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+arXiv:2301.03067v1  [astro-ph.HE]  8 Jan 2023
+Neutron stars in the context of f(T, T ) gravity
+Cl´esio E. Mota1,∗ Luis C. N. Santos2,† Franciele M. da Silva3,‡
+Cesar V. Flores4,5,§ Iarley P. Lobo6,7,¶ and Valdir B. Bezerra2∗∗
+1Departamento de F´ısica, CFM - Universidade Federal de Santa
+Catarina; C.P. 476, CEP 88.040-900, Florian´opolis, SC, Brasil.
+2Departamento de F´ısica, CCEN-Universidade Federal da Para´ıba;
+C.P. 5008, CEP 58.051-970, Jo˜ao Pessoa, PB, Brazil
+3N´ucleo Cosmo–ufes & Departamento de F´ısica, Universidade Federal do Esp´ırito Santo,
+Av.
+Fernando Ferrari, 540, CEP 29.075-910, Vit´oria, ES, Brazil
+4Centro de Ciˆencias Exatas, Naturais e Tecnol´ogicas,
+CCENT - Universidade Estadual da Regi˜ao Tocantina do Maranh˜ao; C.P. 1300,
+CEP 65901-480, Imperatriz, MA, Brasil.
+5Departamento de F´ısica, CCET - Universidade Federal do Maranh˜ao,
+Campus Universit´ario do Bacanga; CEP 65080-805, S˜ao Lu´ıs, MA, Brasil.
+6Department of Chemistry and Physics, Federal University of Para´ıba,
+Rodovia BR 079 - Km 12, 58397-000 Areia-PB, Brazil. and
+7Physics Department, Federal University of Lavras,
+Caixa Postal 3037, 37200-000 Lavras-MG, Brazil.
+In this work, we investigate the existence of neutron stars (NS) in the framework of f(T, T )
+gravity, where T is the torsion tensor and T is the trace of the energy-momentum tensor. The
+hydrostatic equilibrium equations are obtained, however, with p and ρ quantities passed on by
+effective quantities ¯p and ¯ρ, whose mass-radius diagrams are obtained using modern equations of
+state (EoS) of nuclear matter derived from relativistic mean field models and compared with the
+ones computed by the Tolman-Oppenheimer-Volkoff (TOV) equations. Substantial changes in the
+mass-radius profiles of NS are obtained even for small changes in the free parameter of this modified
+theory. The results indicate that the use of f(T, T ) gravity in the study of NS provides good results
+for the masses and radii of some important astrophysical objects, as for example, the low-mass X-ray
+binary (LMXB) NGC 6397 and the pulsar of millisecond PSR J0740+6620. In addition, radii results
+inferred from the Lead Radius EXperiment (PREX-2) can also be described for certain parameter
+values.
+Keywords : general relativity, modified gravity, neutron stars.
+I.
+INTRODUCTION
+In recent years, there have been a growing number
+of ideas exploring modifications and alternative formu-
+lations of General Relativity (GR) emerging of different
+contexts. In fact, GR is a theory well tested, providing
+an interesting description of the space-time nature as a
+dynamical stage where physical phenomena takes place.
+In parallel to the advances in GR, the quantization of
+the gravitational field remains an open problem. With
+respect to this issue, it was pointed out that the action
+for gravity should be constructed with higher-order cur-
+vature terms in the context of renormalization at one
+loop level [1]. In the literature there are some formula-
+tions of gravity where the usual Einstein-Hilbert action
+is supplemented by higher-order curvature terms, as for
+example in the context of the f(R) theory in which case
+∗ [email protected]
+† [email protected]
+‡ [email protected]
+§ [email protected]
+¶ iarley˙lobo@fisica.ufpb.com
+∗∗ [email protected]
+the Ricci scalar R in the action is replaced by a general
+function f(R) [2].
+On the other hand, there are questions concerning the
+content of energy and matter in the universe that, at
+the moment, are not satisfactorily explained in the scope
+of standard theories.
+The observed rotation curves of
+galaxies [3] and the “missing mass” of galaxy clusters
+[4] suggest the dark matter hypothesis, while the ac-
+celerated expansion of the universe observed today can
+be interpreted as an effect of the so-called dark energy
+[5, 6]. Unexpectedly these observations reveals that the
+ordinary baryonic matter corresponds to only 4% of con-
+tent of energy of the universe while the dark matter and
+dark energy correspond to 20% and 76%, respectively. In
+this sense, there are studies considering the possibility of
+modified theories of gravity which may help to alleviate
+the need for dark components of energy of the universe
+beyond the scope of GR.
+The late-time acceleration of the universe can be in-
+terpreted under two points of view. In the first one, it
+is introduced a dark energy sector in the energy content
+of the universe through a type of field. In the second
+one, the gravitational field itself is modified.
+In addi-
+tion, there may be combinations of both approaches de-
+pending on the couplings between gravitational and non-
+
+2
+gravitational sectors of theory [7–10]. Thus, it is expected
+that different formulations of gravity imply that standard
+results in astrophysics suffer modifications. Compact ob-
+jects as neutron stars (NS), have been studied consid-
+ering effects of such modifications [11–20].
+NS in the
+context of f(R) gravity were studied in [21–23] and in
+f(R, T ) gravity in the papers [24–28]. In common, all
+of these works have considered effects on NS due to the
+modification of the gravitational field that include extra
+terms in the action. In the scheme of nonconservative
+gravity, the modification of the gravitational field can be
+done through a reinterpretation of the conservation law,
+as was considered in the papers [29, 30] (for a review on
+non-conservative theories of gravity, see [31]). Usually,
+the non-conservation of the stress-energy tensor is pro-
+portional to the matter density and pressure themselves.
+For this reason, an environment such as a compact ob-
+ject like a NS turns out to be an appealing laboratory for
+testing such theories.
+In the context of modified theories of gravity, the so-
+called f(T,T ) gravity is a class of such theories, free of
+ghosts and instabilities which, when applied to cosmo-
+logical problems, leads to interesting results [32]. In this
+formulation, the action depends on the torsion scalar T
+and on the trace of the energy-momentum tensor T . As
+in the case of f(T) gravity where the action is an ar-
+bitrary function of the torsion, in f(T, T ) gravity, the
+action is a arbitrary function of both the trace of the
+energy-momentum tensor and the torsion scalar.
+In this paper, we study an important context, not yet
+explored in the literature, that are the implications of
+the f(T,T ) gravity on NS. In particular, we obtain the
+mass-radius relation of NS in the context of this modified
+gravity and compare our results with recent astrophysical
+observations and experiments.
+This work is organized as follows: In Section II we ex-
+pose a summary of the f(T,T ) gravity. In Section III we
+derive the equations describing static, spherically sym-
+metric stars in this modified theory of gravity. In Section
+IV we present our results and in Section V we close with
+our final remarks.
+II.
+GRAVITATIONAL FIELD EQUATIONS OF
+f(T, T ) GRAVITY
+Given a line element describing a space-time we want
+to study
+ds2 = gµνdxµdxν = ηABeA
+µeB
+νdxµdxν
+(1)
+where gµν and {eA
+µ} are respectively the metric tensor
+and the components of the tetrad associated to space-
+time geometry, and ηAB = diag(1, −1, −1, −1) is the
+Minkowski metric. The signature (+ − − −) and ge-
+ometrized units, that is, G = c = 1, will be taken into
+account.
+In GR we assume that gravity is associated
+with the curvature of the space-time and thus we use the
+Levi-Civita’s connection
+◦
+Γρ
+µν = 1
+2gρσ (∂νgσµ + ∂µgσν − ∂σgµν)
+(2)
+to compute quantities associated with the curvature such
+as the Ricci scalar, R, that is present in the GR’s action.
