+celerating torque.The experiment also uses a digital tachome-
+(a)
+(b)
+Figure 2: (a) Fabricated turbine and (b) the inlet system
+ter to validate the turbine’s RPM. We conduct our experiment
+at 6 bar of inlet pressure for uni-and bi-directional outlet
+configuration. The detail of the experimental setup can be
+seen in the figure 3.
+(a)
+(b)
+Figure 3: Details of the experimental set up. 1- Compres-
+sor discharge, 2- Tachometer, 3- Turbine, 4- Arduino
+based Tachometer, 5- Pressure gauge, 6- PC for data
+acquisition, 7- Camera, 8- Generator, 9- Multimeters,
+10- Rheostat, 11- Electrical circuit
+III.
+RESULTS AND DISCUSSION
+A. Performance characteristics
+We tabulate the RPM with time with the help of a serial
+monitor and calculate the angular acceleration as a function
+of RPM. Once the turbine reaches its stable RPM, we shut
+off the compressed air sypply and continue to perform data
+acquisition until the turbine becomes stationary. We continue
+the similar process for a uni-directional outlet case. We
+observe from figure 4 that the turbine with a uni-directional
+outlet accelerates faster than the turbine with a bi-directional
+outlet configuration. In addition, for a supply pressure of
+6 bar, we achieve a maximum RPM of 13019 and 11124
+for uni and bi-directional outlet configurations, respectively.
+Figure 4 shows the power variation due to accelerating
+and braking torque. There is an increment of ≈ 38% in
+power due to accelerating torque for a uni-directional outlet
+configuration.
+We integrate the turbine with a 100 W class Fedus
+RS-775 DC electric motor to measure the electrical power
+output. However, the motor is used as a generator to measure
+the output voltage and current. The electrical circuit is
+attached to a rheostat, and the experiment is performed with
+2
+
+t (sec)
+0
+15
+30
+45
+RPM
+0
+5000
+10000
+15000
+RPM
+0
+4500
+9000
+13500
+Power (Watts)
+0
+40
+80
+120
+(a)
+(b)
+Figure 4: Distribution of (a) RPM with t; (b) Power
+due to accelerating torque with RPM for bi-directional
+(dashed line) and uni-directional (solid line) outlet. Dot-
+ted line represents the power due to braking torque.
+the 5-ohm load resistance. The figure 5 illustrates the motor’s
+voltage and power variation at various RPM. The turbine-
+generator can produce a maximum of 78 watts of electrical
+power at 7800 RPM for bi-directional outlet configuration.
+RPM
+0
+4000
+8000
+PowerE (Watts)
+0
+40
+80
+80
+0
+40
+Voltage (Volts)
+Figure 5: Experimental results of voltage (dashed line)
+and electrical power (solid line) at different RPM
+B. Inlet design
+Designing the inlet is the most crucial part of the turbine
+where the maximum loss occurs. Notably, the compressor
+and back pressure at the inlet rotor connection point regu-
+lates the flow across the nozzle. In the present article, we
+only consider the inlet section for analysis. The details of
+the nozzle dimensions are in figure 6.
+1) Numerical methodology and grid Grid sensitivity:
+The Inlet section of the nozzle is a pressure inlet boundary
+where the total pressure is at 6 bar. We consider the outlet
+of the nozzle is at 4 bar. The temperature at the compressor
+discharge and the inlet-rotor junction are 450K and 400K,
+respectively. The surface of the nozzle is a no-slip type
+boundary. Considering these boundary conditions, we solve
+governing compressible Reynolds-averaged Navier-Stokes
+equations using the K − ω SST turbulence model in the
+commercial CFD package Fluent 2021 R2. The numerical
+domain is discretized using the body-fitted tetrahedral grids,
+maintaining a wall y+ ≈ O(1). The table 1 below presents
+the grid sensitivity study. As the desired output shows a
+Figure 6: Schematic diagram of the nozzles. The dotted,
+dashed and solid lines represent nozzle 1, nozzle 2 and
+nozzle 3, respectively.
+variation ≤ 2%, We conduct the subsequent numerical
+simulations using grids with ≈ 1M elements. The inlet
+Table 1: Grid sensitivity of the three inlet configurations
+Grid type
+Elements
+Outlet area averaged Mach no
+Nozzle 1
+Nozzle 2
+Nozzle 3
+Coarse
+≈ 0.5 M
+.5093
+.4993
+.5296
+Medium
+≈ 1 M
+.5089
+.4979
+.5289
+Fine
+≈ 2 M
+.5080
+.4971
+.5284
+design presented in this article is based on a two-pronged
+objective. (a) To Maximize the Mach number peak (b) To
+minimize the disparity in the Mach number peaks through
+every slit. To understand the loss mechanism and the flow
+behavior, we conduct a series of numerical simulations
+Where Nozzle 1 is the replica of the inlet nozzle considered
+for the experiment. Due to the abrupt divergent section and
+the presence of a large recirculation zone enclosed by a
+strong shear layer, we detect significant losses in Mach no.
+The dividing streamlines seen in figure 7 (a) are considered
+when designing the following two inlet systems. The second
+nozzle we design is to eliminate the effect of recirculation.
+We gradually increase the inlet nozzle area until we reach
+the section where the flow bends in the previous observation.
+Despite the improvement in the average peak Mach number
+observed from figure 7 (b), the disparity in the discharge
+Mach number through the slits increased. We consider
+designing the third nozzle as a converging-diverging type
+nozzle where we place the throat section at x/L ≈ 0.6 to
+provide sufficient scope for flow to bend. Figure 8 represents
+the peak discharge Mach number through the nozzles and
+the associated % disparity from the mean peak Mach no
+through each slit. The comparison shows that the third nozzle
+offers maximum peak Mach number along with minimum
+% deviation from the mean peak Mach no. It is necessary
+to note that the differential pressure between the upstream
+and downstream sections, the fluid, and the fluid’s thermo-
+physical characteristics significantly influence the nozzle
+design. The nozzle’s optimal size and shape might differ
+depending on these conditions.
+3
+
+10mm
+40 mm
+4mm
+8
+40
+ww
+20mm
+mm
+mm
+15
+mm
+2
+60mm(a)
+(b)
+(c)
+Figure 7: Distribution of the Mach number for (a) nozzle
+1 (b) nozzle 2 (c) nozzle 3.
+IV.
+CONCLUSIONS
+The article presents a preliminary experimental investiga-
+tion of a centimeter scaled Tesla turbine using compressed
+air as a working medium and suggests an improved inlet
+design that could deliver higher achievable power compared
+to the inlet nozzle considered for this experiment. The key
+findings of the article are as follows;
+1)
+Experimental investigation of the turbine conducted
+for uni and bi-directional outlet configuration at 6
+bar inlet pressure.
+2)
+uni-directional outlet con��guration offered a peak
+power output of ≈ 110 watts at ≈ 7000 RPM.
+Whereas, bi-directional outlet configuration a peak
+output of 80 watts at 6500 RPM.
+3)
+The turbine-generator configuration produced a
+peak electrical power output of 78 watts at 7800
+-0.02
+-0.01
+0
+0.01
+0.02
+M
+0
+0.4
+0.8
+(a)
+%M/
+-20
+-10
+0
+10
+20
+%M'
+-30
+-20
+-10
+0
+10
+20
+%M'
+-15
+-10
+-5
+0
+5
+(b)
+(c)
+(d)
+Figure 8: (a) Comparison of peak Mach number across
+the slits. The dotted, dashed and solid lines represent
+First, second and third nozzle, respectively. (b, c, d) The
+% disparity in peak Mach no at discharge for three
+nozzles.
+RPM.
+4)
+While assessing the inlet, we observe that nozzle 2
+offers better peak Mach number than nozzle 1, but
+due to the losses accounted by the sharp bend, the
+disparity in % deviation of the peak mach numbers
+from mean peak Mach no is nearly 30%.
+5)
+Nozzle 3 performs the best among all three nozzles.
+It offers a maximum peak Mach no with minimum
+% deviation from mean peak Mach no is reduced
+to 12%.
+The turbine’s efficiency depends on several factors, e.g.,
+RPM, disc gap, rotor radius, rotor to casing clearance,
+outlet configurations, fluid properties, and many more.
+Designing an efficient Tesla turbine could bring down the
+cost of a turbine substantially as compared to the other
+existing technologies because of its simplistic design. The
+above findings presented in this article could be helpful in
+the design improvement, thus leaving a plausible scope for
+further investigation.
+NOMENCLATURE
+t
+time
+[sec]
+x/L
+Non-dimensional-axial location
+[–]
+k
+Turbulent kinetic energy
+[J/kg]
+ω
+Specific dissipation rate
+[1/sec]
+M
+Mach no
+–
+M ′
+% Deviation from peak mean Mach no
+–
+4
+
+0.050.1
+0.150.2
+0.25
+0.30.350.40.450.50.55
+0.60.650.7
+0.75
+0.8
+0.85REFERENCES
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+5
+

_dAzT4oBgHgl3EQfhPzZ/content/tmp_files/load_file.txt

@@ -0,0 +1,240 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf,len=239
+page_content='Proceedings of the 9th International and 49th National Conference on Fluid Mechanics and Fluid Power (FMFP)  December 14-16, 2022, IIT Roorkee, Roorkee-247667, Uttarakhand, India  FMFP2022-594  Performance enhancement of a 100 watts class Tesla turbine  Arindam Mandal1, Rajosik Adak1, and Sandeep Saha1  1Department of Aerospace Engineering, IIT Kharagpur, Kharagpur 721302, India  ABSTRACT Tesla turbines are an attractive but less  explored  area  in  low-power  applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  This  article  presents an experimental investigation of a centimeter  scale  Tesla  turbine  in  bi  and  uni-directional  outlet  configuration with compressed air at 6 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' The turbine’s  performance is enhanced by ≈ 38% for a uni-directional  outlet configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' Furthermore, we also investigate the  electrical power of the turbine in a bi-directional outlet  configuration by coupling the turbine with a generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  Despite achieving higher performance in uni-directional  outlet configuration, we observe substantial losses at the  inlet we use for the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' To illustrate and improve  the losses, we numerically investigate the turbine inlet  at a total pressure and temperature difference of 2 bar  and  50◦C,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  Subsequently,  we  design  two  more nozzles and compare their performance with the  nozzle we used in our experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' Our findings suggest  that nozzle 3 performs the best in delivering the highest  Mach no and uniformity across the slits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' This observation  would help optimize the nozzle suitable for the Tesla turbine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  Keywords: Tesla turbine, Friction turbine, Low power  application, Slit nozzle,  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  INTRODUCTION  Small scale low microturbines are crucial since there is a  growing need for them in numerous applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' Examples  of notable applications include waste heat recovery, pico-  hydro power, organic Rankine cycle technologies, biomass,  waste heat recovery, micro GT, and many more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' The increas-  ing need for energy harvesting at these scales poses a variety  of challenges to the performance and manufacturability of  the turbine due to their compact sizes, high rotational speeds,  and increased viscous losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' An alternative expansion de-  vice addressing these challenges can be highly beneficial  regarding its techno-economic feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' The tesla turbine  is one of its kind, which has a uniquely simple design and  a unique mechanism of momentum transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' The turbine  rotor consists of several co-axial discs closely packed with  each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' Each of the rotor discs has outlet ports near the  center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' A casing covering the rotors helps guide the flow  from the inlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' The turbine shaft is attached to the rotor-  casing configuration with the help of two bearings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' Figure 1  shows the different components of the turbine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' After being  injected through the inlet system of the turbine, the fluid  flows spirally inward and exits through the ports located  near the shaft in the axial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' The momentum transfer  from the fluid to the rotor takes place using the adhesion  and viscosity of the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' This mechanism makes the turbine  attractive at small scales where the dominance of the viscous  force becomes significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  This turbine was conceptualized by Nicola Tesla [39]  in the year 1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' Initially, the device was unable to attract  market attention because of the no potential requirement of  low-power harvesting technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' After that, till the twen-  tieth century, a considerable amount of thrust was given to  the possible design modifications [27], [28], [12], improved  seal designs [2] and loss analysis [6], [30] of the turbine  and its ancillary components for enhancing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' In  addition, a few simplified theoretical models were developed  [1], [21] to get an insight into the influence of the parameters  associated to the turbine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  Figure 1: Schematic of the exploded view of the turbine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  a: Casing, b: Rotors, c: Inlet configurations (i,ii and iii),  d: Shaft, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' Bearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' (f) shows the fluid flow between two  consecutive corotating discs  More recently, there has been a surge in interest in  exploring this technology due to its several advantages  over conventional energy harvesting technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' Due  to its simple design and flow mechanics, the turbine can  handle particle-laden fluids and two-phase expansion with  minimal damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' Numerical simulations of the gas flow  and mass transfer between two coaxially co-rotating discs  were performed by Sandilya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' [32] where they found a  satisfactory match between the numerical and experimental  value  of  mass  transfer  coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  Using  numerical  simulations, Ladino [18] presented the load coefficient  curve, efficiency, and degree of reaction variation after  maintaining a constant rotational speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' Upon developing  an analytical model, Deam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' [7] scaled down the turbine  in millimeter size to achieve better efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' Lemma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  [22] investigated the Parasitic, viscous, and other dissipative  losses in the bearings and end walls to mitigate their effect  on decaying the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  An explicit description of a flexible test rig was de-  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='01483v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='flu-dyn]  4 Jan 2023  (i)  (ii)  (ii)  (c)  (p)  (b)  (a)  f  (e)veloped by Hoya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' [14] where they calculated the  low torque at high RPM using the angular acceleration  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' A comparative study in the efficiency offered by  Tesla and small bladed microturbine in a micro power plant  was addressed by Lampart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' [20], [19] to establish  the competitiveness of a Tesla turbine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' To understand the  transport phenomena inside the turbine rotor, a number  of numerical and analytical models solving the Navier-  Stokes equations using different approaches are present in  the literature repository [15], [9], [36], [10], [31], [35], [34],  [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' Aside from that, flow diagnostics using particle tracking  velocimetry by Schosser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' [33] provides an insight into  the component-wise velocity profiles inside the rotor gaps at  different radial locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' In recent years, A wide range of  application based studies of Tesla turbine related to Micro-  air vehicles [23], Organic Rankine cycle [37], [38], [8],  [25], [24], Combined heat plant [3], Pico-hydro applications  [13], [4], [16], [17], Ammonia synthesys [11] have been  investigated or under investigation [26], [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  The recent resurgence in interest indicates the impor-  tance of investigating the turbine further to mitigate the  losses due to the nonuniformity and disturbances associated  with the inflow to rotor and rotor to outflow interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  This article presents the experimental investigation of a  centimeter-scaled Tesla turbine in uni and bi-directional  outlet configuration with compressed air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' We compare the  performance in terms of Mechanical power output for the  two configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' Furthermore, we investigate the losses in  the inlet due to the sharp divergent and the slit configuration  at the inlet rotor junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' Finally, we compare the nozzles’  performance by looking at the peak discharge Mach number  and channel-wise disparity in flow injection to the rotor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' Our  numerical results can be beneficial in coming up with better  inlet configurations for extracting maximum energy from the  fluid, which leaves a further scope for further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  METHODOLOGY AND EXPERIMENT  A lab scale turbine prototype (in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' 2a) is fabricated  for experimentation in the Aerospace Engineering depart-  ment, IIT Kharagpur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' The turbine consists of 10 consecutive  corotating discs having an outer diameter of 10 cm and a  thickness of 2 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' The gap between the consecutive rotating  discs is 2 mm, and the distance between two extreme discs to  the adjacent casing