+On the other hand, in teleparallel theory one assumes
+that gravity is associated to the torsion of the space-time
+and thus the Weizenbock’s connection
+Γλ
+µν = e λ
+A ∂µeA
+ν = −eA
+µ∂νe λ
+A
+(3)
+is used to construct quantities associated with the tor-
+sion, as the torsion scalar T that appears in the telepar-
+allel gravity action. In the modified teleparallel theories
+it is assumed that the action depends on a arbitrary func-
+tion of T. In our case, we are going to consider a modified
+action given by [32]
+S =
+�
+d4x e
+�T + f(T, T )
+16π
++ Lm
+�
+,
+(4)
+where e is the determinant of the tetrads e = det(eA
+µ) =
+√−g and T
+= gµνTµν is the trace of the energy-
+momentum tensor Tµν, which can be obtained from the
+Lagrangian for the matter distribution Lm in the follow-
+ing way
+Tµν = gµνLm − 2∂Lm
+∂gµν .
+(5)
+Let us assume that the function f(T, T ) is given by
+f (T, T ) = ω Tn T − 2Λ ,
+(6)
+where ω, n and Λ are arbitrary constants, specifically ω
+can be interpreted as a coupling constant of geometry
+with matter fields, n is a pure number (assumed to be
+unity here) and Λ can be recognized as the cosmological
+constant as discussed in [32, 33].
+We are interested in matter that can be described by
+a perfect fluid, so that Tµν is given by:
+Tµν = −pgµν + (p + ρ)uµuν,
+(7)
+where p is the pressure and ρ is the energy density of
+the fluid. By varying the action from Equation (4) with
+respect to the tetrad we find the following field equation
+Gµν = 8πT eff
+µν ,
+(8)
+where the effective energy-momentum tensor T eff
+µν
+is
+T eff
+µν
+= gµν
+
+
+�
+− ω(ρ − 3p) + 2Λ
+�
+16π
++ ωp
+8π
+
++Tµν
+�
+1+ ω
+8π
+�
+.
+(9)
+Calculating the covariant derivative of the energy-
+momentum tensor given by Equation (7), we obtain the
+following result
+∇µTνµ =
+1
+�
+4π + (1/2)ω
+�
+�ω
+4 (∂νT ) − ω
+2 ∂νp
+�
+.
+(10)
+
+3
+In a cosmological context, equation 10 can be associated
+to creation or destruction of matter throughout the uni-
+verse evolution. As discussed in [26], the interpretation of
+creation or destruction of matter particles in the NS level
+encounters difficulties in a static framework as occurs
+in the study of the hydrostatic equilibrium expression,
+i.e, the Tolman-Oppenheimer-Volkof equation. Also, it
+usually implies in the presence of a fifth force and non-
+geodesic trajectory for free particles. Naturally, results
+that depend on such imput would also be modified corre-
+spondingly. However, this is not the case analyzed in the
+present paper. In the next section we use Equations (8)
+to (10) to obtain and analyse the mass-radius relation of
+NS in the context of modified teleparallel gravity.
+III.
+STELLAR STRUCTURE EQUATIONS
+In this section, we discuss some of the main procedures
+that leads to the deduction of the hydrostatic equilibrium
+equation in the context of f(T, T ) gravity.
+To study compact stars, such as NS, magnetars and
+other astrophysical structures, we assume these objects
+as being homogeneous, static (no rotation), isotropic and
+spherically symmetric [34]. Therefore, we must use the
+appropriate metric in a convenient coordinate system
+that describes the object being studied. The most gen-
+eral metric describing the space-time under consideration
+is given by the line element
+ds2 = eν(r)dt2 − eλ(r)dr2 − r2(dθ2 + sin θ2dφ2),
+(11)
+where ν and λ are radial functions that we want to de-
+termine based on the field equations (8).
+Thus, using
+Equation (11) and substituting appropriately into Equa-
+tion (8),we obtain the following results
+e−λ�λ′
+r − 1
+r2
+�
++ 1
+r2 = 8π
+
+
+
+
+
+�
+− ω(ρ − 3p) + 2Λ
+�
+16π
++ ωp
+8π
+
+ + ρ
+�
+1 + ω
+8π
+�
+
+ = 8π¯ρ,
+(12)
+e−λ�ν′
+r + 1
+r2
+�
+− 1
+r2 = −8π
+
+
+
+
+
+�
+− ω(ρ − 3p) + 2Λ
+�
+16π
++ ωp
+8π
+
+ − p
+�
+1 + ω
+8π
+�
+
+ = 8π¯p,
+(13)
+e−λ
+4r
+�
+2
+�
+λ′ − ν′�
+−
+�
+2ν′′ + ν′2 − ν′λ′�
+r
+�
+= −8π
+
+
+
+
+
+�
+− ω(ρ − 3p) + 2Λ
+�
+16π
++ ωp
+8π
+
+ − p
+�
+1 + ω
+8π
+�
+
+ = 8π¯p,
+(14)
+where, the prime denotes a derivative with respect to
+the radial coordinate r. The quantities ¯ρ and ¯p are the
+effective pressure and energy density, defined as
+¯ρ = ρ + ωρ
+16π + 5ω p
+16π + Λ
+8π,
+(15)
+¯p = p + ωρ
+16π − 3ω p
+16π − Λ
+8π .
+(16)
+In addition to the field equations, we also need to consider
+the conservation equation (10) in f(T, T ) gravity so that
+we have a complete set of equations to be solved.
+In
+the case we are studying, Equation (10) has the form as
+follows
+−p′− ν′
+2 (ρ+p) =
+1
+�
+4π + (1/2)ω
+�
+�ωρ′
+4
+− 5ω p′
+4
+�
+. (17)
+Redefining the function λ(r) as
+e−λ(r) = 1 − 2M(r)
+r
+,
+(18)
+and rearranging Equations (12) and and (17), we get the
+equations required to describe static spherically symmet-
+ric stellar structures in f(T, T ) gravity theory, which are
+given by
+dM
+dr = 4πr2 ¯ρ,
+(19)
+and
+d¯p
+dr = −M ¯ρ
+r2
+�
+1 + ¯p
+¯ρ
+� �
+1 + 4πr3¯p
+M
+� �
+1 − 2M
+r
+�−1
+.
+(20)
+In the next section we show some results obtained by
+solving Equations (19) and (20) for realistic EoS of NS.
+
+4
+IV.
+RESULTS
+In this section, we present the results obtained from the
+solution of the field equations in the context of f(T, T )
+modified theory of gravity applied to NS.
+As an input to the stellar hydrostatic equilibrium equa-
+tions, we use two realistic EoS obtained from a relativis-
+tic mean field (RMF) approach. Firstly, we consider the
+IU-FSU [35] parametrization because it is able to explain
+reasonably well both nuclear [36] and stellar matter prop-
+erties [37]. We then compare the IU-FSU results with the
+ones obtained with a stiffer EoS calculated with a model
+of coupling of mesons and quarks, the quark–meson cou-
+pling (QMC) model [38]. (For the EoS with the QMC
+model, we refer the reader to refs. [38–42].) It is well
+known that a stiffer EoS leads to a bigger NS maximum
+mass in contrast to a softer one. In fact, using the EoS
+QMC as an input to the stellar equilibrium equations
+yields a maximum mass greater than 2.0 M⊙, and, there-
+fore, we want to verify that we get the same qualitative
+behavior for macroscopic properties (such as mass and
+radius) with parameterizations that are substantially dif-
+ferent. For the NS crust, we use the full BPS [43] EoS.
+After defining the EoS, some boundary conditions are
+required to solve the equations (19) and (20) along the
+radial coordinate r, from the center towards the surface
+of the star. At the star’s center r = 0 we take:
+M(0) = 0 ;
+¯ρ(0) = ¯ρc ;
+¯p(0) = ¯pc.
+(21)
+The radius of the star (r = R) is determined as the
+point where the pressure vanishes, i.e, ¯p(R) = 0.
+At
+this point, the interior solution connects softly with the
+Schwarzschild vacuum solution, indicating that the po-
+tential metrics of the interior and the exterior metric are
+related as eν(R) =
+1
+eλ(R) = 1 − 2M/R, being M the total
+mass of the star.
+Let us discuss and compare our results with recent
+astrophysical observations and nuclear physics experi-
+ments. At first, the NS in LMXB NGC 6397, depicted as
+a green shaded area in all figures, provides a constraint
+at 68% confidence level over the possible values of the
+masses and corresponding radii of the NS [44, 45]. Simi-
+larly, the millisecond pulsars are among the most useful
+astrophysical objects in the Universe for testing funda-
+mental physics, because they impose some of the most
+stringent constraints on high-density nuclear physics in
+the stellar interior [46].
+Recent measurements coming
+from the Neutron Star Interior Composition Explorer
+(NICER) mission reported pulsar observations for canon-
+ical (1.4 M⊙) and massive (2.0 M⊙) NS. The mass mea-
+surement and radius estimates provided for these objects,
+are 11.80 km ≤ R1.4 ≤ 13.1 km for the 1.4M⊙ NS PSR
+J0030+0451 (horizontal line segment in red colour shown
+in all Figures) and 11.60 km ≤ R ≤ 13.1 km for a NS with
+mass between 2.01M⊙ ≤ M ≤ 2.15M⊙ PSR J0740+6620
+(the rectangular region in orange colour shown in all Fig-
+ures).
+However, the authors of Ref.
+[47] used the re-
+cent measurement of neutron skin on 208Pb by PREX-2
+ 0.5
+ 1
+ 1.5
+ 2
+ 2.5
+ 4
+ 6
+ 8
+ 10
+ 12
+ 14
+ 16
+IU-FSU
+M /MO•
+R(km)
+ϖ = 0.0
+ϖ = 0.01
+ϖ = 0.02
+ϖ = 0.08
+ϖ = 0.1
+ϖ = 0.2
+ϖ = - 0.01
+ϖ = - 0.02
+ϖ = - 0.2
+ 1.9
+ 1.95
+ 10.5
+ 11
+ 11.5
+ 12
+ 12.5
+IU-FSU
+FIG. 1. Mass-radius relation for families of NS’s described
+by the IU-FSU EoS. We analyse the effect of varying the pa-
+rameter ω of the f(T, T ) theory. The red and green line seg-
+ment represent the radius range of the 1.4M⊙ NS for PSR
+J0030 + 0451 and PREX-2, respectively. The orange rectan-
+gular region corresponds to the range of radius estimates for
+2.08 ± 0.07M⊙ NS PSR J0740+6620. Similarly, the blue and
+pink horizontal lines stand, respectively, for the mass mea-
+surements of NS PSR J1614 + 2230 and NS PSR J0348 +
+0432.
+The purple solid line curve is solution for the usual
+TOV equation from GR.
+to constrain the radius of NS, which leads to a predic-
+tion of the radius of the canonical 1.4 M⊙ of 13.25 km
+≲ R1.4 ≲ 14.26 km (horizontal line segment in green
+colour shown in all Figures).
+Likewise, we also com-
+pare our results with two massive stars that had been
+discovered in 2010 and 2013, namely, PSR J1614+2230
+[48] with mass 1.97 ± 0.04 M⊙ (horizontal line in blue
+colour shown in all Figures) and PSR J0348+0432 [49]
+with mass 2.01 ± 0.04 M⊙ (horizontal line in pink colour
+shown in all Figures). Our results are discussed in the
+next paragraphs.
+We modelled the function f(T, T ) according to equa-
+tion (6). This function model has already been used in
+recent works as, for example, in [32, 33]. We explore the
+values of the parameter ω which range from −0.2 to 0.2.
+On the other hand, we check that the Λ parameter has no
+significant effect on the mass-radius profiles of NS, since
+it appears as a constant in the f(T, T ) function that we
+have chosen. Therefore, we use Λ = 0. Note that we
+recover the GR solution from f(T, T ) theory by assum-
+ing that ω = Λ = 0. These plots are represented by the
+continuous purple lines in the Figures.
+In Figure 1 we show the effects of f(T, T ) theory on
+NS properties obtained with the IU-FSU EoS. We can
+see that the value of ω has a very small influence on the
+maximum mass of the stars. The radius of the canon-
+ical NS (M = 1.4M⊙) is considerably affected. Note a
+bigger (smaller) radius for the most positive (negative)
+values of ω. We can observe that the results of PREX-2
+
+5
+ 0.5
+ 1
+ 1.5
+ 2
+ 2.5
+ 4
+ 6
+ 8
+ 10
+ 12
+ 14
+ 16
+QMC
+M /MO•
+R(km)
+ϖ = 0.0
+ϖ = 0.01
+ϖ = 0.02
+ϖ = 0.08
+ϖ = 0.1
+ϖ = 0.2
+ϖ = - 0.01
+ϖ = - 0.02
+ϖ = - 0.2
+ 1.92
+ 2
+ 2.08
+ 2.16
+ 10.5
+ 12
+ 13.5
+QMC
+FIG. 2. Mass-radius relation for families of NS’s described by
+the QMC EoS. We analyse the effect of varying the parameter
+ω of the f(T, T ) theory. The red and green line segment repre-
+sent the radius range of the 1.4M⊙ NS for PSR J0030 + 0451
+and PREX-2, respectively. The orange rectangular region cor-
+responds to the range of radius estimates for 2.08 ± 0.07M⊙
+NS PSR J0740+6620. Similarly, the blue and pink horizontal
+lines stand, respectively, for the mass measurements of NS
+PSR J1614 + 2230 and NS PSR J0348 + 0432. The purple
+solid line curve is the solution for the usual TOV equation
+from GR.
+cannot be described with IU-FSU EoS in the GR, but in
+f(T, T ) theory the solutions with ω = 0.08 and ω = 0.1
+produce mass and radius that agree with this constraint.
+However, the solutions obtained with IU-FSU EoS can-
+not describe the mass and radius of PSR J0740+6620,
+PSR J1614+2230 and NS PSR J0348+0432 neither on
+GR nor on f(T, T ) theory.
+In Figure 2 we show the mass-radius relation obtained
+for QMC EoS in f(T, T ) gravity. Again, the effect of the
+parameter ω is to increase the radius when its values in-
+crease positively and to decrease the radius when its val-
+ues increase negatively. At the same time, the maximum
+mass changes very little with the variation of ω. We can
+also see that the solutions obtained with the QMC EoS
+in f(T, T ) can accommodate almost all the constraints
+we are taking into consideration, and with a smaller ra-
+dius than in GR, if we take ω = −0.01 or ω = −0.02.
+The exception is NS PSR J0030+0451 which only can be
+described with QMC EoS in f(T, T ) gravity if we take
+ω = −0.2. We can note that for both EoS analysed we
+could not find a configuration that satisfies all the con-
+straints at the same time.
+We can see that for both EoS’s the value of ω has a
+very small influence on the maximum mass of the stars,
+on the other hand, the value of the radius of the star with
+maximum mass increases when we increase the value of
+ω and decreases when ω decreases. Also for both EoS’s,
+the case ω = −0.2 produces mass-radius curves that are
+typical of quark stars.
+V.