wall is 1 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' The turbine has four outlet  holes near the shaft, having a center distance of 2 cm from  the shaft center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' The shaft and the exhaust ports’ diameters  are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='5 cm and 1 cm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' The distance between the  shroud and the disc’s edge is 1 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' The inlet system of the  turbine is designed to connect the compressor outlet port of  6mm dia to the turbine having an inlet of 4 cm thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  The area of the inlet is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='4 cm2, which is segregated using  slit configurations to guide air to each gap with minimum  interaction with the peripheral walls of the rotor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  The turbine inlet is connected to the compressor by  Polyeurathane pneumatic pipes with an installed pres-  sure gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' The RPM of the turbine is measured using  an Arduino-based tachometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' The angular acceleration-  deceleration approach computes the net accelerating and de-  celerating torque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='The experiment also uses a digital tachome-  (a)  (b)  Figure 2: (a) Fabricated turbine and (b) the inlet system  ter to validate the turbine’s RPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' We conduct our experiment  at 6 bar of inlet pressure for uni-and bi-directional outlet  configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' The detail of the experimental setup can be  seen in the figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  (a)  (b)  Figure 3: Details of the experimental set up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' 1- Compres-  sor discharge, 2- Tachometer, 3- Turbine, 4- Arduino  based Tachometer, 5- Pressure gauge, 6- PC for data  acquisition, 7- Camera, 8- Generator, 9- Multimeters,  10- Rheostat, 11- Electrical circuit  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  RESULTS AND DISCUSSION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' Performance characteristics  We tabulate the RPM with time with the help of a serial  monitor and calculate the angular acceleration as a function  of RPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' Once the turbine reaches its stable RPM, we shut  off the compressed air sypply and continue to perform data  acquisition until the turbine becomes stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' We continue  the similar process for a uni-directional outlet case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' We  observe from figure 4 that the turbine with a uni-directional  outlet accelerates faster than the turbine with a bi-directional  outlet configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' In addition, for a supply pressure of  6 bar, we achieve a maximum RPM of 13019 and 11124  for uni and bi-directional outlet configurations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  Figure 4 shows the power variation due to accelerating  and braking torque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' There is an increment of ≈ 38% in  power due to accelerating torque for a uni-directional outlet  configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  We integrate the turbine with a 100 W class Fedus  RS-775 DC electric motor to measure the electrical power  output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' However, the motor is used as a generator to measure  the output voltage and current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' The electrical circuit is  attached to a rheostat, and the experiment is performed with  2  t (sec)  0  15  30  45  RPM  0  5000  10000  15000  RPM  0  4500  9000  13500  Power (Watts)  0  40  80  120  (a)  (b)  Figure 4: Distribution of (a) RPM with t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' (b) Power  due to accelerating torque with RPM for bi-directional  (dashed line) and uni-directional (solid line) outlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' Dot-  ted line represents the power due to braking torque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  the 5-ohm load resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' The figure 5 illustrates the motor’s  voltage and power variation at various RPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' The turbine-  generator can produce a maximum of 78 watts of electrical  power at 7800 RPM for bi-directional outlet configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  RPM  0  4000  8000  PowerE (Watts)  0  40  80  80  0  40  Voltage (Volts)  Figure 5: Experimental results of voltage (dashed line)  and electrical power (solid line) at different RPM  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' Inlet design  Designing the inlet is the most crucial part of the turbine  where the maximum loss occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' Notably, the compressor  and back pressure at the inlet rotor connection point regu-  lates the flow across the nozzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' In the present article, we  only consider the inlet section for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' The details of  the nozzle dimensions are in figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  1) Numerical methodology and grid Grid sensitivity:  The Inlet section of the nozzle is a pressure inlet boundary  where the total pressure is at 6 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' We consider the outlet  of the nozzle is at 4 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' The temperature at the compressor  discharge and the inlet-rotor junction are 450K and 400K,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' The surface of the nozzle is a no-slip type  boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' Considering these boundary conditions, we solve  governing compressible Reynolds-averaged Navier-Stokes  equations using the K − ω SST turbulence model in the  commercial CFD package Fluent 2021 R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' The numerical  domain is discretized using the body-fitted tetrahedral grids,  maintaining a wall y+ ≈ O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' The table 1 below presents  the grid sensitivity study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' As the desired output shows a  Figure 6: Schematic diagram of the nozzles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' The dotted,  dashed and solid lines represent nozzle 1, nozzle 2 and  nozzle 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  variation ≤ 2%, We conduct the subsequent numerical  simulations using grids with ≈ 1M elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' The inlet  Table 1: Grid sensitivity of the three inlet configurations  Grid type  Elements  Outlet area averaged Mach no  Nozzle 1  Nozzle 2  Nozzle 3  Coarse  ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='5 M  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='5093  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='4993  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='5296  Medium  ≈ 1 M  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='5089  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='4979  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='5289  Fine  ≈ 2 M  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='5080  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='4971  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='5284  design presented in this article is based on a two-pronged  objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' (a) To Maximize the Mach number peak (b) To  minimize the disparity in the Mach number peaks through  every slit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' To understand the loss mechanism and the flow  behavior, we conduct a series of numerical simulations  Where Nozzle 1 is the replica of the inlet nozzle considered  for the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' Due to the abrupt divergent section and  the presence of a large recirculation zone enclosed by a  strong shear layer, we detect significant losses in Mach no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  The dividing streamlines seen in figure 7 (a) are considered  when designing the following two inlet systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' The second  nozzle we design is to eliminate the effect of recirculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  We gradually increase the inlet nozzle area until we reach  the section where the flow bends in the previous observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  Despite the improvement in the average peak Mach number  observed from figure 7 (b), the disparity in the discharge  Mach number through the slits increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' We consider  designing the third nozzle as a converging-diverging type  nozzle where we place the throat section at x/L ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='6 to  provide sufficient scope for flow to bend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' Figure 8 represents  the peak discharge Mach number through the nozzles and  the associated % disparity from the mean peak Mach no  through each slit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' The comparison shows that the third nozzle  offers maximum peak Mach number along with minimum  % deviation from the mean peak Mach no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' It is necessary  to note that the differential pressure between the upstream  and downstream sections, the fluid, and the fluid’s thermo-  physical characteristics significantly influence the nozzle  design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' The nozzle’s optimal size and shape might differ  depending on these conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  3  10mm  40 mm  4mm  8  40  ww  20mm  mm  mm  15  mm  2  60mm(a)  (b)  (c)  Figure 7: Distribution of the Mach number for (a) nozzle  1 (b) nozzle 2 (c) nozzle 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  CONCLUSIONS  The article presents a preliminary experimental investiga-  tion of a centimeter scaled Tesla turbine using compressed  air as a working medium and suggests an improved inlet  design that could deliver higher achievable power compared  to the inlet nozzle considered for this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' The key  findings of the article are as follows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  1)  Experimental investigation of the turbine conducted  for uni and bi-directional outlet configuration at 6  bar inlet pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  2)  uni-directional outlet configuration offered a peak  power output of ≈ 110 watts at ≈ 7000 RPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  Whereas, bi-directional outlet configuration a peak  output of 80 watts at 6500 RPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  3)  The turbine-generator configuration produced a  peak electrical power output of 78 watts at 7800  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='01  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='02  M  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content="8  (a)  %M/  20  10  0  10  20  %M'  30  20  10  0  10  20  %M'  15  10  5  0  5  (b)  (c)  (d)  Figure 8: (a) Comparison of peak Mach number across  the slits." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' The dotted, dashed and solid lines represent  First, second and third nozzle, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' (b, c, d) The  % disparity in peak Mach no at discharge for three  nozzles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  RPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  4)  While assessing the inlet, we observe that nozzle 2  offers better peak Mach number than nozzle 1, but  due to the losses accounted by the sharp bend, the  disparity in % deviation of the peak mach numbers  from mean peak Mach no is nearly 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  5)  Nozzle 3 performs the best among all three nozzles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  It offers a maximum peak Mach no with minimum  % deviation from mean peak Mach no is reduced  to 12%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  The turbine’s efficiency depends on several factors, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=',  RPM, disc gap, rotor radius, rotor to casing clearance,  outlet configurations, fluid properties, and many more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  Designing an efficient Tesla turbine could bring down the  cost of a turbine substantially as compared to the other  existing technologies because of its simplistic design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' The  above findings presented in this article could be helpful in  the design improvement, thus leaving a plausible scope for  further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  NOMENCLATURE  t  time  [sec]  x/L  Non-dimensional-axial location  [–]  k  Turbulent kinetic energy  [J/kg]  ω  Specific dissipation rate  [1/sec]  M  Mach no  –  M ′  % Deviation from peak mean Mach no  –  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='650.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='85REFERENCES  [1]  James Hal Armstrong, An investigation of the performance of a  modified tesla turbine, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' thesis, Georgia Institute of Technology,  1952.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='  [2]  John Spencer Caldwell, The efficiency of a viscous flow compressor,  Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
+page_content=' thesis, Georgia Institute of Technology, 1973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'}
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atAyT4oBgHgl3EQfv_nZ/content/tmp_files/2301.00642v1.pdf.txt

@@ -0,0 +1,1820 @@
+arXiv:2301.00642v1  [math.CA]  30 Dec 2022
+SPECIAL POLYNOMIALS AND NEW REAL-ROOTEDNESS RESULTS
+AURELIEN GRIBINSKI
+EPFL
+Abstract. In this paper, we show that for some orthogonal polynomials P (z)
+n (x) showing up
+in physics, namely Laguerre and Gegenbauer, P (z)
+n (x) are realrooted in z for x in the support of
+orthogonality. As an application we show realrootedness in x and interlacing properties of ∂k
+z P (z)
+n (x)
+for k ≤ n for z > 0.
+1.
+Introduction
+1
+2.
+General strategy
+1
+3.
+Laguerre polynomials
+2
+4.
+Gegenbauer polynomials
+7
+5.
+Applications to realrootedness in x
+13
+References
+16
+1. Introduction
+Orthogonal polynomials like generalized Laguerre and Gegenbauer polynomials have long been
+studied, and show up in all fields of maths and physics. However little has been said about the
+properties of such polynomials when we vary the underlying parameter (see [1]). We study families
+of generalized orthogonal polynomials P (z)
+n (x) depending on a parameter z from a new angle, that
+is, as bivariate polynomials Pn(x, z). We fix the usual variable x and consider them instead as
+polynomials in z. We show that they are real-rooted in z for x in the support of the underlying
+measure of orthogonality, and monotonous. Furthermore we show that when we differentiate these
+orthogonal polynomials with respect to z > 0 and consider them as polynomials in x, then they are
+realrooted in x. Such polynomials (derivatives with respect to the parameter) seem to have many
+nice properties similar to their corresponding orthogonal polynomials from which they are derived
+and yet have never been studied.
+2. General strategy
+Consider P (z)
+n (x), a family of polynomials depending on a parameter z. We want to show that they
+are real-rooted in z for a fixed x in a given interval.
+The strategy is as follows
+• We check that for x at one of the extreme point of the interval it is real-rooted.
+• We show that locally around this extreme point the roots in z are all monotonous.
+• We show that there can’t be a shared zero in z for ∂xPn(x, z) and Pn(x, z) or equivalently
+for Pn−1(x, z) and Pn(x, z).
+• The roots in z are therefore monotonous when they are well-defined.
+Date: January 3, 2023.
+1
+
+2
+AURELIEN GRIBINSKI EPFL
+• We show that the roots in z of Pn(x, z) and simultaneously Pn−1(x, z) and ∂xPn(x, z) inter-
+lace as long as they are well-defined.
+• We extend the local properties by exhibiting an ODE to which all roots in z are solutions
+and show that there has to be explosion at the other extreme point of the interval, all roots
+evolving monotonously along the way.
+First we derive a way to locally prove the existence of rel-rootedness if it is known at some extreme
+point of the interval.
+Lemma 2.1 (Local existence of the roots and smoothness). Take a ∈ R. Assume P(x, z) is a
+bivariate polynomial such that P(a, z) is realrooted in z of degree j and has only simple roots in
+z (let’s call them zi(a) for i = 1....j). Assume that P(x, z) has degree less than j for all x.Then
+for x in a neighborhood of a, P(x, z) is also realrooted in z of degree j with simple roots zi(x).
+Furthermore, the roots zi(x) are C∞ functions of x.
+Proof. Consider the equation P(x, z) = 0, around the points
+�
+a, zi(a)
+�
+. We have ∂zP(x, z)|x=a,z=zi(a) ̸=
+0 as the roots are simple (can’t be a root of the derivative in z). Then using the implicit function
+theorem, we can find in the neighborhood of each point a C∞ function zi(x) which will be the only
+solution to the equation P(x, z) = 0 on this neighborhood. Therefore we have found j roots, and
+it is the maximal number of roots for a fixed x, as the polynomial is of degree less than j.
+□
+3. Laguerre polynomials
+Let L(z)
+n (x) be the Laguerre polynomials with complex parameter z. It is a polynomial in R[x, z].
+Theorem 3.1. For fixed x0 ∈ [0, +∞[, L(z)
+n (x0) is a real-rooted polynomial in z of degree n. Fur-
+thermore, its roots in z are strictly increasing to +∞ when x0 moves along [0, +∞[.
+Proof. Let’s recall the hypergeometric confluent definition
+L(z)
+n (x) =
+�n + z
+n
+�
+M(−n, z + 1, x)
+=
+n
+�
+l=1
+z + n + 1 − l
+l
+n
+�
+k=0
+(−1)kn!
+(n − k)! �k−1
+l=0 (z + 1 + k)
+xk
+=
+n
+�
+k=0
+(−1)k �n
+l=1(z + l)
+(n − k)! �k
+l=1(z + l)
+xk
+=
+n
+�
+k=0
+(−1)k �n
+j=k+1(z + j)
+(n − k)!
+xk
+= 1
+n!zn +
+��n
+j=1 j
+n!
+−
+x
+(n − 1)!
+�
+zn−1 + Rn−2(z, x)
+where Rn−2(z, x) is of degree lower than n−2 in z. We will henceforth write Ln(x, z) as it is clearly
+a bivariate polynomial of degree n in z with front coefficient
+1
+n!.
+Let’s furthermore decompose
+Ln(x, z) using a priori complex roots λi(x):
+Ln(x, z) = 1
+n!