+FINAL REMARKS
+We have investigated the effects of f(T, T ) gravity on
+NS assuming these compact objects as being homoge-
+neous, static and isotropic. In this way, we have consid-
+ered a spherically symmetric space-time and solved the
+field equations and the hydrostatic equilibrium equation
+in the context of this modified theory of gravity. This
+type of system can be transformed into a system with
+effective pressure and energy density which permitted
+that the hydrostatic equilibrium equation was obtained
+through known techniques. For the choice of the f(T, T )
+function used here, we obtained that this theory can pre-
+dict NS with almost the same mass and smaller radius
+than in GR, for a given EoS, that is an interesting result
+in view of the recent observations. Considering the low-
+mass X-ray binary (LMXB) NGC 6397 and the pulsar of
+millisecond PSR J0740+6620, the results obtained using
+the modified hydrostatic equilibrium equations present
+good agreement with the observed masses and radii.
+We particularize f(T, T ) gravity according to equation
+(6).
+The good results obtained in comparison to GR
+suggest future extensions of this work, as for example, by
+taking into consideration different choices of the f(T, T )
+function, which should be done in a near future. It can be
+interesting to test, for example, high powers in T besides
+and new couplings between T and T . In addition, we
+can use different EoS as input to the stellar hydrostatic
+equilibrium equations along the aforementioned choices
+of f(T, T ) function.
+ACKNOWLEDGEMENTS
+L.C.N.S. would like to thank Conselho Nacional de
+Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq) for
+partial financial support through the research Project
+No.
+164762/2020-5 and F.M.S. would like to thank
+CNPq for financial support through the research Project
+No.
+165604/2020-4.
+I. P. L. was partially supported
+by the National Council for Scientific and Technologi-
+cal Development - CNPq grant 306414/2020-1 and by
+the grant 3197/2021, Para´ıba State Research Foundation
+(FAPESQ). I. P. L. would like to acknowledge the contri-
+bution of the COST Action CA18108. V.B.B. is partially
+supported by CNPq through the Research Project No.
+307211/2020-7.
+[1] R. Utiyama and B. S. DeWitt, “Renormalization of a
+classical gravitational field interacting with quantized
+matter fields,” Journal of Mathematical Physics, vol. 3,
+no. 4, pp. 608–618, 1962.
+
+6
+[2] T. P. Sotiriou and V. Faraoni, “f(R) Theories Of Grav-
+ity,” Reviews of Modern Physics, vol. 82, pp. 451–497,
+2010.
+[3] V. C. Rubin and W. K. Ford Jr, “Rotation of the An-
+dromeda nebula from a spectroscopic survey of emission
+regions,” The Astrophysical Journal, vol. 159, p. 379,
+1970.
+[4] F. Zwicky, “Die rotverschiebung von extragalaktischen
+nebeln,” Helvetica Physica Acta, vol. 6, pp. 110–127,
+1933.
+[5] S. Perlmutter, G. Aldering, G. Goldhaber, R. A. Knop,
+P. Nugent, P. G. Castro, S. Deustua, S. Fabbro, A. Goo-
+bar, D. E. Groom, et al., “Measurements of Ω and Λ from
+42 high-redshift supernovae,” The Astrophysical Journal,
+vol. 517, no. 2, p. 565, 1999.
+[6] V. Sahni, “5 dark matter and dark energy,” The Physics
+of the Early Universe, pp. 141–179, 2004.
+[7] S. Capozziello and M. De Laurentis, “Extended theories
+of gravity,” Physics Reports, vol. 509, no. 4-5, pp. 167–
+321, 2011.
+[8] A. De Felice and S. Tsujikawa, “f(R) theories,” Living
+Reviews in Relativity, vol. 13, no. 1, pp. 1–161, 2010.
+[9] S. Nojiri and S. D. Odintsov, “Unified cosmic history
+in modified gravity: from F(R) theory to Lorentz non-
+invariant models,” Physics Reports, vol. 505, no. 2-4,
+pp. 59–144, 2011.
+[10] F. S. N. Lobo, “The dark side of gravity: Modified the-
+ories of gravity,” arXiv preprint arXiv:0807.1640, 2008.
+[11] T. Harada, “Neutron stars in scalar-tensor theories of
+gravity and catastrophe theory,” Physical Review D,
+vol. 57, no. 8, p. 4802, 1998.
+[12] M. Orellana, F. Garc´ıa, F. A. T. Pannia, and G. E.
+Romero, “Structure of neutron stars in R-squared grav-
+ity,” General Relativity and Gravitation, vol. 45, no. 4,
+pp. 771–783, 2013.
+[13] D.
+Momeni
+and
+R.
+Myrzakulov,
+“Tol-
+man–Oppenheimer–Volkoff
+equations
+in
+modified
+Gauss–Bonnet
+gravity,”
+International
+Journal
+of
+Geometric Methods in Modern Physics, vol. 12, no. 02,
+p. 1550014, 2015.
+[14] A. Oliveira, H. Velten, J. Fabris, and L. Casarini, “Neu-
+tron stars in Rastall gravity,” Physical Review D, vol. 92,
+no. 4, p. 044020, 2015.
+[15] S. Hendi, G. Bordbar, B. E. Panah, and S. Panahiyan,
+“Modified TOV in gravity’s Rainbow: properties of neu-
+tron stars and dynamical stability conditions,” Journal of
+Cosmology and Astroparticle Physics, vol. 2016, no. 09,
+p. 013, 2016.
+[16] K. N. Singh, F. Rahaman, and A. Banerjee, “Einstein’s
+cluster mimicking compact star in the teleparallel equiv-
+alent of general relativity,” Physical Review D, vol. 100,
+no. 8, p. 084023, 2019.
+[17] S. K. Maurya and F. Tello-Ortiz, “Charged anisotropic
+compact star in f(R, T ) gravity: A minimal geometric
+deformation gravitational decoupling approach,” Physics
+of the Dark Universe, vol. 27, p. 100442, 2020.
+[18] C. E. Mota, L. C. N. Santos, G. Grams, F. M. da Silva,
+and D. P. Menezes, “Combined Rastall and Rainbow
+theories of gravity with applications to neutron stars,”
+Physical Review D, vol. 100, no. 2, p. 024043, 2019.
+[19] C. E. Mota, L. C. N. Santos, F. M. da Silva, C. V. Flo-
+res, T. J. N. da Silva, and D. P. Menezes, “Anisotropic
+compact stars in Rastall–Rainbow gravity,” Classical and
+Quantum Gravity, vol. 39, no. 8, p. 085008, 2022.
+[20] F. M. da Silva, L. C. N. Santos, and C. C. Bar-
+ros, “Rapidly rotating compact stars in Rastall’s grav-
+ity,” Classical and Quantum Gravity, vol. 38, no. 16,
+p. 165011, 2021.
+[21] A. Cooney, S. DeDeo, and D. Psaltis, “Neutron stars
+in f(R) gravity with perturbative constraints,” Physical
+Review D, vol. 82, no. 6, p. 064033, 2010.
+[22] S. Capozziello, M. De Laurentis, R. Farinelli, and S. D.
+Odintsov, “Mass-radius relation for neutron stars in f(R)
+gravity,” Physical Review D, vol. 93, no. 2, p. 023501,
+2016.
+[23] S. Arapo˘glu, C. Deliduman, and K. Y. Ek¸si, “Constraints
+on perturbative f(R) gravity via neutron stars,” Journal
+of Cosmology and Astroparticle Physics, vol. 2011,
+no. 07, p. 020, 2011.
+[24] P. H. R. S. Moraes, J. D. V. Arba˜nil, and M. Malheiro,
+“Stellar equilibrium configurations of compact stars in
+f(R, T ) theory of gravity,” Journal of Cosmology and
+Astroparticle Physics, vol. 2016, no. 06, p. 005, 2016.
+[25] J. M. Z. Pretel, S. E. Jor´as, R. R. R. Reis, and J. D. V.
+Arba˜nil, “Neutron stars in f(R, T ) gravity with con-
+served energy-momentum tensor:
+Hydrostatic equilib-
+rium and asteroseismology,” Journal of Cosmology and
+Astroparticle Physics, vol. 2021, no. 08, p. 055, 2021.
+[26] S. I. dos Santos, G. A. Carvalho, P. H. R. S. Moraes,
+C. H. Lenzi, and M. Malheiro, “A conservative energy-
+momentum tensor in the f(R, T ) gravity and its impli-
+cations for the phenomenology of neutron stars,” The
+European Physical Journal Plus, vol. 134, no. 8, pp. 1–8,
+2019.
+[27] M. Sharif and A. Waseem, “Anisotropic quark stars
+in f(R, T ) gravity,” The European Physical Journal C,
+vol. 78, no. 10, pp. 1–10, 2018.
+[28] D. Deb, S. V. Ketov, M. Khlopov, and S. Ray, “Study
+on charged strange stars in f(R, T ) gravity,” Journal of
+Cosmology and Astroparticle Physics, vol. 2019, no. 10,
+p. 070, 2019.
+[29] P. Rastall,
+“Generalization of the Einstein theory,”
+Physical Review D, vol. 6, no. 12, p. 3357, 1972.
+[30] C. E. Mota, L. C. N. Santos, F. M. da Silva, G. Grams,
+I. P. Lobo, and D. P. Menezes, “Generalized Rastall’s
+gravity and its effects on compact objects,” International
+Journal of Modern Physics D, vol. 31, no. 04, p. 2250023,
+2022.
+[31] H. Velten and T. R. P. Caramˆes, “To conserve, or not to
+conserve: A review of nonconservative theories of grav-
+ity,” Universe, vol. 7, no. 2, p. 38, 2021.
+[32] T. Harko, F. S. N. Lobo, G. Otalora, and E. N. Saridakis,
+“f(T, T ) gravity and cosmology,” Journal of Cosmology
+and Astroparticle Physics, vol. 2014, no. 12, p. 021, 2014.
+[33] I. G. Salako, M. Khlopov, S. Ray, M. Arouko, P. Saha,
+and U. Debnath, “Study on anisotropic strange stars in
+f(T, T ) gravity,” Universe, vol. 6, no. 10, p. 167, 2020.
+[34] S. Carroll, Spacetime and Geometry. Cambridge Univer-
+sity Press, 2019.
+[35] F. J. Fattoyev, C. J. Horowitz, J. Piekarewicz, and
+G. Shen, “Relativistic effective interaction for nuclei, gi-
+ant resonances, and neutron stars,” Physical Review C,
+vol. 82, no. 5, p. 055803, 2010.
+[36] O. Louren¸co, M. Dutra, C. H. Lenzi, C. V. Flores, and
+D. P. Menezes, “Consistent relativistic mean-field models
+constrained by GW170817,” Physical Review C, vol. 99,
+no. 4, p. 045202, 2019.
+
+7
+[37] M. Dutra, O. Louren¸co, and D. P. Menezes, “Stel-
+lar properties and nuclear matter constraints,” Physical
+Review C, vol. 93, no. 2, p. 025806, 2016.
+[38] P. A. M. Guichon, “A possible quark mechanism for the
+saturation of nuclear matter,” Physics Letters B, vol. 200,
+pp. 235–240, 1988.
+[39] K. Saito and A. W. Thomas, “A quark-meson coupling
+model for nuclear and neutron matter,” Physics Letters
+B, vol. 327, no. 1-2, pp. 9–16, 1994.
+[40] K. Saito and A. W. Thomas, “Composite nucleons in
+scalar and vector mean fields,” Physical Review C,
+vol. 52, no. 5, p. 2789, 1995.
+[41] S. Pal,
+M. Hanauske,
+I. Zakout,
+H. St¨ocker,
+and
+W. Greiner, “Neutron star properties in the quark-meson
+coupling model,” Physical Review C, vol. 60, no. 1,
+p. 015802, 1999.
+[42] G. Grams, A. M. Santos, and D. P. Menezes, “Equation
+of State Grid with the Quark-Meson-Coupling Model,”
+Brazilian Journal of Physics, vol. 46, no. 1, pp. 111–119,
+2016.
+[43] G. Baym, C. Pethick, and P. Sutherland, “The Ground
+state of matter at high densities: Equation of state and
+stellar models,” The Astrophysical Journal, vol. 170,
+pp. 299–317, 1971.
+[44] F. ¨Ozel and P. Freire, “Masses, Radii, and the Equation
+of State of Neutron Stars,” Annual Review of Astronomy
+and Astrophysics, vol. 54, pp. 401–440, 2016.
+[45] A. W. Steiner, C. O. Heinke, S. Bogdanov, C. K. Li,
+W. C. Ho, A. Bahramian, and S. Han, “Constrain-
+ing the mass and radius of neutron stars in globular
+clusters,” Monthly Notices of the Royal Astronomical
+Society, vol. 476, no. 1, pp. 421–435, 2018.
+[46] H. T. Cromartie, E. Fonseca, S. M. Ransom, P. B.
+Demorest, Z. Arzoumanian, H. Blumer, P. R. Brook,
+M. E. DeCesar, T. Dolch, J. A. Ellis, et al., “Relativistic
+Shapiro delay measurements of an extremely massive mil-
+lisecond pulsar,” Nature Astronomy, vol. 4, no. 1, pp. 72–
+76, 2020.
+[47] B. T. Reed,
+F. J. Fattoyev,
+C. J. Horowitz,
+and
+J. Piekarewicz, “Implications of PREX-2 on the equa-
+tion of state of neutron-rich matter,” Physical Review
+Letters, vol. 126, no. 17, p. 172503, 2021.
+[48] P. B. Demorest, T. Pennucci, S. M. Ransom, M. S. E.
+Roberts,
+and J. W. T. Hessels,
+“A two-solar-mass
+neutron star measured using Shapiro delay,” Nature,
+vol. 467, no. 7319, pp. 1081–1083, 2010.
+[49] J. Antoniadis, P. C. C. Freire, N. Wex, T. M. Tauris,
+R. S. Lynch, M. H. Van Kerkwijk, M. Kramer, C. Bassa,
+V. S. Dhillon, T. Driebe, et al., “A massive pulsar in a
+compact relativistic binary,” Science, vol. 340, no. 6131,
+p. 1233232, 2013.
+

4tFAT4oBgHgl3EQfFByL/content/tmp_files/2301.08425v1.pdf.txt

@@ -0,0 +1,488 @@
+Graphene as Infrared Light Sensor Material 
+Ahalapitiya H. Jayatissaa) and Madhav Gautam  
+Mechanical, Industrial, and Manufacturing Engineering (MIME) Department 
+The University of Toledo, OH 43606, USA 
+a)Correspondence:  [email protected] 
+ 
+ 
+Abstract: The infrared (IR) photoresponse of graphene synthesized by atmospheric chemical vapor 
+deposition (CVD) system using a mixture of hydrogen and methane gases was studied. The IR sensor 
+devices were fabricated using graphene films transferred on to a SiO2 substrate by a lift off process. The 
+quality of graphene was investigated with the Raman spectroscopy and optical microscopy. The 
+photoresponse was recorded under the illumination of IR light of wavelength 850 nm and intensity of 
+around 2.16 µW/mm2. The effects of temperature and hydrogenation on photoconductivity were also 
+studied. It was found that the transient response and recovery times decreased with the increase of the 
+temperature. Hydrogenation effect also caused the significant decrease in the photoresponse of the device.  
+Although the net change in the photoresponse for IR light was lower at low illumination intensity levels, 
+the transient responses were observed around 100 times faster than the recently reported CNT-based IR 
+sensors. 