+n
+�
+i=1
+�
+z − λi(x)
+�
+where we order the roots by decreasing module: |λ1(x)| ≥ |λ2(x)| ≥ ... ≥ |λn(x)|.
+Lemma 3.2 (Local realrootedness). Ln(x, z) is real rooted of degree n in z with simple roots in a
+neighborhood of x = 0.
+
+SPECIAL POLYNOMIALS AND NEW REAL-ROOTEDNESS RESULTS
+3
+Proof. We have
+Ln(0, z) =
+�n
+l=1(z + l)
+n!
+so that we can apply Lemma 2.1 with a = 0.
+□
+Lemma 3.3 (Local increasing property). The roots of Ln(x, z) in z are all strictly increasing when
+x is, in a neighborhood of x = 0.
+Proof. To prove this, we need some information on the derivatives with respect to x of the roots in
+the neighborhood of 0, which we know are going to be real by the previous lemma.
+Lemma 3.4. dlλi(x)
+dxl
+|x=0 = 0 for 1 ≤ l < i, and diλi(x)
+dxi
+|x=0 > 0.
+Proof. We get, for all 1 ≤ l ≤ n,
+∂l
+xLn(x, z)|x=0 = l!(−1)l
+�n
+j=l+1(z + j)
+(n − l)!
+Notice that λi(0) = −i for i = 1...n, so we see that for l ≤ i − 1,
+∂l
+xLn
+�
+0, λi(0)
+�
+= l!(−1)l
+�n
+j=l+1(λi(0) + j)
+(n − l)!
+= 0
+And
+∂i
+xLn
+�
+0, λi(0)
+�
+= i!(−1)i
+�n
+j=i+1(λi(0) + j)
+(n − i)!
+= i!(−1)i
+On the other hand,
+∂zLn
+�
+0, λi(0)
+�
+= 1
+n!
+n
+�
+l=1,l̸=i
+(−i + l) = i!(−1)i−1(n − i)!
+n!
+Now we have Ln
+�
+x, λi(x)
+�
+= 0 for all i = 1...n, by definition, so differentiating with respect to x,
+we get
+dλi(x)
+dx
+= −∂xLn
+∂zLn
+�
+x, λi(x)
+�
+Note that the denominator is nonzero as the roots in z are simple at 0 ( so they won’t be roots of
+the derivative in z). Using Leibniz’s formula and induction on l, we get for i > l ≥ 1,
+dlλi(x)
+dxl
+|x=0
+= 0
+And
+diλi(x)
+dxi
+|x=0
+= −∂i
+xLn
+∂zLn
+�
+− 1, λi(0)
+�
+=
+n!
+(n − i)! > 0
+□
+We conclude by a Taylor expansion around 0 as
+λi(x) = −i + xi
+i!
+n!
+(n − i)! + o(xi)
+□
+Lemma 3.5 (Distinct roots, degree and derivative wise). Assume λi(x) is real for x ∈]0, bi[, bi > 0.
+Then ∂xLn(x, λi(x)) can’t be zero for x ∈]0, bi[. Therefore it has a constant sign on this interval.
+Equivalently, Ln−1(x, λi(x)) can’t be zero either: that is we can’t have a nontrivial shared real root
+for Ln−1(x, z) and Ln(x, z).
+
+4
+AURELIEN GRIBINSKI EPFL
+Proof. Traditional results on simplicity of the roots can’t be used because they are true only for
+z ≥ 0. By definition, Ln
+�
+x, λi(x)
+�
+= 0. Then the usual differential euqation still holds
+(1)
+x∂xLn(x, z) = nLn(x, z) − (n + z)Ln−1(x, z)
+Let’s assume by contradiction that ∂xLn
+�
+x0, λi(x0)
+�
+= 0 for some i and x0 ∈]0, bi[. As ∂xLn(x, λi(x))
+is nonzero in a neighborhood of x = 0, x > 0(local monotonicity above), then we can assume x0 is
+the smallest x > 0 such that ∂xLn
+�
+x, λi(x)
+�
+= 0. Therefore as
+dλi(x)
+dx
+= −∂xLn
+∂zLn
+�
+x, λi(x)
+�
+we have that on ]0, x0], λi(x) is strictly increasing in x. As λi(0) ≥ −n = λn(0) for all i, then
+�
+n − λi(x0)
+�
+> 0, and we get that the statement is equivalent to Ln−1
+�
+x0, λi(x0)
+�
+= 0 using
+Equation 1. But then, as the following recurrence relations are still valid
+(n + 1)Ln+1(x, z) = (−x + 2(n + 1) + z)Ln(x, z) − (n + z)Ln−1(x, z)
+we also get by induction Ln+k(x0, λi(x0) + 1) = 0 for all k ∈ N. Using then the equality
+∂xLn+k(x, z) = −Ln+k−1(x, z + 1)
+we get that Ln+k−1(x0, λi(x0) + 1) = 0 for all k ∈ N.
+Using induction, applying successively
+the previous recurrence equations, we then get that Ln+k−1(x0, λi(x0) + j) = 0 for all j ∈ N.
+For j ≤ n, the polynomials Ln+k−1(x, λi(x) + j) are standard Laguerre polynomials (positive
+parameter). It would mean that successive Laguerre polynomials of parameter λi(x0) + j have
+the root x0 in common, so that their derivatives share this root too, which is absurd as the roots
+of Laguerre polynomials are simple by classical orthogonality.
+We conclude that Ln−1(x, λi(x))
+as well as ∂xLn
+�
+x0, λi(x0)
+�
+can’t be zero, and therefore ∂xLn
+�
+x0, λi(x0)
+�
+has a constant sign for
+x ∈]0, bi[.
+□
+Corollary 3.6 (Extended monotonicity). Assume λi(x) is real for x ∈]0, bi[, bi > 0. Then it
+follows from the previous proof that for x ∈]0, bi[
+dλi(x)
+dx
+> 0
+Theorem 3.7 (Interlacing roots, degreewise, simple roots). Consider an interval I = [0, b[ such
+that Ln(x, z) has real roots in z on I, then the same will be true of Ln−1(x, z) and their roots
+interlace. Furthermore, the interlacing is strict for x > 0 and both polynomials have simple roots
+on I.
+Proof. Let’s write
+Ln(x, z) = 1
+n!
+n
+�
+i=1
+�
+z − λn
+i (x)
+�
+Ln−1(x, z) =
+1
+(n − 1)!
+n−1
+�
+i=1
+�
+z − λn−1
+i
+(x)
+�
+and show that all x ∈ I, all i ≤ n − 1:
+λn
+i (x) ≥ λn−1
+i
+(x) ≥ λn
+i+1(x)
+with strict inequalities for x > 0. We first check the property locally, that is a neighborhood of 0.
+λn
+i (0) = −i
+λn−1
+i
+(0) = −i
+λn
+i+1(0) = −(i + 1)
+
+SPECIAL POLYNOMIALS AND NEW REAL-ROOTEDNESS RESULTS
+5
+We have
+diλn
+i (x)
+dxi
+|x=0
+=
+n!
+(n − i)!
+=
+n
+n − i
+(n − 1)!
+(n − 1 − i)!
+=
+n
+n − i
+diλn−1
+i
+(x)
+dxi
+|x=0
+As
+n
+n−i > 1, we conclude that for all i ≤ n − 1, diλn
+i (x)
+dxi
+|x=0 > diλn−1
+i
+(x)
+dxi
+|x=0.
+We can then do a Taylor expansion around x = 0:
+λn
+i (x) = −i + (x + 1)i
+i!
+diλn
+i (x)
+dxi
+|x=−1
++ o((x + 1)i)
+λn−1
+i
+(x) = −i + (x + 1)i
+i!
+diλn−1
+i
+(x)
+dxi
+|x=−1
++ o((x + 1)i)
+It is then clear that in a neighborhood of 0, λn
+i (x) > λn−1
+i
+(x).
+As λn−1
+i
+(−1) − λn
+i+1(−1) = 1, we also get λn−1
+i
+(x) > λn
+i+1(x) in a neighborhood of 0. Now as
+for all i λn
+i (x), λn−1
+i
+(x), λn
+i+1(x) are continuous functions of x, if by contradiction such inequalities
+where to fail for some x ∈ I, then there would exist x0 such that λn
+i (x0) = λn−1
+i
+(x0) or λn−1
+i
+(x0) =
+λn
+i+1(x0).But then this would mean that λn−1
+i
+(x0) is a root of Gn(x0, z) and Gn−1(x0, z), which is
+impossible by Lemma 4.5. Therefore we conclude that the inequality
+λn
+i (x) > λn−1
+i
+(x) > λn
+i+1(x)
+holds for allx ∈ I, x > 0 and i ≤ n − 1.
+□
+Theorem 3.8 (Interlacing roots, derivative). Consider an interval I = [0, b[ such that Ln(x, z) has
+real roots in z on I, then the same will be true of ∂xLn(x, z) and the roots of the two polynomials
+interlace and are simple.
+Proof. We bring ourselves back to a variant of the previous theorem by using the equality
+∂xLn(x, z) = −Ln−1(x, z + 1)
+The roots being simple results from Lemma 3.7. So it amounts to proving that Ln−1(x, z + 1) and
+Ln(x, z) interlace. We want to show more precisely that for all x ∈ I, with x > 0, all i ≤ n − 1
+λn
+i (x) ≥ λn−1
+i
+(x) − 1 ≥ λn
+i+1(x)
+With strict inequalities for x > 0. First we check such inequalities in a neighborhood of 0. We
+can check that the inequality λn
+i (x) > λn−1
+i
+(x) − 1 is going to be true in a neighborhood of 0 as
+λn
+i (0) = λn−1
+i
+(0). So the nontrivial one is the other one, λn−1
+i
+(x) − 1 > λn
+i+1(x) for x > 0. We have
+equality at the origin as λi(0) − 1 = λi+1(0). Then we look at the Taylor expansions around x = 0:
+λn−1
+i
+(x) − 1 = λi+1(0) + (x + 1)i
+i!
+diλn−1
+i
+(x)
+dxi
+|x=0
++ o((x + 1)i)
+λn
+i+1(x) = λi+1(0) + (x + 1)i+1
+(i + 1)!
+di+1λn
+i+1(x)
+dxi+1
+|x=0
++ o((x + 1)i+1)
+It is clear then that locally λn−1
+i
+(x) − 1 > λn
+i+1(x) as (x + 1)i+1 << (x + 1)i. We extend the
+inequality to the whole interval I by noticing again that if the inequalities where not valid anymore,
+then there would have to be some equality λn
+i (x) = λn−1
+i
+(x) − 1 or λn−1
+i
+(x) − 1 = λn
+i+1(x), which
+would mean ∂xLn(x, λn−1
+i
+(x)) = 0 and as Ln(x, λn−1
+i
+(x)) = 0, we would again get a contradiction
+by Lemma 4.5.
+□
+
+6
+AURELIEN GRIBINSKI EPFL
+Lemma 3.9 (Global extension through ODE). The local property is in fact true over the whole
+interval: Ln(x, z) is real rooted in z with simple (distinct) roots for x ∈ [0, +∞[ , and the roots are
+all increasing to +∞ when x goes to +∞.
+Proof. Denote by Fn(x, z) := − ∂xLn
+∂zLn
+�
+x, z
+�
+. Consider a rectangular domain D such that ∂zLn(x, z)
+is nonzero on the domain. Fn is continuous in x and z in the the domain D. Indeed, it is a rational
+fraction whose denominator is nonzero and it is therefore C∞ in both variables by theorem of
+composition. As Ln(0, z) is realrooted in z with simple roots, ∂zLn
+�
+0, λi(0)
+�
+̸= 0 and by continuity
+we can find small rectangles Di := [0, ǫ] × [λi(0) − δ, λi(0) + δ] such that Ln(x, z) is nonzero on
+Di. A strong version of Picard’s theorem tells us that there is a maximal interval Imax
+i
+= [0, ηi
+max[
+(where ηi
+max ∈
+¯
+R+) for which the roots λi(x) (i = 1, 2...n) are the unique solutions of the initial
+value ODE
+dz
+dx = Fn(x, z),
+z(−1) = −i
+Note that on Imax
+i
+, ∂zLn
+�
+x, λi(x)
+�
+̸= 0 (the denominator is nonzero, so that the differential equa-
+tion is well defined).
+Let’s prove that Imax
+i
+= [0, +∞[ (for all i) and that there is explosion at +∞ (roots going to
+infinity), the roots increasing constantly to +∞.
+By Corollary 3.6, Fn(x, λi(x)) > 0 on Imax
+i
+.
+According to Picard’s theorem, we either have
+λi(x) →x→ηimax +∞ (explosion), or ηi
+max is such that limx→ηimax Fn(x, λi(x)) is not well defined
+(we leave the domain of definition).
+Now, explosion can’t happen if ηi
+max < +∞. Indeed, we have using the hypergeometric expansion
+above
+n
+�
+i=1
+λi(x) = −n!
+�n(n + 1)
+2
+1
+n! −
+x
+(n − 1)!
+�
+= n
+�
+− n + 1
+2
++ x
+�
+so the sum of roots is bounded above for x < ηi
+max and there can be no explosion (necessarily to+∞
+by monotonicity).
+We can leave the domain of definition only if limx→ηimax ∂zLn
+�
+x, λi(x)
+�
+= 0. If this is the case
+and if by contradiction ηi
+max < +∞, ∂zLn(ηi
+max, z) would be of degree n in z.
+Therefore it
+means that limx→ηimax λi(x) = µ where µ is a root of ∂zLn(ηi
+max, z).
+But then it means that
+we can extend by continuity λi(x) at x = ηi
+max with λi(ηi
+max) = µ. We check by continuity that
+Ln
+�
+ηi
+max, λi(ηi
+max)
+�
+= ∂zLn
+�
+ηi
+max, λi(ηi
+max)
+�
+= 0 so that in fact λi(ηi
+max) is a real double root in z
+of Ln
+�
+ηi
+max, z
+�
+. Using Lemma 3.8, as there is a root of ∂xLn(x, z) between any two roots of Ln(x, z)
+in z by interlacing, it follows that necessarily ∂xLn
+�
+ηi
+max, λi(ηmax)
+�
+= 0. But this is impossible
+according to Lemma 4.5. Therefore, we have necessarily ηi
+max = +∞ for all i = 1...n.
+Furthermore, assume by contradiction that there is no explosion for some index i at +∞. As λi(x)
+is monotonous for x ∈ [0, +∞[, then we have necessarily that limx→+∞ λi(x) = µ exists and is
+finite. By continuity we have Ln(x, µ) ∼x→∞ (−1)nxn, and ∂xLn(x, µ) ∼x→∞ n(−1)nxn−1, as well
+as ∂zLn(x, µ) ∼x→∞ (−1)n−1xn−1 using
+L(z)
+n (x) =
+n
+�
+k=0
+(−1)k �n
+j=k+1(z + j)
+(n − k)!
+xk
+so that
+dλi(x)
+dx
+→x→+∞ n
+and clearly we would have λi(x) → +∞, which is a contradiction.
+
+SPECIAL POLYNOMIALS AND NEW REAL-ROOTEDNESS RESULTS
+7
+□
+□
+4. Gegenbauer polynomials
+Let G(z)
+n (x) be the Gegenbauer polynomial with complex parameter z. It is a polynomial in R[x, z].
+Theorem 4.1. For fixed x0 ∈ [−1, 1], G(z)
+n (x0) is a real-rooted polynomial in z of degree at most n
+(exactly n except for x0 = 0). Furthermore, its roots in z then they are increasing for x ∈ [−1, 0[,
+and decreasing for x ∈]0, 1], with an explosion to infinity at 0.
+Proof. As the Gegenbauer polynomials are even or odd in x, that is G(z)
+n (−x) = (−1)nG(z)
+n (x), it is
+enough to prove our statement for x ∈ [−1, 0[. We prove it in an incremental way moving x from
+−1 to 0. Let’s recall the hypergeometric expression:
+G(z)
+n (x) =
+n−1
+�
+l=0
+(2z + l)
+n
+�
+k=0
+(−1)k
+1
+k!(n − k)!
+�k−1
+i=0 (2z + n + i)
+�k−1
+i=0 (z + 1/2 + i)2k (1 − x)k
+=
+n
+�
+k=0
+(−1)k 2n�n
+k
+�
+n!
+�n+k−1
+i=0
+(z + i/2)
+�k−1
+i=0 (z + (2i + 1)/2)
+(1 − x)k
+=
+n
+�
+k=0
+2n�n
+k
+�
+n!
+n+k−1
+�
+i=2k
+(z + i/2)
+k−1
+�
+i=0
+(z + i)(x − 1)k
+We will henceforth write Gn(x, z) as it is clear from the previous expression that it is indeed a
+bivariate polynomial and not a rational fraction in z.
+We start dealing with the extreme boundary. We have
+(2)
+Gn(x, z) = (−1)nGn(−x, z) =
+n
+�
+k=0
+2n�n
+k
+�
+n!
+(−1)n+k
+n+k−1
+�
+i=2k
+(z + i/2)
+k−1
+�
+i=0
+(z + i)(x + 1)k
+So that Gn(−1, z) = (−1)n 2n
+n!
+�n−1
+i=0 (z +i/2), which is clearly realrooted in z with simple roots. We
+can also check this property for x = 0:
+Gn(0, z) =
+Γ(n/2 + z)
+Γ(z)Γ(n/2 + 1) =
+1
+(n/2)!
+j=n/2−1
+�
+j=0
+(z + j)
+when n is even
+Gn(0, z) = 0
+when n is odd
+Also, each bivariate polynomial in the sum is of degree n in z so the sum is of degree at most n in
+z. It is in fact of degree exactly n for x ̸= 0 by inspection of the coefficient of zn in the sum, which
+is equal to
+(−1)n 2n
+n!
+n
+�
+k=0
+�n
+k
+�
+(−1)k(1 + x)k = (−1)n 2n
+n!
+�
+1 − (1 + x)
+�n = (−1)n 2n
+n! (−x)n = (2x)n
+n!
+Notice that we can write, if n is even,
+Gn(x, z) =
+� n/2−1
+�
+j=0
+(z + j)
+�
+˜
+Gn(x, z)
+and if n is odd,
+Gn(x, z) =
+� (n−1)/2
+�
+j=0
+(z + j)
+�
+˜
+Gn(x, z)
+
+8
+AURELIEN GRIBINSKI EPFL
+So that in all cases for x ̸= 0, we can write
+Gn(x, z) =
+� ⌈n/2⌉−1
+�
+j=0
+(z + j)
+�
+˜
+Gn(x, z)
+=
+� ⌈n/2⌉−1
+�
+j=0
+(z − µj)
+�(2x)n
+n!
+⌊n/2⌋
+�
+i=1
+�
+z − λi(x)
+�
+where µj = −j and the λi(x) are a priori complex roots defined only forx ̸= 0. But it simplifies
+greatly for x = −1:
+˜
+Gn(x, z)|x=−1 = 2n
+n! (−1)n
+⌊n/2⌋
+�
+i=1
+(z + 1/2 + i − 1)
+so that λi(−1) = −1/2−(i−1) for i = 1, 2...⌊n/2⌋ are all real and distinct. That is, approximately
+half of the roots in z, depending on the oddness, are constant when x is moving. We can therefore
+investigate instead the evolution of the roots of Ln(x, z) to avoid considering roots that remain
+constant (and real). First, let’s explain why roots will be indeed smooth and real for x close to −1.
+Corollary 4.2 (Local realrootedness).
+˜
+Gn(x, z) is real rooted of degree ⌊n/2⌋ in z with simple roots
+in a neighborhood of x = −1.
+Proof. We have
+˜
+Gn(x, z)|x=−1 = 2n
+n! (−1)n
+⌊n/2⌋
+�
+i=1
+(z + 1/2 + i − 1)
+which has ⌈n/2⌉ simple roots in z, so we just have to apply Lemma 2.1 with a = −1.
+□
+Lemma 4.3 (Local increasing property). The roots of ˜
+Gn(x, z) in z are all strictly increasing when
+x is, in a neighborhood of x = −1.
+To prove this, we need some information on the derivatives with respect to x of the roots in the
+neighborhood of −1.
+Lemma 4.4. If we denote by λi(x) the roots of ˜
+Gn(x, z) in decreasing order, then dlλi(x)
+dxl
+|x=−1 = 0
+for 1 ≤ l < i, and diλi(x)
+dxi
+|x=−1 > 0.
+Proof. Using Equation 2 we get that
+∂l
+xGn(x, z)|x=−1 = l!2n�n
+l
+�
+n!
+(−1)n+l
+n+l−1
+�
+j=2l
+(z + j/2)
+l−1
+�
+j=0
+(z + j)
+So that
+∂l
+x ˜
+Gn(x, z)|x=−1 = 2n�n
+l
+�
+l!
+n!
+(−1)n+l
+⌈n/2⌉−1
+�
+j=l
+(z + 1/2 + j)
+n+l−1
+�
+j=n+1
+(z + j/2)
+So we see that
+∂l
+x ˜
+Gn(x, z)
+�
+− 1, λi(−1)
+�
+= 0 for i ≥ l + 1
+and
+(−1)n+i∂i
+x ˜
+Gn(x, z)
+�
+− 1, λi(−1)
+�
+= 2n�n
+i
+�
+i!
+n!
+⌈n/2⌉−1
+�
+j=i
+�
+j − (i − 1)
+� n+i−1
+�
+j=n+1
+�
+(j − 1)/2 − (i − 1)
+�
+> 0
+as i ≤ ⌊n/2⌋.
+
+SPECIAL POLYNOMIALS AND NEW REAL-ROOTEDNESS RESULTS
+9
+Now we have ˜
+Gn
+�
+x, λi(x)
+�
+= 0 for all i, by definition, so differentiating with respect to x, we get:
+dλi(x)
+dx
+= −∂x ˜
+Gn
+∂z ˜
+Gn
+�
+x, λi(x)
+�
+Note that the denominator is nonzero as the roots in z are simple at −1 ( so they won’t be roots
+of the derivative in z). Using Leibniz’s formula and induction on l, we get for i > l ≥ 1,
+dlλi(x)
+dxl
+|x=−1
+= 0
+And
+diλi(x)
+dxi
+|x=−1
+= −∂i
+x ˜
+Gn
+∂z ˜
+Gn
+�
+− 1, λi(−1)
+�
+Now, we have ˜
+Gn(x, z)|x=−1 = (−1)n 2n
+n!
+�⌈n/2⌉−1
+j=0
+(z + 1/2 + j) so that
+∂z ˜
+Gn(x, z)
+�
+−1, λi(−1)
+�
+= (−1)n 2n
+n!
+⌈n/2⌉−1
+�
+j=0,j̸=i−1
+�
+j−(i−1)
+�
+= (−1)n 2n
+n! (−1)i−1
+i−2
+�
+j=0
+�
+(i−1)−j
+� ⌈n/2⌉−1
+�
+j=i
+�
+j−(i−1)
+�
+and it follows that (−1)n+i−1∂z ˜
+Gn(x, z)
+�
+− 1, λi(−1)
+�
+> 0. Therefore
+diλi(x)
+dxi
+|x=−1
+> 0
+as claimed.
+□
+Then Lemma 4.3 follows easily by Taylor expansion of the roots in x around 0 , as by some
+asymptotic expansion at x = −1,
+λi(x) = λi(−1) + (x + 1)i
+i!
+diλi(x)
+dxi
+|x=−1
++ o((x + 1)i)
+We have proved the ”initial condition” : now we need to look at the evolution of roots from a
+differential equation point of view. First, let’s prove some intermediate results.
+Lemma 4.5 (Simple roots). Assume λi(x) is real for x ∈] − 1, bi[, bi ≤ 0 ( such a bi > −1 exists
+according to the previous local existence). Then ∂x ˜
+Gn(x, λi(x)) and a fortiori ∂xGn(x, λi(x)) ( for
+i = 1, 2...⌊n/2⌋) can’t be zero for x ∈] − 1, bi[. Therefore it has a constant sign on this interval.
+Equivalently, Gn−1(x, λi(x)) can’t be zero either: that is we can’t have a nontrivial shared root for
+Gn−1(x, z) and Gn(x, z).
+Proof. We have ∂xGn(x, λi(x)) = �⌈n/2⌉−1
+j=0
+�
+λi(x) + j
+�
+∂x ˜
+Gn
+�
+x, λi(x)
+�
+. We can’t use directly the
+results on the monotonicity of the roots of Gegenbauer polynomials when the paremeter is mov-
+ing, or the simplicity of the roots in x, because the parameter here is negative and orthogonal-
+ity results don’t apply. Let’s assume by contradiction that ∂x ˜
+Gn(x0, λi(x0)) = 0, and therefore
+∂xGn(x0, λi(x0)) = 0 for some i and x0 ∈] − 1, bi[. As ∂xGn(x, λi(x)) is nonzero in a neighborhood
+of x = −1, x > −1(local monotonicity), then we can assume x0 is the smallest x > −1 such that
+∂xGn(x, λi(x)) = 0. Therefore on ] − 1, x0], λi(x) is strictly increasing in x because
+dλi(x)
+dx
+= −∂x ˜
+Gn
+∂z ˜
+Gn
+�
+x, λi(x)
+�
+As λi(−1) ≥ −(n − 1)/2 for all i, then (n + 2λi(x0) − 1) > 0, and using the differential equation
+(1 − x2
+0)∂xGn(x0, λi(x0)) = −nxGn(x0, λi(x0)) + (n + 2λi(x0) − 1)Gn−1(x, λi(x0))
+
+10
+AURELIEN GRIBINSKI EPFL
+and the fact that Gn(x0, λi(x0)) = 0 (by definition), it would lead us to Gn−1(x0, λi(x0)) = 0. Then,
+using the recurrence relation (we have a fortiori 2n + 2λi(x0) > 0)
+n + 1
+2n + 2λi(x0)Gn+1(x0, λi(x0)) = xGn(x, λi(x)) − n + 2λi(x0) − 1
+2n + 2λi(x0) Gn−1(x0, λi(x0))
+we get successively by induction that Gn+k(x0, λi(x0)) = 0 for all k ∈ N, and using again the
+differential equation we get that ∂xGn+k(x0, λi(x0)) = 0 for all k ∈ N. But we have
+∂xGn+k(x0, λi(x0)) = 2λi(x0)Gn+k−1(x0, λi(x0) + 1)
+so that Gn+k−1(x0), λi(x0) + 1) = 0 for all k ∈ N. Then it is easy to show by induction that
+Gn+k−1(x0, λi(x0) + j) = 0 for all j ∈ N, and for j larger than (n − 1)/2, the parameter is
+positive, and we are brought back to classical Gegenbauer polynomials. This means that successive
+Gegenbauer polynomials with parameter λi(x0)+j have a root in common, so that their derivatives
+share these roots too, which is absurd as their roots are simple by orthogonality. We conclude that
+∂xGn(x, λi(x)) has a constant sign for all x ∈] − 1, bi[.
+□
+Theorem 4.6 (Interlacing roots, degree). Consider an interval I = [−1, b[ such that ˜
+Gn(x, z) has
+simple real roots in z on I, then the same will be true of ˜Gn−1(x, z) and their roots interlace.
+Proof. Let’s write
+˜
+Gn(x, z) = (2x)n
+n!
+⌊n/2⌋
+�
+i=1
+�
+z − λn
+i (x)
+�
+˜Gn−1(x, z) = (2x)n−1
+(n − 1)!
+⌊(n−1)/2⌋
+�
+i=1
+�
+z − λn−1
+i
+(x)
+�
+and show that for all x ∈ I, all i ≤ ⌊(n − 1)/2⌋, λn
+i (x) > λn−1
+i
+(x) > λn
+i+1(x). We first check the
+property locally, that is a neighborhood of −1, above −1. We have
+diλn
+i (x)
+dxi
+|x=−1
+= −∂i
+x ˜
+Gn
+∂z ˜
+Gn
+�
+− 1, λi(−1)
+��
+= −
+(−1)n+i 2n(n
+i)i!
+n!
+�⌈n/2⌉−1
+j=i
+�
+j − (i − 1)
+� �n+i−1
+j=n+1
+�
+(j − 1)/2 − (i − 1)
+�
+2n
+n! (−1)n+i−1 �i−2
+j=0
+�
+(i − 1) − j
+� �⌈n/2⌉−1
+j=i
+�
+j − (i − 1)
+�
+=
+�n
+i
+�
+i! �n+i−1
+j=n+1
+�
+(j − 1)/2 − (i − 1)
+�
+�i−2
+j=0
+�
+(i − 1) − j
+�
+=
+n
+n − i
+�
+(n + i − 1)/2 − (i − 1)
+�
+�
+(n − 1)/2 − (i − 1)
+�
+�n − 1
+i
+�
+i!
+�n+i−2
+j=n
+�
+(j − 1)/2 − (i − 1)
+�
+�i−2
+j=0
+�
+(i − 1) − j
+�
+=
+n
+n − i
+�
+(n + i − 1)/2 − (i − 1)
+�
+�
+(n − 1)/2 − (i − 1)
+�
+diλn−1
+i
+(x)
+dxi
+|x=−1
+As
+n
+n−i
+�
+(n+i−1)/2−(i−1)
+�
+�
+(n−1)/2−(i−1)
+�
+> 1, we conclude that for all i, diλn
+i (x)
+dxi
+|x=−1 > diλn−1
+i
+(x)
+dxi
+|x=−1.
+As λn
+i (−1) = λn−1
+i
+(−1) = λi(−1) = −1/2 − (i − 1), we can do a Taylor expansion around x = −1:
+λn
+i (x) = λi(−1) + (x + 1)i
+i!