+ 
+Key words: CVD graphene, single layer, Infra-Red light, photoconductivity, 2D sensor materials 
+ 
+ 
+1. Introduction 
+ 
+Optoelectronic devices working in near infra-red (NIR) (800 - 2000 nm) are always demanding for 
+different applications [1-4]. There has been significant works reported on the fabrication of optoelectronic 
+devices using NIR materials [5-12]. In recent years, single walled carbon nanotubes (SWCNTs) have 
+been investigated extensively as a semiconducting material for IR sensors because of its strong absorption 
+behavior in NIR region [7-12]. One of the key challenges in developing NIR detectors is the finding of 
+ultra fast optical response in the sensor material [5-8]. Recently, strong absorption behavior in NIR region 
+has been reported for thermally reduced graphene oxides [1,2]. This provides a pathway to use graphene 
+as an optoelectronic material for IR detection. Although the optical properties of graphene in visible 
+region have been reported by many researchers [13-15], we have not found any research work related to 
+the photoresponse of graphene in IR region of the spectrum. In this paper, photoresponse of graphene film 
+on macro-scale has been reported in different conditions.  
+Graphene is a monolayered carbon film with a film thickness of around 0.32Å [13 - 15], where carbon 
+atoms are arranged in a two-dimensional hexagonal lattice structure. It can be thought of as a single layer 
+peeled off from the graphite stack.  It has evolved as an interesting material due to its unique physical and 
+electrical properties [16]. This material is different from most of the conventional semiconductors because 
+of its zero bandgap semi-conducting behavior [17]. For example, graphene-based transistor devices may 
+operate very faster than traditional silicon devices due to high intrinsic carrier mobility (~ 2x105 cm2v-1s-1) 
+[1, 2, 18]. Being the material of high mechanical stress and low density (2.2 gm/cm3), it may lead to the 
+application in nano-robotics [19, 20].  
+We have investigated the photoconductivity of graphene layers synthesized in atmospheric chemical 
+vapor deposition (CVD) of CH4 on a copper substrate. The devices were fabricated by transferred CVD 
+graphene onto a SiO2/Si substrate. The investigations were carried out to understand the temperature 
+dependence and hydrogenation effect on photoconductivity of graphene in NIR region. Although the net 
+change in the photoresponse for IR light was lower at low illumination intensity levels (2.16 µW/mm2), 
+
+the transient responses were observed around 100 times faster than photoconductivity of CNT for NIR 
+lights. 
+ 
+2. Experimental Procedures 
+  
+The growth of graphene films was carried out on a copper (Cu) substrate (25 µm thick) in an alumina 
+tube furnace system under the flow of methane (CH4) and hydrogen (H2) gases. Copper substrate 
+(99.999% pure, Alfa Aesar) was heated in a tube furnace under the 150 standard cubic centimeters per 
+minute (sccm) flow of mixture of hydrogen and Argon (10% H2, 90% Ar) and annealed at 1100 0C for 
+one hour. After annealing, graphene deposition was carried out by passing a mixture of methane and 
+argon (5% CH4, 95% Ar) followed by the immediate cooling. Graphene deposited on copper by CVD 
+method was transferred to SiO2/Si substrate by wet etching of Cu [15, 21-23]. The thickness of the 
+thermally-grown SiO2 was 118 nm as confirmed by UV spectrometry [24].  The Raman spectra of these 
+films were recorded with the excitation wavelength of 530 nm. 
+ In order to fabricate the IR sensors, a thin layer of gold (about 100 nm) was coated onto the 
+transferred graphene film by a vacuum evaporation method. The gold electrodes were patterned by 
+lithography followed by etching of gold with aqueous KI/I2 solution. The spacing and the length of these 
+electrodes were 6 mm and 4 mm, respectively.  Fig. 1 shows the schematic diagram of the fabricated IR 
+sensor and photoresponse measurement circuit. The device was biased with a constant voltage (1.0 V) 
+during collection of the data.  To understand the reflection of light from graphene, reflectance from bi-
+layer substrate (SiO2/Si) and tri-layer substrate (graphene/SiO2/Si) were measured with a double beam 
+UV/Visible spectrometer (Shimadzu). The reflectance spectra were investigated in the spectral range 300-
+1100 nm. 
+ 
+Au
+A
+V0
+Graphene
+SiO2
+IR 
+Light
+Si
+ 
+Fig.1: Schematic of photoresponse measurement system (V0= 1.0 V). 
+ 
+3. Results and Discussions 
+ 
+3.1. Surface Characterization 
+The Raman spectroscopy has been used to characterize the quality of graphene. The Raman spectrum 
+of Graphene gives for main bands corresponding to the vibration mode of graphene. Fig. 2 shows the as-
+measured Raman spectra of graphene films produced on SiO2 surface. The spectrum was normalized with 
+respect to the intensity level of 2D band. The peak at around 1580 cm-1 and 2660 cm-1, respectively, 
+indicate the G band and the 2D band, which are characteristics Raman peaks of graphene. It has been 
+reported that the defect free monolayer graphene can be identified with characteristic features of Raman 
+band intensities [25]. The intensity of 2D band is ~2 times larger than the intensity of G band suggesting 
+that the presence of less defective graphene on SiO2 surface. This fact is also supported by the weak 
+intensity of D-band (1350 cm-1). 
+
+ 
+ 
+Fig.  2:  Raman spectra of graphene transferred to silicon wafer (SiO2 + Si) scaled with respect to 
+the maximum peak. 
+ 
+ 
+3.2. Photoconductivity 
+ 
+3.2.1. Dynamic response 
+Fig. 3 shows the dynamic response of photoconductivity of graphene film for the NIR light at room 
+temperature. Fig. 3(a) shows the response and recovery of the device when the IR light was turned on and 
+off, respectively, whereas Fig. 3(b) indicates the same characteristic for one cycle only. The intensity of 
+the IR light source used was 2.16 µW/mm2 at the device surface. Although the intensity level was very 
+low, a clear photoresponse of device was measured. The photogeneration of carriers can be primarily 
+attributed to the creation of bands at the defect of graphene sheets. When graphene is deposited on a 
+copper plate, defects are developed at the grain boundary of polycrystalline copper films. We believe that 
+these defects are responsible for the creation of localized photoactive regions, which contribute to the 
+photogeneration of carriers [26,27]. The photoresponse could be characterized with a time step function. 
+In both the photocurrent increase and drop cases, the experimental data were fitted well into the 
+exponential form as [10],  
+ 
+ 
+                    
+
+
+
+
+
+ −
++
+=
+
+t
+A
+I
+I
+o
+o
+exp
+.   
+ 
+ 
+ 
+ 
+ 
+ 
+         (1)  
+ 
+Here, I is the current, t is the response time and Io,  and A0 are constants. Fig. 4(a) and 4(b) show the fit 
+of the response in the form explained above. The data analysis indicated that the time constants were 10 
+ms and 31 ms for rise and fall of the photocurrent, respectively.  
+ 
+
+1.2
+2D
+nsityRatio (ll)
+1
+8'0
+0.6
+G
+0.4
+0.2
+D
+G
+人
+0
+1000
+1500
+2000
+2500
+3000
+Raman Shift (anl) 
+Fig. 3: The photoresponse of the device due to IR light for (a) different cycles and (b) for one cycle. 
+ 
+ 
+ 
+ Fig. 4: The photoresponse of the device due to IR light for (a) response and (b) recovery.  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. 5: The photoresponse of the device due to IR light at (a) 50 0C and (b) 100 0C. 