+diλn
+i (x)
+dxi
+|x=−1
++ o((x + 1)i)
+λn−1
+i
+(x) = λi(−1) + (x + 1)i
+i!
+diλn−1
+i
+(x)
+dxi
+|x=−1
++ o((x + 1)i)
+It is then clear that in a neighborhood of −1 and above −1, λn
+i (x) > λn−1
+i
+(x). As λn−1
+i
+(−1) −
+λn
+i+1(−1) = 1, we also get λn−1
+i
+(x) > λn
+i+1(x) in a neighborhood of −1. Now as for all i λn
+i (x), λn−1
+i
+(x), λn
+i+1(x)
+are continuous functions of x, if by contradiction such inequalities where to fail for some x ∈ I,
+then there would exist x0 such that λn
+i (x0) = λn−1
+i
+(x0) or λn−1
+i
+(x0) = λn
+i+1(x0).But then this would
+
+SPECIAL POLYNOMIALS AND NEW REAL-ROOTEDNESS RESULTS
+11
+mean that λn−1
+i
+(x0) is a root of Gn(x0, z) and Gn−1(x0, z), which is impossible by Lemma 4.5.
+Therefore we conclude that the inequality
+λn
+i (x) > λn−1
+i
+(x) > λn
+i+1(x)
+holds for allx ∈ I and i ≤ ⌊(n − 1)/2⌋. Notice that according to n, the polynomial ˜Gn(x, z) can be
+of the same degree than ˜Gn−1(x, z), or of degree one more.
+□
+Theorem 4.7 (Interlacing roots, derivative). Consider an interval I = [−1, b[ such that ˜Gn(x, z)
+has simple real roots in z on I, then the same will be true of ∂x ˜Gn(x, z) and the roots of the two
+polynomials interlace.
+Proof. We bring ourselves back to a variant of the previous theorem by using the equality
+∂xGn(x, z) = 2zGn−1(x, z + 1)
+As
+∂xGn(x, z) =
+� ⌈n/2⌉−1
+�
+j=0
+(z + j)
+�
+∂x ˜Gn(x, z)
+Gn−1(x, z + 1) =
+� ⌈(n−1)/2⌉−1
+�
+j=0
+(z + j + 1)
+�
+˜Gn−1(x, z + 1)
+We get
+∂x ˜Gn(x, z) =
+�⌈(n−1)/2⌉−1
+j=0
+(z + j + 1)
+�⌈n/2⌉−1
+j=0
+(z + j)
+2z ˜Gn−1(x, z + 1)
+And
+∂x ˜Gn(x, z) = 2(z + n/2) ˜Gn−1(x, z + 1)
+if n is even
+∂x ˜
+Gn(x, z) = 2 ˜Gn−1(x, z + 1)
+if n is odd
+So as −n/2 < mini,x λi(x) for all i and x, it amounts to proving that
+˜
+Gn−1(x, z + 1) and ˜
+Gn(x, z)
+interlace.
+We want to show that for all x ∈ I, with x > −1, all i ≤ ⌊(n − 1)/2⌋, λn
+i (x) >
+λn−1
+i
+(x) − 1 > λn
+i+1(x).
+First we check this in a neighborhood of −1.
+We can check that the
+inequality λn
+i (x) > λn−1
+i
+(x) − 1 is going to be true in a neighborhood of −1 as λn
+i (−1) = λn−1
+i
+(−1).
+So the nontrivial one is the other one, λn−1
+i
+(x) − 1 > λn
+i+1(x) for x > −1. We have equality at the
+origin as λn
+i (−1) = λn−1
+i
+(−1) := λi(−1) and λi(−1) − 1 = λi+1(−1). Then we look at the Taylor
+expansions around x = −1:
+λn−1
+i
+(x) − 1 = λi+1(−1) + (x + 1)i
+i!
+diλn−1
+i
+(x)
+dxi
+|x=−1
++ o((x + 1)i)
+λn
+i+1(x) = λi+1(−1) + (x + 1)i+1
+(i + 1)!
+di+1λn
+i+1(x)
+dxi+1
+|x=−1
++ o((x + 1)i+1)
+It is clear then that locally λn−1
+i
+(x) − 1 > λn
+i+1(x) as (x + 1)i+1 << (x + 1)i. We extend the
+inequality to the whole interval I by noticing again that if the inequalities where not valid anymore,
+then there would have to be some equality λn
+i (x) = λn−1
+i
+(x) − 1 or λn−1
+i
+(x) − 1 = λn
+i+1(x), which
+would mean ∂x ˜
+Gn(x, λn−1
+i
+(x)) = 0 and as ˜
+Gn(x, λn−1
+i
+(x)) = 0, we would again get a contradiction
+by Lemma 4.5.
+□
+Lemma 4.8 (Global extension through ODE). The local property is in fact true over the whole
+interval:
+˜
+Gn(x, z) is real rooted in z with simple (distinct) roots for for x ∈ [−1, 0[ , and they are
+all increasing to +∞ when x goes to zero.
+
+12
+AURELIEN GRIBINSKI EPFL
+Proof. Denote by Fn(x, z) := − ∂x ˜
+Gn
+∂z ˜
+Gn
+�
+x, z
+�
+. Consider a rectangular domain D such that ∂z ˜
+Gn(x, z)
+is nonzero on the domain.
+Fn is continuous in x and z in the the domain D.
+Indeed, it is a
+rational fraction whose denominator is nonzero and it is therefore C∞ in both variables by theorem
+of composition. As ˜
+Gn(−1, z) is realrooted in z with simple roots, ∂z ˜
+Gn(−1, λi(−1)) ̸= 0 and by
+continuity we can find small rectangles Di := [−1, −1 + ǫ] × [λi(−1) − δ, λi(−1) + δ] such that
+∂z ˜
+Gn(x, z) is nonzero on Di. A strong version of Picard’s theorem tells us that there is a maximal
+interval Imax
+i
+= [−1, ηi
+max[ for which the roots λi(x) (i = 1, 2...⌊n/2⌋) are the unique solutions of
+the initial value ODE
+dz
+dx = Fn(x, z),
+z(−1) = −1/2 − (i − 1)
+Note that on Imax
+i
+, ∂z ˜
+Gn(x, λi(x)) ̸= 0 (the denominator is nonzero, so that the differential equa-
+tion is well defined). Let’s prove that Imax
+i
+= [−1, 0[ (for all i) and that there is explosion at 0
+(roots going to infinity), the roots increasing constantly to +∞. The local Lemma 4.3 tell us that
+on a neighborhood of −1, Fn(x, λi(x)) > 0, and as by Lemma 4.5, the numerator is of constant
+sign and the denominator doesn’t vanish, then Fn(x, λi(x)) > 0 on Imax
+i
+.
+According to Picard’s theorem, we either have λi(x) →x→ηimax +∞ (explosion), or ηi
+max is such
+that lim Fn(x, λi(x)) is not well defined (we leave the domain of definition).
+Now, explosion can’t happen if ηi
+max < 0. Indeed, we have that
+�
+i
+λi(x) +
+�
+j
+µj = −Pn−1(x)
+(2x)n
+n!
+where Pn−1(x) is a polynomial of degree n − 1 as well as the coefficient of zn−1 in the expansion
+of Gn(x, z). So the sum of roots is bounded above by a constant, so there can be no explosion
+(necessarily to+∞ by monotonicity).
+We can leave the domain of definition only if limx→ηimax ∂z ˜
+Gn
+�
+x, λi(x)
+�
+= 0. If this is the case and
+if by contradiction ηi
+max < 0, we have seen that ∂z ˜
+Gn(ηi
+max, z) would be of degree exactly ⌊n/2⌋ − 1
+in z. Therefore it means that limx→ηimax λi(x) = µ where µ is a root of ∂z ˜
+Gn(ηi
+max, z). But then
+it means that we can extend by continuity λi(x) at x = ηi
+max with λi(ηmax) = µ. We check by
+continuity that ˜
+Gn
+�
+ηi
+max, λi(ηmax)
+�
+= ∂z ˜
+Gn
+�
+ηi
+max, λi(ηmax)
+�
+= 0 so that in fact λi(ηmax) is a real
+double root in z of ˜
+Gn
+�
+ηi
+max, z
+�
+. Using Lemma 4.7, as there is a root of ∂x ˜
+Gn(x, z) between any two
+roots of ˜
+Gn(x, z) in z by interlacing, it follows that necessarily ∂x ˜
+Gn
+�
+ηi
+max, λi(ηmax)
+�
+= 0. But this
+is impossible according to Lemma 4.5. Therefore, we have necessarily ηi
+max = 0 for all i = 1...⌊n/2⌋.
+Furthermore, assume by contradiction that there is no explosion for some index i at 0. As λi(x) is
+monotonous for x ∈] − 1, 0[, then we have necessarily that limx→0 λi(x) = µ exists and is finite. By
+continuity we have ˜
+Gn(0, µ) = 0. Let’s distinguish according to the parity of n. If n is even, we
+have ˜
+Gn(0, z) =
+Gn(0,z)
+�n/2−1
+j=0
+(z+j) =
+1
+(n/2)! for all z, which shows the contradiction right away. If n is odd,
+then ∂z ˜
+Gn
+�
+x, λi(x)
+�
+= xQn
+�
+x, λi(x)
+�
+where Qn(0, z) = 2(−1)⌊n/2⌋
+(n−1)/2! . For x ∈ [−1, 0], xFn(x, λi(x)) is
+therefore bounded above as a continuous function on a compact. It is always nonpostive, and the
+maximum can’t be zero because it would mean that for an x ∈ [−1, 0], ∂xLn
+�
+x, λi(x)
+�
+= 0, which
+is impossible according to Lemma 4.5. Therefore it is always smaller than −K < 0.
+dλi(x)
+dx
+= 1
+xxFn(x, λi(x)) > −K
+x
+λi(x) − λi(−1) > −K log(|x|)
+
+SPECIAL POLYNOMIALS AND NEW REAL-ROOTEDNESS RESULTS
+13
+It would follow that λi(x) →x→0 +∞ which is contrary to the assumptions.
+We conclude that there is explosion for all i = 1...⌊n/2⌋.
+□
+□
+5. Applications to realrootedness in x
+We start by recalling a well-known monotonicity result. In all the following we will consider z > 0.
+Lemma 5.1 (Monotonicity of the roots with respect to the parameter, from [1]). If xi(z) are
+the roots of L(z)
+n (x) (Laguerre), then
+d
+dzxi(z) ≥ 0, i ≤ n. Also, the positive roots yi of G(z)
+n (x)
+(Gegenbauer) are such that
+d
+dzyi(z) ≥ 0 and the negative symmetric such that
+d
+dzyi(z) ≤ 0.
+Lemma 5.2. For a fixed z, we have that the roots of ∂zLn(x, z) in x (of degree n − 1) are real and
+interlace those of Ln(x, z).
+Proof.
+Ln(x, z) =
+n
+�
+k=0
+(−1)k �n
+j=k+1(z + j)
+(n − k)!
+xk
+(3)
+From this expression it follows that ∂zLn(x, z) is of degree n − 1 in x.
+d
+dz xi = −∂zLn
+∂xLn
+(xi(z), z)
+∂zLn(xi(z), z) = −∂xLn(xi(z), z) d
+dz xi
+(4)
+∂xLn(xi(z), z) changes sign when we increment i because Ln(xi(z), z) = 0 so the derivative changes
+sign when we go from one root to the next. As
+d
+dzxi ≥ 0, We get that ∂zLn(x, z) changes sign
+n − 1 times and therefore we have n − 1 real zeros between zeros of Ln. Therefore all the zeros of
+∂zLn(x, z) have been found and are interlacing with the zeros of Ln(x, z).
+□
+Theorem 5.3. ∂zLn(x, z) and more generally ∂k
+z Ln(x, z) for all k ≤ n are real-rooted in x, and they
+form an interlacing family of decreasing degree, in the sense that ∂k+1
+z
+Ln(x, z) interlaces ∂k
+z Ln(x, z)
+and the roots are monotonously increasing in z.
+Proof. We show inductively the following property: ∂k+1
+z
+Ln(x, z) is realrooted, the roots of ∂k+1
+z
+Ln(x, z)
+interlace the roots of ∂k
+z Ln(x, z) and are increasing in z. We start with the initial condition. Using
+5.2, we get the real-rootedness and interlacing property for ∂zLn(x, z). Now we need to prove that
+the roots ˜xi(z) (i = 1...n − 1) of ∂zLn(x, z) also share the monotonicity property.
+∂zLn( ˜xi(z), z) = 0
+Which lead to
+∂zLn
+Ln
+( ˜xi(z), z) = 0
+d(∂zLn
+Ln ( ˜xi(z), z)
+�
+dz
+= ∂x
+�∂zLn
+Ln
+( ˜xi(z), z)
+�d ˜xi
+dz + ∂z
+�∂zLn
+Ln
+( ˜xi(z), z)
+�
+= 0
+(5)
+We want to show that
+d ˜xi
+dz ≥ 0
+On the one hand,
+∂zLn
+Ln
+( ˜xi(z), z) =
+n
+�
+j=1
+∂zLn
+∂xLn
+(xj, z)
+1
+˜xi − xj
+So that
+∂x
+�∂zLn
+Ln
+( ˜xi(z), z)
+�
+= −
+n
+�
+j=1
+∂zLn
+∂xLn
+(xj, z)
+1
+( ˜xi − xj)2 =
+n
+�
+j=1
+dxj
+dz
+1
+( ˜xi − xj)2 ≥ 0
+(6)
+
+14
+AURELIEN GRIBINSKI EPFL
+On the other hand, using the real-rootedness in z, as ˜xi ∈ [0, +∞[ by the interlacing property, then
+∂z
+�∂zLn
+Ln
+( ˜xi(z), z)
+�
+≤ 0
+using Laguerre inequality for realrooted polynomials, stating that ∂zzLnLn − (∂zLn)2 ≤ 0. We
+conclude by gathering the two inequalities.
+The induction is proven using exactly the same method, given that
+∂z
+�∂k+1
+z
+Ln
+∂kz Ln
+�
+≤ 0
+Because ∂k
+z Ln(x, z) is realrooted in z as the derivative of a realrooted polynomial, and x in the
+appropriate interval.
+□
+Lemma 5.4. 0 is a root of ∂k
+z Gn(x, z) for all k ≤ n, when n is odd.
+Proof. It comes directly from the formula
+Gn(x, z) =
+⌊n/2⌋
+�
+k=0
+(−1)k Γ(n − k + z)
+Γ(z)k!(n − 2k)!(2x)n−2k
+□
+Lemma 5.5. For a fixed z > 0, we have that the roots of ∂zGn(x, z) in x (of degree n) are real
+and the positive ones interlace those of Gn(x, z) by below (that is the largest root in module belongs
+to Gn(x, z)).
+Proof.
+Gn(x, z) =
+⌊n/2⌋
+�
+k=0
+(−1)k Γ(n − k + z)
+Γ(z)k!(n − 2k)!(2x)n−2k
+(7)
+From this expression it follows that ∂zGn(x, z) is of degree n in x, as the coefficient of xn is 2n Γ(n+z)
+Γ(z)n! .
+Let’s denote the roots of Gn(x, z) by yi, then by differentiating the equality Gn(x, z) = 0 with
+respect to z we get
+d
+dz yi = −∂zGn
+∂xGn
+(yi(z), z)
+∂zGn(yi(z), z) = −∂xGn(yi(z), z) d
+dz yi
+(8)
+∂xGn(yi(z), z) changes sign when we increment i because Gn(yi(z), z) = 0 so the derivative −∂xGn(yi(z), z)
+changes sign when we go from one root to the next (the roots are simple). Let’s distinguish between
+the even and odd cases. In the even case,
+d
+dzyi ≥ 0 when i = 1..n/2, and it gives us by the sign rule
+i = 1..n/2 − 1 positive roots. Now there is still one root missing, and we can check that the sign
+of ∂zGn(yn/2(z), z) is opposite to the sign of ∂zGn(0, z), and as ∂zGn(−yn/2(z), z) has the same
+sign by symmetry there has to be a root between both, and actually two, one positive and one
+negative by symmetry. We get n roots overall. In the odd case, 0 is a root, and we have again two
+roots missing. But if yl denotes the positive smallest root, then ∂zGn(yl(z), z) has the same sign as
+∂zGn(−yl(z), z) and so as 0 is a root we need to have at least two other roots by some easy change
+of sign argument. Therefore all the zeros of ∂zGn(x, z) have been found and the positive ones are
+interlacing with the zeros of Gn(x, z).
+□
+Theorem 5.6. ∂zGn(x, z) and more generally ∂k
+z Gn(x, z) for all k ≤ n are real-rooted in x for
+z > 0, and the positive roots are monotonously increasing in z, and their symmetric negative
+counterpart monotonously decreasing in z.
+
+SPECIAL POLYNOMIALS AND NEW REAL-ROOTEDNESS RESULTS
+15
+Proof. We have
+Gn(−x, z) = (−1)nGn(x, z)
+∂zGn(−x, z) = (−1)n∂zGn(x, z)
+(9)
+So that if we have a root ˜y of ∂zGn(x, z), then its symmetric −˜y will also be a root of ∂zGn(x, z). In
+other terms, ∂zGn(x, z) and more generally , ∂k
+z Gn(x, z) have symmetric roots. We show inductively
+the following property: ∂k+1
+z
+Gn(x, z) is realrooted, the positive roots of ∂k+1
+z
+Gn(x, z) interlace the
+positive roots of ∂k
+z Gn(x, z) and the positive roots are increasing with z. We start with the initial
+condition. Using 5.2, we get the real-rootedness and interlacing property for ∂zGn(x, z). Now we
+need to prove that the positive roots ˜yi(z) of ∂zGn(x, z) also share the monotonicity property.
+∂zGn( ˜yi(z), z) = 0
+which leads to
+∂zGn
+Ln
+( ˜yi(z), z) = 0
+d(∂zGn
+Gn ( ˜yi(z), z)
+�
+dz
+= ∂x
+�∂zGn
+Gn
+( ˜yi(z), z)
+�d ˜yi
+dz + ∂z
+�∂zGn
+Gn
+( ˜yi(z), z)
+�
+= 0
+(10)
+We want to show that for the positive roots,
+d ˜yi
+dz ≥ 0
+On the one hand,
+∂zGn
+Gn
+( ˜yi(z), z) =
+n
+�
+j=1
+∂zGn
+∂xGn
+(yj, z)
+1
+˜yi − yj
+so that
+∂x
+�∂zGn
+Ln
+( ˜yi(z), z)
+�
+= −
+n
+�
+j=1
+∂zGn
+∂xGn
+(xj, z)
+1
+( ˜yi − yj)2 =
+n
+�
+j=1
+dyj
+dz
+1
+( ˜yi − yj)2
+(11)
+=
+⌊n/2⌋
+�
+j=1
+dyj
+dz
+�
+1
+( ˜yi − yj)2 −
+1
+( ˜yi + yj)2
+�
+= 4˜yi
+⌊n/2⌋
+�
+j=1
+dyj
+dz yj
+( ˜yi − yj)2( ˜yi + yj)2
+(12)
+As we have ˜yi > 0 and dyi
+dz ≥ 0 as well as yj > 0 we get
+∂x
+�∂zGn
+Ln
+( ˜yi(z), z)
+�
+≥ 0
+On the other hand, using the real-rootedness in z, as ˜yi ∈ [−1, 1] by the interlacing property, then
+∂z
+�∂zGn
+Gn
+( ˜yi(z), z)
+�
+≤ 0
+using Laguerre inequality for realrooted polynomials, stating that ∂zzGnGn − (∂zGn)2 ≤ 0. We
+conclude by gathering the two inequalities.
+The induction is proven using exactly the same method, given that
+∂z
+�∂k+1
+z
+Gn
+∂kz Gn
+�
+≤ 0
+Because ∂k
+z Gn(x, z) is realrooted in z as the derivative of a realrooted polynomial, and x in the
+appropriate interval ([−1, 1]).
+□
+
+16
+AURELIEN GRIBINSKI EPFL
+References
+[1] G. Szego. Orthogonal polynomials. AMS, Providence, RI MR 51 (1975): 8724.
+

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@@ -0,0 +1,1409 @@
+Thermodynamics
+in Stochastic Conway’s Game of Life
+Krzysztof Pomorski1A,2 and Dariusz Kotula1B
+1 Cracow University of Technology
+A: Faculty of Electrical and Computer Engineering
+B: Faculty of Computer Science and Telecommunications
+2 Quantum Hardware Systems (www.quantumhardwaresystems.com)
+Abstract. Cellular automata can simulate many complex physical phenomena using the
+power of simple rules. The presented methodological platform expresses the concept of pro-
+grammable matter in which Newton’s laws of motion are one of examples. Energy has been
+introduced as the equivalent of the ”Game of Life” mass, which can be treated as first level of
+approximation. The temperature presence and propagation was calculated for various lattice
+topology and boundary conditions by using the Shannon entropy measure. The conducted
+study provides strong evidence that despite not fulfillment the principle of mass and energy
+conservation, the entropy, mass distribution and temperatures approaches thermodynamic
+equilibrium. In addition, the described cellular automata system transits from positive to a
+negative temperatures that stabilizes and can be treated as a signature of system dynami-
+cal equilibrium. Furthermore, the system dynamics was presented in case of few species of
+cellular automata competing for maximum presence on given lattice with different boundary
+conditions.
+1
+Introduction to Classical Conway’s Game of Life (CCGoL)
+A cellular automaton is a system consisting of cells arranged most often on a one, two or three
+dimensional regular lattice, which at a given moment are in one of M-th possible states expressing
+M-valued logic. The dynamics of the model depends on the definition of individual cell states and
+the rules of transitions between them [13]. One of the simplest examples is a one-dimensional cellular
+automaton. Suppose that the cells placed on the lattice can be in one of two states, which are marked
+with white (default assigned to dead state or logical zero) or black color (default assigned to alive
+state or logical one). We define a rule that if a given cell is black, then the cell to the right of it will
+change its state. This situation is depicted in a Figure 1. We can observe how the system changes
+in the subsequent steps of the simulation. The parameter determining the change of the cell state
+is the state of the left neighbour of a given cell. There are many other possibilities to choose such a
+parameter, e.g. the condition of the state of both neighboring cells or having the nearest neighbors
+with opposite states. In order to determine system dynamics, we must have a defined initial cellula
+automata states (information about initial system dynamical state) and a specific set of deterministic
+or probabilistic rules.
+arXiv:2301.03195v1  [nlin.CG]  9 Jan 2023
+
+Fig. 1: Evolution of a one-dimensional cellular automaton in successive cycles with left side partial
+logical negation rule (if the state of nearest left cell is alive then given automata cell change its state
+to opposite).
+The Conway’s Game of Life [3] is an example of a cellular automaton with a deterministic rules. It
+was proposed in 1970 by John Conway and cellular automaton system consists of cells located on
+a two-dimensional lattice, which can be in one of two states: alive or dead. The rules specify the
+required number of neighbors and cell states that are taken into account to determine their state in
+the next cycle. Given the neighborhood through the 8 closest cells, we can write the 3 main rules of
+the Classical Conway’s Game of Life (CCGoL):
+1. If a dead cell has exactly 3 neighbors, it comes alive in the next cycle.
+2. If a living cell has 2 or 3 neighbors, it survives in the next cycle.
+3. If a cell has a different number of neighbors than stated above, it will be dead in the next cycle.
+The rules defined in this way allow for the generation of various types of structure topologies with
+automaton state set to 1, as shown in Figure 2. The most common are ”unstable structures”, which
+change in successive cycles, but do not return to their initial state. A single cell cannot survive on the
+lattice, because it has less than 2 neighbors. Dead cell surrounded by alife cells cannot come to life,
+because it has number of neighbors is different from 3. If the simulation is continuing sufficiently
+long time, on the lattice usually remain structures that are unchanging in time - ”still lifes” (an
+example is the ”block” shown in the Figure 2) or changing in time in periodic way, so they return to
+their original shape after k cycles - ”oscillators” (an example is the ”blinker” shown in the Figure 2).
+There are also structures that move in a certain direction - ”gliders” (Figure 3) that are analogical
+to 1st Newton dynamics preserving momentum (speed and direction of propagation in this case),
+leave a trace of ”blinkers” - ”star ships” and objects, which periodically eject ”gliders” - ”guns” [8].
+Figure 4 shows one among many existing in the Conway’s Game of Life oscillators during successive
+iterations of the system simulation.
+The ”toad” oscillator has a period of 2, which means that it has continuous switching between 2
+different fixed configurations. Each 2-dimensional discrete lattice field has a specified number that
+indicates the number of neighbors of the given cell. If the number is green, the cell will be alive in
+the next cycle. If the number is blue, that cell will be dead in the next cycle.
+
+step 0
+step 1
+step 2
+step 3Fig. 2: Evolution of various topologies of cellular automata structures in time with deterministic
+rules of CCGoL. One can identify two dynamically unstable structures and two structures that have
+dynamical stability with time.
+(a)
+(b)
+(c)
+(d)
+(e)
+Fig. 3: Evolution of the ”glider” configuration of cellular automata having propagation property with
+time in deterministic CCGoL.
+Fig. 4: Evolution of the ”toad” configuration of cellular automata being an oscillator in successive
+cycles in CCGoL.
+
+step 0
+step 1
+step 2
+Unstable structure 1
+Unstable structure 2
+Still life "block"
+Oscilator "blinker"3
+1
+2
+2
+3
+2
+2
+2
+3
+2
+3
+3
+2
+3
+3
+4
+3
+3
+m
+2
+2
+2
+3
+2
+2
+2
+72
+Introduction to Stochastic Classical Conway’s Game of Life (SCCGoL)
+The Stochastic Classical Conway’s Game of Life (SCCGoL) was created by an addition of a cell spon-
+taneous change probability to the rules that were initially deterministic, so stochastic determinism
+was achieved. With a given prefixed spontaneous probability value, the state of the cell can change
+regardless of the number of neighbors. Cell states have values between 0 and 1 and are called mass.
+Due to the fact that SCCGoL have different rules from CCGoL, cells have almost never exactly two
+or three neighbors. A condition for given cell to come to life from a dead cell state (creationism of
+alife cell) is by having a number of neighbors in a certain range of values. Similar rules applies to
+a living cell justifying its alife or dead state in next time iteration. By setting standard intervals of
+allowed/forbidden number of neighbours values, in which cell is alife/dead and by adding the addi-
+tional spontaneous rule probability for cell to change in the next iteration (probability of changing
+the state of a cell regardless of the number of neighbors), we are able to create a simulator, where the
+cells never die or it is very difficult from a probabilistic point of view for a given cell to stay alive.
+If, maintaining standard neighbours condition for being dead/alife in previously defined intervals
+and having a selected initial cell alife automata configuration, we can systematically increase the
+spontaneous probability level from 0% to 100%, and we result in the graph as depicted in Figure
+5. Simulations were conducted for a lattice size 10 by 10 with the initial condition of a cellular
+(a)
+(b)
+(c)
+Fig. 5: (a) Schematic view of initial conditions in SCCGoL. (b-c) Dependence of automata population
+average cycle lifetime (over 1000 trials) on probability of spontaneous change of cell state from life
+to dead and reversely with preservation of standard rules in Conway’s Game of Life.
+automaton 2 by 2 (Figure 5a) with limited maximum number of cycles set to 1000 in conducted
+simulations. For a probability of changing the state of a cell regardless of the number of neighbors
+equal to zero, there is full determinism, therefore it is a situation of CCGoL with changed numbers of
+neighbors. As the probability increases (in range from 0 to 2%), the life expectancy of the population
+decreases due to too few neighbors. Starting simulation from a probability level close to two percent
+
+Life expectancy of the population
+cycleofl
+Aoverage denth
+240
+0
+0
+24
+4
+140
+The probability of changing the stabe of a cell
+regardless of the number of neighbors (%]Life expectancy of the population
+Aaverage deeth cycle of the populatior
+10F
+102
+10~2
+10-
+100
+101
+The probability pf changing the stabe of a cell
+regardless of the number of neighbor's (%](probability of changing the state of a cell regardless of the number of neighbors), the average life
+expectancy of the population begins to rise, which is caused by the more frequent appearances of
+living cells. Conducting the simulation with a probability level above eighteen percent we observe
+that the population practically never dies and keeps average cycle life time being at least 1000 or
+more time iterations.
+3
+Generalization of Stochastic Classical Conway’s Game of Life
+(SCCGoL) to the case of N competing cellular automata species
+The created simulation platform enables to create N different cellular automata species that com-
+petes between themselves for the resources enabling them to replicate. The given automata species
+has better replication properties within its own community and worse replication properties in case
+of neighbors being of different species. Such situation is normally encountered in human population
+when people of one homogenous identity prefer to collaborate more than in the case of people hav-
+ing much different identity (being from different tribes). Due to that fact soft antagonistic relation
+between different species is introduced indirectly by means of higher level of tolerance (or effective-
+ness of replication) towards neighbors of own species than neighbors of other species (other different
+species are treated in the same way). In real way this is an analogy of members collaboration of a
+given nation or culture within given culture or community versus different culture or community.
+Let us consider N = 4 (number of different automata species), so we have the following determined
+formula for number of effective existing neighbors Neeffective,k for given k-th tribes as:
+Neeffective,k = a1,kNe1 + a2,kNe2 + ... + ak,kNek + ... + ak,NNeN.