+ 
+ 
+3.2.2. The effect of temperature on photoconductivity 
+Fig. 6 shows the effect of temperature on the photoconductivity of graphene. The photoconductivity 
+was tested at 50 0C and 100 0C, respectively. During the experiment, the device was heated to the desired 
+
+(a)1,252
+1.2515
+b)
+1.251
+1.2505
+1.25
+1.2495
+1.249
+0
+50
+100
+150
+200
+250
+300
+Time (ms):(a)(b)1.2888
+(a)
+1.2882
+(vu)
+1.2864
+20
+40
+60
+80
+100
+120
+140
+Time (ms)(b)temperature for 30 minutes to ensure the thermal equilibrium. Transient responses of the device were 
+10.26 ms and 6.57 ms and the transient recovery times were 12.55 ms and 5.91 ms at 50 0C and 100 0C, 
+respectively. A significant difference in transient response of the device was not found when the device 
+temperature was increased from room temperature to 50 0C and transient response time decreased by 40% 
+when the temperature was changed from 50 0C to 100 0C. Similarly, the transient recovery time decreased 
+by 60% when the temperature was changed from room temperature to 50 0C and it decreased by 50% 
+when the temperature was changed from 50 0C to 100 0C.  
+On the other hand, the amplitude of the photocurrent didn’t show any significant difference when the 
+temperature was changed from room temperature to 50 0C whereas it decreased by 50% when the 
+temperature was changed from 50 0C to 100 0C. A slight change in photocurrent at high temperature 
+measurement (100 0C) from low temperature (50 0C) can be attributed to the career generation is 
+influenced by thermal effect associated with defects. Furthermore, the increase in current due to the 
+thermal effect of IR light is less pronounced at elevated temperatures because the change in the 
+temperature by IR heating is negligible. Therefore, the total photocurrent generation can be attributed to 
+the photo generation of carriers in the graphene.  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. 6: The photoresponse of the device in IR light due to hydrogenation at 100sccm of hydrogen 
+flow for (a) difference cycle and (b) one cycle. 
+ 
+On the other hand, the amplitude of the photocurrent didn’t show any significant difference when the 
+temperature was changed from room temperature to 50 0C whereas it decreased by 50% when the 
+temperature was changed from 50 0C to 100 0C. Smaller change in low temperature gradient can be 
+attributed to the fact that small bandgap in graphene. Furthermore, the increase in current due to the 
+thermal effect of IR light is less pronounced at elevated temperatures because the change in the 
+temperature by IR heating is negligible. Therefore, the total photocurrent generation can be attributed to 
+the photo generation of carriers in the graphene.  
+ 
+3.2.3. The effect of hydrogenation on photoconductivity 
+The effect of hydrogenation on photoresponse of the device was tested at 100 0C for different 
+concentrations of hydrogen flow rates. The device was heated at 100 0C for 30 min to ensure the thermal 
+equilibrium followed by the constant hydrogen flow for more than one hour until reach of the saturation 
+of surface of graphene by hydrogen by adsorption. The saturation was confirmed by monitoring resistance 
+changes against time using two-point probe method.  
+ 
+ 
+ 
+ 
+ 
+
+0.15026
+LtzosT'o
+(a)
+0.150234
+0.150221
+0.150208
+0.150195
+400
+600
+800
+10000.15026
+(b)
+(mA)
+0.150221
+0.150208
+0.150195
+460
+500
+Time (ms)Fig. 7 shows the photoresponse of the device at different flow rates of hydrogen. Transient responses 
+of the device were 6.05 ms and 7.27 ms in 50 sccm and 100 sccm flow rate of hydrogen gas, respectively, 
+and the corresponding values during recovery process were 7.1 ms and 7.81 ms, respectively. The 
+transient response of the device was found to differ by 17% in going from 50 to 100 sccm of hydrogen 
+flow rates. Table 1 lists the transient response and recovery times at different temperatures to compare the 
+effect of hydrogenation. 
+ 
+ 
+ 
+Fig. 7: The photoresponse of the device in IR light due to hydrogenation at (a) 50 sccm 
+and (b) 100 sccm flow rate of hydrogen gas at 100 0C. 
+ 
+Table 1: Transient response and recovery times at different temperatures. 
+Temperature 
+(0C) 
+Transient response (1) 
+(ms) 
+Transient recovery (2) 
+(ms) 
+    In vacuum 
+In hydrogen 
+(100 sccm) 
+    In vacuum In hydrogen 
+(100 sccm) 
+Room Tem. 
+ 10.04 
+13.90 
+31.26 
+44.29 
+       100  
+   6.57 
+  7.24 
+  5.91 
+  7.81 
+  
+The photoresponse of the device in hydrogen was also calculated and compared with that of the 
+device in vacuum at different temperatures. Response of the device was calculated using the formula 
+given by [25],  
+ 
+%
+100
+*
+2
+2
+1
+
+
+
+
+
+
+−
+=
+I
+I
+I
+S
+. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(2) 
+ 
+Where, I1 and I2 are the currents with and without IR light, respectively. Generally, response is calculated 
+in percentage. 
+Fig. 8 shows the comparison of the responses due to hydrogenation effect at 100 0C. The response 
+was found to decrease by around 57% when the device was hydrogenated at 50 sccm flow rate of 
+hydrogen gas while it decreased by around 68% when the flow rate was increased to 100 sccm. The effect 
+of hydrogenation was even seen substantial at room temperature compared with hydrogenation at 100 0C. 
+The flow of hydrogen was continued during cooling. The decrease in the response of the device due to 
+hydrogenation effect was observed as expected. The semiconducting Behaviour of graphene is attributed 
+
+1.4933
+1.4932
+(a)
+1.4931
+L.493
+mo
+1.4929
+1.4928
+1.492
+1L.4926
+0
+10
+28
+30
+40
+Time (ws)1.5025
+1.5024
+(b)
+1.5021
+1.502
+1.5019
+.0
+10
+20.
+30
+40
+50
+Tine (ms)to the formation of bands at the defect sites [26]. When hydronation is occurred, the conductivity can be 
+reduced to a certain extent due to the passivation of defect sites with hydrogen.  
+ 
+ 
+ 
+Fig. 8: The photoresponse of the device in IR light at 100 0C in (a) hydrogenation at 100 sccm of 
+hydrogen flow and (b) in ambient condition. 
+ 
+4. Conclusion    
+In this paper, a graphene-based IR sensor was investigated in different conditions in terms of the 
+photoresponse in the presence of light. The device was fabricated between electrode materials and the 
+presence of a monolayer of graphene was confirmed by Raman Spectroscopy. The effect of temperature 
+on photoconductivity was recorded at different temperature conditions. The photoconductivity of 
+graphene films was interpreted as due to the creation of localized bands in defect sites at the gran 
+boundaries of CVD graphene. The device exhibited a temperature-dependent effect on the photoresponse 
+behavior. The transient response and recovery times were seen reduced in the high-temperature region, 
+indicating that the thermal effect due to heating was more pronounced than the heating effect caused by 
+the IR light. It also revealed the fact that the net photocurrent change due to IR light decreases as the 
+charge carriers responsible for conduction are already excited to the conduction band due to thermal 
+heating before IR light was used. The hydrogenation effect on photoconductivity was also studied. The 
+hydrogenation caused a significant decrease in the photoresponse of the device at high temperature as 
+expected because the hydrogen ions were believed to be adsorbed at the grain boundaries and passivate 
+the defects that are responsible for photoconductivity. As the device was illuminated with a low intensity 
+(~ 2.16 µW/mm2) of IR light, the net change in the photocurrent was not significant. However, the 
+transient responses were observed around 100 times faster than the recently reported CNT-based IR 
+sensor, which may lead to the application of graphene towards ultra-fast optical response devices. 
+ 
+Acknowledgements: This research was supported by a grant (Grant #: ECCS 0925783) from National 
+Science Foundation (NSF) of USA.   
+ 
+References 
+[1] 
+S A McDonald et al. Nat. Mater. 4 (2005) 138. 