+(1)
+Previously defined replication rules promoting its own species can be formally expressed by following
+condition (max(as,k) = ak,k and ak,k > as,k if s ̸= k). In the conducted simulations all tribes have
+assigned the same value (a1,1 = a2,2 = a3,3 = a4,4 = 1
+2) and another same value for (a1,2 = a1,3 =
+a1,4 = a2,1 = a2,3 = a2,4 = a3,1 = a3,2 = a3,4 = a4,1 = a4,2 = a4,3 =
+1
+3) what simply means
+that automata tribes promotes its own tribe and only partly promotes other tribes with no special
+distinction on which different tribe it is pointing to. Cellular automata species distribution for k-th
+tribe across two-dimensional lattice is generated with use of a two-dimensional Gaussian distribution
+given as follows:
+f(x, y)k = Ake
+−
+�
+(x−x0,k)2
+2σ2
+x,k
++
+(y−y0,k)2
+2σ2
+y,k
+�
+,
+(2)
+where f(x, y)k is a mass function depending on the cell coordinates, (x0,k; y0,k) the center of the
+given cellular automaton species. Given parameters A0,k and σx,k, σy,k can control the mass of
+species and spread across the ”Globe” as depicted in Figure 6. In the case of several species on
+the lattice, SCCGoL simulation results promote the formation of new cells among the species that
+has the most mass in the neighborhood as in accordance in the Figure 7. During life cycle always
+newly formed cell has a mass that is more dependent on cells of the same species than other species
+of its neighbors. We observed that automata species fight each other in order to spread across the
+lattice and exclude other species what is an expression of some form of biological Darwinism being
+a consequence of formula 1.
+
+(a)
+(b)
+Fig. 6: (a) Case of mass distribution of cellular automata of one species using a two-dimensional
+Gaussian function being isotropic. (b) Example of initial distribution of three cellular automata tribes
+with the case of boundary existence between each separate tribe with a rule that one geometrical
+place on the lattice is occupied by cellular automata of given species with dominant mass.
+(a)
+(b)
+Fig. 7: Evolution of map distribution of 4 cellular automata populations of different species with 100
+by 100 lattice size. One can spot effective tendency of each species to occupy maximum possibly
+territory at the cost of other species in 4-species SCCGoL (SCCGoL4S) what can be understand as
+occurrence of weak antagonistic relation
+4
+Methodology of description of SCCGoL dynamics by tools of classical
+statistical physics
+Adding probability to the initially deterministic Game of Life and changing the interaction rules
+between neighboring automata brought the necessity for a new quantity called mass that has con-
+tinuous real values (other than 0 and 1 present in deterministic Game of Life). At this stage vividness
+of the whole cellular automata population can be understood and approximated by the population
+total energy (where simply mass is equal to energy). The level of the automata distribution order is
+expressed by the entropy. Entropy was introduced by Rudolf Clausius in 1865 as a thermodynamic
+state function [11]. If the entropy at the initial state and the entropy of the final state are devoted
+by Si and Sf respectively, then we have:
+Sf − Si =
+� f
+i
+dE
+T , dS
+dE = 1
+T
+(3)
+
+Very last formula gives definition of a temperature under assumption that energy and entropy is
+defined that is the case. We use instead of thermodynamic entropy the Shannon entropy measure
+given as:
+SShannon = −
+�
+i,j
+p(xi, yj) log p(xi, yj) = E [− log p(x, y)]
+(4)
+In order to calculate the probability in the equation 4, the SCCGoL simulation is repeated several
+hundred times, so one determines an average population cell mass in successive cycles. The calcula-
+tion of the temperature, which is a measure of thermal state was achieved with an equation 3 with
+a changed form:
+T(x, y, t) =
+dE(x,y,t)
+dt
+dSShannon(x,y,t)
+dt
+=
+dE(x, y, t)
+dSShannon(x, y, t)
+(5)
+The temperature defined by last formula can be calculated by two different methods. The first
+method relies on the calculation of the energy and entropy for the entire system, and then numerical
+calculation of the derivative of energy in function of entropy. The second calculation approach is
+to differentiate mass and entropy with respect to simulation time for each cell and as a result of
+corresponding ratio we obtain a temperature map. The example of evolution of mass and entropy
+for SCCGoL is depicted in Figure 8 and shows maximization and saturation of entropy, what is
+confirmation of Second Thermodynamic Law that is valid in physical systems, while we are dealing
+with cellular automata system. On another hand the mass of a system saturates and increases to
+certain critical value as given in left part of Figure 8.
+Fig. 8: Evolution of mass and entropy in successive cycles in SCCGoL manifesting maximization with
+saturation and approaching stationary thermodynamical equilibrium with initial condition depicted
+in Figure 9.
+5
+Numerical analysis of SCCGoL dynamics with methodology of
+classical statistical physics
+Numerical simulations were carried out for 4 topologies cellular automata (case of Figures 9a, 11a,
+13a, 15a). We preinpose such a rule that given alive cell has 20% probability of changing its state
+to dead state and that initially dead cell has 20% probability of changing its state to alive with
+a mass choose randomly from an interval value in (0,0.5). Still given cell state is depended on its
+neighbors, since it has 80% probability of changing its state due to state of neighbors. In order
+to obtain cellular automata probability map dynamics, simulations average of cellular automata
+positions was conducted. Figure 10 shows results that were obtained for a lattice size 100 by 100
+with a one cell alive as initial lattice state (case of Figure 9a). With subsequent cycles, the cells
+
+Mass
+Entropy
+25000
+800
+20000
+600
+15000
+400
+10000
+200
+5000
+0
+0
+0
+25
+50
+75
+100
+125
+150
+175
+200
+0
+25
+50
+75
+100
+125
+150
+175
+200
+Cycle
+Cycle(a)
+(b)
+Fig. 9: Evolution of diffusion process in cellular automata system for a limited lattice size 100 by 100
+in SCCGoL (averaged over 1000 trials). It shows final saturation of mass and entropy (as depicted
+in Figure 8) what implies approaching thermodynamical equilibrium with characteristic fluctuations
+of mass and entropy around effective stationary values.
+occupy more and more space on the lattice, which can be seen as increase the mass of the entire
+system (possible mass creationism is inherent feature of Conway’s Game of Life). The sum of the
+masses of all cells in successive cycles is depicted in Figure 8 with a comparison of the entropy
+change of the entire system. As we observe in Figure 10e, high entropy occurs at the edges of the
+population, which is caused by the entropy wave that meets area with almost zero cell occurrence.
+Before equilibrium is established, the entropy of the system slightly decreases due to the loss of this
+extra entropy at the edges as can be seen in the right part of Figure 8. Having established those two
+quantities, we conduct their differentiating with respect to time, and by formula 3 one can establish
+the whole effective temperature of system by dividing change of mass at given time by change of
+the entropy. Figure 10k shows the greater susceptibility of the entropy change to the constraints
+associated with the limited size of the simulation lattice that is ended with impenetrable walls. The
+dependence of mass derivative with respect to time simulation (case of Figure 10j) does not have
+such large oscillations as in the case of entropy derivative with simulation simulation (case of Figure
+10k). The temperature calculated by such procedure gives values just above zero (slightly positive)
+up to the 50’th cycle. We can see a large down peak caused by a slowdown in entropy increase.
+From about the 75’th cycle, the temperature of the system goes from positive to negative values.
+A Figure 10l) describes anomalous thermalization process in SCCGoL with a case of approaching
+temperature and entropy equilibrium. Surprisingly, the thermodynamic equilibrium is achieved for
+a case for negative temperatures. A second possible approach in determination the temperature is
+by use the derivatives of the mass and entropy with respect to time of the individual cells and by
+obtaining a temperature map of all cells. The temperature depicted in Figure 10i is mostly negative
+and steady, what corresponds to a situation where mass and entropy have come to equilibrium. We
+can distinguish two regions in the simulation, the first one with no temperature gradient and zero
+negative temperature and the second region with non-zero temperature gradient that also include
+positive temperatures. The place, where time derivatives of mass and entropy have noticeable values
+is at the edges of automata population, where we observe slightly positive temperature, which
+corresponds to the situation in Figure 10l before the 75’th cycle.
+
+■(a) Mass at t=4
+(b) Mass at t=26
+(c) Mass at t=92
+(d) Entropy at t=4
+(e) Entropy at t=26
+(f) Entropy at t=92
+(g) T(x,y,t=4)
+(h) T(x,y,t=26)
+(i) T(x,y,t=92)
+(j) dm
+dt with time
+(k) dS
+dt with time
+(l) Temperature with time
+Fig. 10: Dynamics of thermodynamical parameters (mass, entropy and temperature) with simula-
+tion time in SCCGoL (lattice size 100 by 100) also given in Figure 8. Achieved thermodynamical
+equilibrium is accompanied with final distribution of negative temperature (starting from positive
+temperature distribution as in accordance with formula 3) as experimentally observed necessary
+criteria for final thermodynamical stability. One can spot various similarities of statistical behaviour
+of SCCGoL with physical systems described by classical thermodynamics (maximization and satu-
+ration of entropy, decay of temperature gradients and final thermalization, uniform distribution of
+mass and energy).
+
+0
+0.10
+20 
+0.08
+40
+0.06
+y
+60 
+0.04
+80
+0.02
+0.00
+0
+20
+40
+60
+08
+x0
+0.10
+20
+0.08
+40
+0.06
+60
+0.04
+80
+0.02
+0.00
+0
+20
+40
+60
+80
+X0.10
+20
+0.08
+40
+0.06
+60
+0.04
+80
+0.02
+20
+40
+60
+80
+X0
+5
+20
+4
+40
+3
+y
+60 
+2
+80
+1
+20
+40
+09
+0
+0
+80
+x0
+8
+7
+20
+6
+40
+5
+4
+60
+3
+2
+80
+1
+20
+40
+60
+80
+0
+X0
+7
+20
+6
+40
+5
+60
+4
+80
+3
+0
+20
+40
+60
+80
+X0
+0.00
+20
+-0.02
+40
+-0.04
+y
+60 
+-0.06
+80
+-0.08
+0
+20
+40
+09
+80
+x0
+0.00
+20
+-0.02
+-0.04
+40
+y
+-0.06
+60
+-0.08
+80
+-0.10
+0
+20
+40
+60
+80-0.02
+20
+-0.04
+40
+-0.06
+60
+-0.08
+80
+-0.10
+0
+20
+40
+60
+80
+XMass derivative with respect to time
+15.0
+12.5
+dm/dt
+14.0
+7.5
+5.D
+25
+MAAy
+0.0
+2.5
+0
+21
+41
+140
+120
+CycleEntropy derivative with respect to time
+40
+3P/SP
+240
+0-
+200
+21
+4
+140
+120
+CycleTemperature
+0.5
+Temperature
+0.D
+0.5
+1.0
+1.5
+-2.0
+0
+21
+140
+120
+Cycle6
+Numerical study of one species cellular automata with various
+boundary conditions
+Next family of simulations were conducted with use of barriers impenetrable by cellular automata
+cells via which cellular automata cannot interact with each other. We consider a single cellular
+(a)
+(b)
+(c)
+Fig. 11: Diffusion process in SCCGoLp20L100b100 (lattice size 100 by 100, 20% probability of
+spontaneous change of cell state from life to dead and reversely with preservation of standard rules
+in Conway’s Game of Life as also given in Figure 5) case of system of two weekly interconnected
+chambers by means of two small holes in barrier. Two stages of diffusion can be spotted in mass and
+entropy dynamics that has two consecutive processes: full diffusion in the left chamber is leading to
+full diffusion in the right chamber. Further details of thermodynamical parameters space dependence
+evolution with time are given by Figure 12.
+automata (single automata seed) placed in empty chamber with impenetrable walls with two small
+holes that link it to another empty chamber as depicted in the Figure 11a. We observe a diffusion of
+cellular automata with simulation time that consists of two main processes: creation and diffusion
+of cellular automata in the first chamber and diffusion of cellular automata from the first chamber
+into second chamber accompanied with creation new automata in the second chamber. Those two
+consequent processes are accompanied by effective slowdown in diffusion that is seen in left part
+of Figure 11c. At the same time we observe slowdown in entropy increase as in an accordance
+to the right part of Figure 11c. Once mass saturation was obtained in the simulation we observe
+maximization and small drop in entropy that later stabilizes and saturates, as in the right part of
+Figure 11c. Entropy unlike mass is characterized by a large variation of values in the middle of the
+population and at the edges of the cellular automata population. In the situation with barrier (case of
+
+Mass
+Entropy
+25000
+1200
+1000
+20000
+800 
+15000
+600
+10000
+400
+5000
+200
+0
+0
+0
+25
+50
+75
+100
+125
+150
+175
+200
+0
+25
+50
+75
+100
+125
+150
+175
+200
+Cycle
+Cycle(a) Mass at t=5
+(b) Mass at t=70
+(c) Mass at t=100
+(d) Entropy at t=5
+(e) Entropy at t=70
+(f) Entropy at t=100
+(g) T(x,y,t=5)
+(h) T(x,y,t=70)
+(i) T(x,y,t=100)
+(j) dm
+dt with time
+(k) dS
+dt with time
+(l) Temperature with time
+Fig. 12: Dynamics of thermodynamical parameters with simulation time in SCCGoLp20L100b100
+with two weekly interconnected chambers by two small holes in barrier that were initially depicted
+in Figure 11.
+
+0
+0.14
+20 
+0.12
+0.10
+40
+0.08
+y
+60 
+0.06
+0.04
+80 
+0.02
+40
+60 
+0.00 
+20
+80
+x0.175
+20
+0.150
+0.125
+40
+0.100
+60
+0.075
+0.050
+80
+0.025
+000'0
+20
+40
+60
+800.200
+0.175
+20
+-0.150
+40 
+0.125
+0.100
+60
+0.075
+0.050
+80
+0.025
+000'0
+600
+20
+40
+60 
+80
+0
+20
+40
+60
+8020
+40
+60
+80
+0
+20
+40
+8020
+40
+60
+80
+20
+40
+500
+0.025
+20
+0.000
+0.025
+40
+y
+0.050
+60 
+0.075
+0.100
+80 
+0.125
+0
+20
+40
+60
+80
+x0.00
+0.02
+20
+0.04
+0.06
+40
+0.08
+60
+0.10
+0.12
+80 
+0.14
+0.16
+90
+60
+800.000
+0.025
+20
+0.050
+40 
+0.075
+0.100
+60
+0.125
+80 
+0.150
+0.175
+80vass derivative with resoect to time
+15
+1
+dm/dt
+0
+25
+54
+S
+14D
+125
+150
+175
+CycleEntropy derivative with respect to time
+500
+310
+p/Sp
+240
+10
+0-
+100
+200
+0
+25
+50
+75
+140
+125
+150
+175
+CycleTemperature
+3.D
+25
+2D
+15
+LD
+0.5
+0.0
+0.5
+1.0
+0
+25
+50
+75
+140
+125
+150
+175
+CycleFigure 12e) this is particularly evident after the cells pass through the gaps. This particular process
+is due to the logarithm function dependence of Von Neumann entropy, which tends to negative
+infinity for arguments going to zero from the right. Observed criteria of an equilibrium is fact that
+mass and entropy have steady values and that temperature have negative value in thermodynamical
+equilibrium. Almost always before the equilibrium moment is achieved, time derivatives of mass and
+entropy have positive values, and still the temperature is positive. From the moment the equilibrium
+is achieved, we are dealing with small fluctuations of entropy and temperature. We observe the
+correlation stating that slight increase in mass results in slight decrease in entropy and vice versa. In
+an analogical way to the system with one barrier, simulations were conducted on the system with two
+barriers and the initial structure as depicted in Figure 13a. We consider a single cellular automata
+(a)
+(b)
+(c)
+Fig. 13: Dynamics of diffusion process for a system SCCGoLp20L100b100 with two barriers with
+two small holes in each barrier (generalization of situation from Figure 12) creating three weakly
+interconnected chambers perturbed by mutual interactions mediated by holes. Monotonicity in in-
+crease of entropy is twice shortly interrupted by small decline, what is associated with automata cells
+”colliding” with barriers and experiencing short lasting slowing down in its propagation. Details on
+space depended thermodynamical parameter evolution with time are given by Figure 14.
+placed in empty chamber with impenetrable walls with two small holes that link it to the second
+empty chamber, which is connected to a third empty chamber by impenetrable walls with two small
+holes as depicted in the Figure 13a. We observe a diffusion of cellular automata with simulation time
+that consists of three main processes: creation and diffusion of cellular automata in the first chamber,
+diffusion of cellular automata from the first chamber into second chamber accompanied with creation
+new automata in the second chamber and diffusion of cellular automata from the second chamber
+into third chamber accompanied with creation new automata in the third chamber. Twice cells try
+
+Mass
+Entropy
+1200
+25000
+1000
+20000
+800
+15000
+600
+10000
+400
+5000
+200
+0
+0
+25
+50
+75
+100
+125
+150
+175
+200
+0
+25
+50
+75
+100
+125
+150
+175
+200
+Cycle
+Cycleto reach impenetrable barriers, we observe small drops in entropy. After the second time entropy
+stabilizes and saturates, as in the right part of Figure 13c. In a system with two barriers, it lasts
+longer for mass and entropy to reach equilibrium than in the case of a system with only one barrier.
+As depicted in Figure 14k, due to the cells approaching the barriers and losing the extra entropy at
+the edges of the population, we observe significant fluctuations in the time derivative of the entropy.
+This results in large peaks seen in the Figure 14l.
+
+(a) Mass at t=30
+(b) Mass at t=70
+(c) Mass at t=110
+(d) Entropy at t=30
+(e) Entropy at t=70
+(f) Entropy at t=110
+(g) T(x,y,t=30)
+(h) T(x,y,t=70)
+(i) T(x,y,t=110)
+(j) dm
+dt with time
+(k) dS
+dt with time
+(l) Temperature with time
+Fig. 14: Space depended dynamics of thermodynamical parameters with simulation time in SCC-
+GoLp20L100b100 with three weekly interconnected chambers by four small holes also depicted in
+Figure 13.
+
+0
+0.175
+0.150
+20 
+0.125
+40
+0.100
+y
+60
+0.075
+0.050
+80
+0.025
+0.000
+0
+20
+40
+09
+08
+x0.175
+20
+0.150
+0.125
+40
+0.100
+60
+0.075
+0.050
+80
+0.025
+0.000
+0
+20
+40
+60
+80
+X0.175
+20
+0.150
+0.125
+40
+0.100
+60
+0.075
+0.050
+80
+0.025
+0.000
+20
+40
+60
+80
+X0
+6
+20
+5
+40
+4
+3
+60
+2
+80
+1
+20
+40
+60
+80
+0
+X0
+6
+20
+5
+40
+4
+3
+60
+2
+80
+1
+0
+20
+40
+60
+80
+Xh
+20
+5
+40
+4
+60
+3
+80
+0
+20
+40
+60
+80
+X0
+0.00
+-0.02
+20
+-0.04
+-0.06
+40
+-0.08
+60
+-0.10
+-0.12
+80
+-0.14
+-0.16
+0
+20
+40
+60
+80
+x0.00
+-0.02
+20
+-0.04
+40
+-0.06
+-0.08
+60
+-0.10
+-0.12
+80
+-0.14
+0
+20
+40
+60
+80
+X0.00
+-0.02
+20
+-0.04
+0.06
+40
+-0.08
+60
+-0.10
+-0.12
+80
+-0.14
+0.16
+0
+20
+40
+60
+80vass derivative with resoect to time
+15 
+dm/dt
+25
+54
+75
+140
+125
+150
+175
+CycleEntropy derivative with respect to time
+D
+P/SP
+24D
+200
+0
+25
+50
+14D
+125
+150
+175
+CycleTemperature
+t0
+Temperature
+0.2
+0.0
+-0.2
+0.4 
+0.6
+0.8
+1.0
+0
+25
+50
+75
+140
+125
+150
+175
+Cycle7
+Numerical study of two species cellular automata in perturbative
+interaction by narrow constriction
+Further simulations for the case of two species cellular automata were carried out for a system divided
+into two reservoirs separated by two impenetrable barriers with one small hole (case of Figure 15a)
+that implies perturbative interaction between tribes. In the left upper corner of the first part of
+the system there have been located cells of the first cellular automata tribe. At a closer distance
+to the hole in impenetrable wall, but in the second right reservoir there have been located cells of
+the second cellular automata tribe. As depicted in Figure 15c, we observe similar final masses and
+their dynamics in case of both tribes, but in accordance to Figure 15d, different entropy dynamics.
+Very last is due to the distance of the cellular automata tribes from the small hole in impenetrable
+wall: a tribe located further from that hole needs more time to propagate and occupy its natural
+neighborhood and first left chamber of the system, and thus this tribe has a lower probability of
+taking over the territory of the other tribe. However the noticeable fact is that mass and entropy
+of both tribes achieves equilibrium and finally tribes end up in bit same geometrical and dynamical
+situation. As depicted in Figure 15f, the right tribe closer to the small hole occupies its nearest
+neighborhood territory more quickly, resulting in an attempt to occupy a rival tribe territory. As
+depicted in Figure 16 we observe large oscillations in the time derivatives of mass and entropy of
+both cellular automata tribes. In contrast to previously conducted simulations, the temperature of
+the system after reaching equilibrium is not only characterized by negative values. Tribes existential
+competition is the reason of occurrence of both positive and negative temperatures.
+
+(a)
+(b)
+(c)
+(d)
+(e) Mass at t=3
+(f) Mass at t=34
+(g) Mass at t=200
+Fig. 15: Diffusion process in case of system with two cellular automata tribes weekly interacting
+with each other via a small hole in double barrier as depicted in (a), where initial configuration
+is presented. After long thermodynamic equilibrium is achieved as given by (b) so two cellular
+automata tribes coexist in two different geometrical domains effectively geographically separated.
+In case of both tribes mass and entropy saturates having tendency to oscillate in thermodynamical
+equilibrium.
+
+Mass of the first tribe
+Mass of the second tribe
+140
+140
+120
+120
+100
+100
+80 
+80
+60
+60
+40
+40
+20 
+20
+0-
++0
+25
+50
+75
+100
+125
+150
+175
+200
+0
+25
+50
+75
+100
+125
+150
+175
+200
+Cycle
+CycleEntropy of the first tribe
+Entropy of the second tribe
+7000
+8000
+6000
+5000
+6000
+4000
+4000
+000m
+2000
+2000
+1000
+0
+0
+25
+50
+75
+100
+125
+150
+175
+200
+0
+25
+50
+75
+100
+125
+150
+175
+200
+Cycle
+Cycle0
+10
+20
+y
+30
+40
+0
+10
+20
+30
+40
+x0
+10
+20
+y
+30
+40
+0
+10
+20
+30
+40
+x0
+10
+20
+30
+40
+0
+10
+20
+30
+40(a) dm
+dt with time for first and second automata tribe
+(b) dS
+dt with time for first and second automata tribe
+(c) Temperature with time for first and second automata tribe
+Fig. 16: Dynamics of thermodynamical variables for the case of two competing cellular automata
+tribes (first tribe is placed on the left and second tribe is placed on the right).
+
+Derivative of mass with resoect to time of the first tribe
+Derivative of mass with resoect to time of the second tribe
+5
+3
+4
+2
+3
+2
+0
++
+-1
+-1
+25
+50
+75
+100
+125
+150
+175
+200
+0
+25
+50
+75
+100
+125
+150
+175
+200
+Cycle
+CycleDerivative of entropy with respect to time of the first tribe
+Derivative ofentropy with respect to time of the second tribe
+400
+300
+200
+200
+100
+0
+0
+-200
+-100
+-400
+-200
+0
+25
+50
+75
+100
+125
+150
+175
+200
+0
+25
+50
+75
+100
+125
+150
+175
+200
+Cycle
+CycleTemperature of the first tribe
+Temperature of the second tribe
+1.4
+2
+1.2
+0
+1.0
+-2
+0.8
+-4
+0.6
+-6
+0.4 
+-8
+0.2
+-10
+0.0
+-12
+0.2
+-14
+0
+25
+50
+75
+100
+125
+150
+175
+200
+0
+25
+50
+75
+100
+125
+150
+175
+200
+Cycle
+Cycle8
+Conclusions and future perspectives
+Cellular automata can simulate many complex physical phenomena using the power of simple rules
+as it was shown in the case of cellular automata diffusion dynamics confirmed by various simulations
+for one and many automata species. Certain type of automata Darwinism was spotted by studying 4
+automata species dynamics as given by Fig.7. The SCCGoL dynamics study provides strong evidence
+that despite the fact that the principle of conservation of mass is not fulfilled, since we have creation-
+ism and annihilation of automata, the entropy and temperature comes to equilibrium. In conducted
+various simulations of Stochastic Conway’s Game of Life dynamics we report transition from positive
+to negative values of temperatures and we are aware that there is maximum level of mass and energy
+density allowed for cellular automata, since otherwise they would die due to overpopulation. The fact
+that the temperature can be negative is known in condensed matter physics, but with assumption
+that the energy is top-bounded. In most ”normal” situations this is impossible, but in a rare cases
+in solid state physics approximately it can be achieved by inverting the population state. Obviously
+there is a such limitation on top-bounded energy value (mass density value) in Stochastic Conway’s
+Game of Life. Therefore, it is still consistent with thermodynamics methodology, as it was pointed
+by professor Adam Bednorz (Faculty of Physics, University of Warsaw).
+Following conclusions were derived basing on conducted simulations and described methodology:
+1. Identification of thermodynamically defined temperature as proper measure of system evolution
+with ’-’ sign (case of Figures 10, 12, 14).
+2. Identification of mass as effective energy of system (in first approximation) (case of Figures 8,
+11, 13, 15).
+3. Identification of Shannon Entropy as effective system entropy (in first approximation) (case of
+Figures 8, 11, 13, 15).
+4. Generalization of Stochastic Conway Game of Life of N tribes (approximated analogy to N-body
+Quantum Physics can be conceptionally drawn) as depicted in Figures 7, 15.
+5. Confirmation validity of second law of thermodynamics in SCCGoL (entropy maximises and
+saturates, case of Figures 8, 11, 13).
+6. Identification of short lasting Shannon entropy peak that later minimizes and saturates in SGoL
+(case of Figures 8, 11). Monotonicity in increase of entropy is twice shortly interrupted by small
+decline, what is associated with automata cells ”colliding” with barriers and experiencing short
+lasting slowing down in its propagation (case of Figure 13).
+Conducted analysis of Stochastic Game of Life allows to treat such system as mathematical object
+well described by methodology of classical statistical physics. Obtained numerical results by various
+simulations suggest that we shall introduce another definition of temperature in Stochastic Conway’s
+Game of Life system by adding ’minus’ sign to temperature known in statistical physics, so we obtain
+the following formula:
+TemperatureConway−P omorski−Kotula = −dE
+dS
+(6)
+TemperatureStatisticalP hysics = +dE
+dS
+Having such a definition of Conway-Pomorski-Kotula temperature we can use tools of statistical
+physics in Stochastic Game of Life preserving most classical physics thermodynamical intuition about
+various situations we can come across. There are well-known inherent analogous between classical
+statistical physics [12][7][2][5][4] and quantum mechanics [1]. Therefore further research perspectives
+in study of Stochastic Classical Conway’s Game of Life assume the usage of quantum mechanics
+
+being able to simulate classical statistical physics (as expressed by epidemic model or stochastic
+finite-state machine) as explicitly represented by tight-binding model [9][10] or Schroedinger model
+directly proposing structures implemented in semiconductor single-electron devices. Noticeable var-
+ious obtained map of probability of Stochastic Conway’s Game of Life can be parameterized by
+non-linear Schrodinger equation and especially by Ginzburg-Landau model [6], what can be the
+base for quantization of Stochastic Classical Conway’s Game of Life.
+9
+Acknowledgment
+We would like to express our acknowledgment to professor Adam Bednorz (University of Warsaw),
+Adam Chochla (Cracow University of Technology) and to doctor Lukasz Stepien (The Pedagogical
+University in Cracow). The consultations with them on manuscript has allowed to improve it. The
+Authors have no conflict of interests and equally contributed to this work with 50 percent of contri-
+bution on each side. First Author proposed methodological and conceptual framework for this work,
+while Second Author conducted all numerical simulations. The interpretations of obtained results is
+equally assigned to each Author.
+References
+1. Baez, J.C., Pollard, B.S.: Quantropy. arXiv:1311.0813 (2013)
+2. Feynman, R.P.: Statistical mechanics. Westview (1972)
+3. Gardner, M.: Mathematical games - the fantastic combinations of john conway’s new solitaire game
+”life”. Scientific American (1970)
+4. Huang, K.: Statistical mechanics. John Wiley & Sons (1963)
+5. Huang, K.: Introduction to statistical physics. CRC Press (2001)
+6. Kotula, D., Pomorski, K.: Thermodynamics of Stochastic Conway Game of Life. ShanghaiAI Lectures,
+https://youtu.be/kLOB9VlF-R4 (2022)
+7. Mishin, Y.: Thermodynamic theory of equilibrium fluctuations. Elsevier (2015)
+8. Peitgen, H.O., J¨urgens, H., Saupe, D.: Chaos and Fractals. Springer (1983)
+9. Pomorski, K.: Equivalence between classical epidemic model and quantum tight-binding model. Springer
+(2022)
+10. Pomorski, K.: Equivalence between finite state stochastic machine, non-dissipative and dissipative tight-
+binding and schroedinger model. arXiv:2208.09758 (2022)
+11. Shannon, C.: A mathematical theory of communication. Bell System Technical Journal (1948)
+12. Velazquez Abad, L.: Principles of classical statistical mechanics: A perspective from the notion of com-
+plementarity. Annals of Physics 327(6), 1682–1693 (2012)
+13. Wolfram, S.: Statistical mechanics of cellular automata. Reviews of Modern Physics 55 (1983)
+