+[2]    B Pradhan, K Setyowati, H Liu, D H Waldeck and J Chen Nano Letters 8 (2008) 1142. 
+[3] 
+M Acik, G Lee, C Mattevi, M Chhowalla, K Cho and Y J Chabal Nature Mater. 9 (2010) 840. 
+[4] 
+M E Itkis, F Borondics, A Yu and R C Haddon Science 312  (2003) 413. 
+[5] 
+K Yoshino et al. Adv. Mater. 11 (1999) 1382. 
+[6] 
+C J Barber, C Winder, N S Sariciftci, J C Hummelen, A Dhanabalan and P A Hal Adv. Funct. Mater. 12 
+(2002) 709. 
+[7] 
+Y Yao, Y Liang, V Shrotriya, S Xiao, L Yu and Y Yang Adv. Mater 19  (2007) 3979. 
+[8] 
+K Wai, C Lai, N Xi, K Carmen, F Fung, H Chen and T Tarn Appl. Phys. Lett. 95  (2009) 221107. 
+[9] 
+M Freitag, Y Martin, J A Misewich, R Martel and P Avouris Nano Letters 3  (2003) 1067. 
+
+0.035
+(t)
+0.02
+0.015
+0.01
+0.005
+0
+20
+40
+60
+80
+Time.(ms)0.06
+(b)
+Response
+0.04
+0.02
+0
+30
+60
+06
+120
+150
+Time (ms)[10] S Lu and B Panchapakesan Nanotechnolgy 17 (2003) 1843. 
+[11] X Qiu, M Freitag, V Perebeinos and P Avouris  Nano Letters 5 (2005) 749. 
+[12] B Pradhan, R R Kohlmeyer, K Setyowati, H A Owen and J Chen Carbon 47  (2009) 1686. 
+[13] E Song at al. Appl. Phys. Let. 96  (2001) 081911. 
+[14] Z H Ni at al. Nano Letters l  (2007) 2758. 
+[15] M Gautam, Z Shi, and AH Jayatissa, Solar Energy Materials and Solar Cells 163 (2017) 1. 
+[16] L Xuesong et al. Science 324 (2009) 1312. 
+[17] O K Varghese et al. Sensors and Actuators B 81  (2001) 32. 
+[18] T Gupta and A H Jayatissa, Third IEEE Conference on Nanotechnology, IEEE-NANO. 2 (2003) 469. 
+[19] L Dong and Q Chen Front. Mater. Sci. China 4  (2010) 45. 
+[20] A H Neto, F Guinea, N M R Peres, K S Novoselov and A K Geim Rev. Modern Phys. 81 (2009) 45. 
+[21] A Reina et al. J. Phys. Chem. C 112  (2008) 17741. 
+[22] D Wei, Y Liu, Y Wang, H Zhang, L Huang and G Yu Nano Letters 9 (2009) 1752. 
+[23] L Xuesong et al. Nano letters 9  (2009) 4359. 
+[24] A. Ferrari et al. Phys. Rev. Let. 97 (2006) 187401. 
+[25] M Gautam, AH Jayatissa, Materials Science and Engineering: C 31 (2011) 1405 
+[26] L Liu, M Qing, Y Wang and S Chen J. Mater. Science & Technol. 31 (2015) 599 . 
+[27] J Sun, N Lin, Z Li, H Ren, C Tang and X Zhao Royal Soc. of Chemistry Adv. 6 (2016) 1090. 
+ 
+ 
+

4tFAT4oBgHgl3EQfFByL/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf,len=289
+page_content='Graphene as Infrared Light Sensor Material Ahalapitiya H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Jayatissaa) and Madhav Gautam Mechanical, Industrial, and Manufacturing Engineering (MIME) Department The University of Toledo, OH 43606, USA a)Correspondence:  ahalapitiya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='jayatissa@utoledo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='edu  Abstract: The infrared (IR) photoresponse of graphene synthesized by atmospheric chemical vapor deposition (CVD) system using a mixture of hydrogen and methane gases was studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The IR sensor devices were fabricated using graphene films transferred on to a SiO2 substrate by a lift off process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The quality of graphene was investigated with the Raman spectroscopy and optical microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The photoresponse was recorded under the illumination of IR light of wavelength 850 nm and intensity of around 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='16 µW/mm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The effects of temperature and hydrogenation on photoconductivity were also studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' It was found that the transient response and recovery times decreased with the increase of the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Hydrogenation effect also caused the significant decrease in the photoresponse of the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Although the net change in the photoresponse for IR light was lower at low illumination intensity levels, the transient responses were observed around 100 times faster than the recently reported CNT-based IR sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='  Key words: CVD graphene, single layer, Infra-Red light, photoconductivity, 2D sensor materials  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Introduction  Optoelectronic devices working in near infra-red (NIR) (800 - 2000 nm) are always demanding for different applications [1-4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' There has been significant works reported on the fabrication of optoelectronic devices using NIR materials [5-12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' In recent years, single walled carbon nanotubes (SWCNTs) have been investigated extensively as a semiconducting material for IR sensors because of its strong absorption behavior in NIR region [7-12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' One of the key challenges in developing NIR detectors is the finding of ultra fast optical response in the sensor material [5-8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Recently, strong absorption behavior in NIR region has been reported for thermally reduced graphene oxides [1,2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' This provides a pathway to use graphene as an optoelectronic material for IR detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Although the optical properties of graphene in visible region have been reported by many researchers [13-15], we have not found any research work related to the photoresponse of graphene in IR region of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' In this paper, photoresponse of graphene film on macro-scale has been reported in different conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Graphene is a monolayered carbon film with a film thickness of around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='32Å [13 - 15], where carbon atoms are arranged in a two-dimensional hexagonal lattice structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' It can be thought of as a single layer peeled off from the graphite stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' It has evolved as an interesting material due to its unique physical and electrical properties [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='  This material is different from most of the conventional semiconductors because of its zero bandgap semi-conducting behavior [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' For example, graphene-based transistor devices may operate very faster than traditional silicon devices due to high intrinsic carrier mobility (~ 2x105 cm2v-1s-1) [1, 2, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Being the material of high mechanical stress and low density (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='2 gm/cm3), it may lead to the application in nano-robotics [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' We have investigated the photoconductivity of graphene layers synthesized in atmospheric chemical vapor deposition (CVD) of CH4 on a copper substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The devices were fabricated by transferred CVD graphene onto a SiO2/Si substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The investigations were carried out to understand the temperature dependence and hydrogenation effect on photoconductivity of graphene in NIR region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Although the net change in the photoresponse for IR light was lower at low illumination intensity levels (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='16 µW/mm2),  the transient responses were observed around 100 times faster than photoconductivity of CNT for NIR lights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Experimental Procedures  The growth of graphene films was carried out on a copper (Cu) substrate (25 µm thick) in an alumina tube furnace system under the flow of methane (CH4) and hydrogen (H2) gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Copper substrate (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='999% pure, Alfa Aesar) was heated in a tube furnace under the 150 standard cubic centimeters per minute (sccm) flow of mixture of hydrogen and Argon (10% H2, 90% Ar) and annealed at 1100 0C for one hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' After annealing, graphene deposition was carried out by passing a mixture of methane and argon (5% CH4, 95% Ar) followed by the immediate cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Graphene deposited on copper by CVD method was transferred to SiO2/Si substrate by wet etching of Cu [15, 21-23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The thickness of the thermally-grown SiO2 was 118 nm as confirmed by UV spectrometry [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The Raman spectra of these films were recorded with the excitation wavelength of 530 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' In order to fabricate the IR sensors, a thin layer of gold (about 100 nm) was coated onto the transferred graphene film by a vacuum evaporation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The gold electrodes were patterned by lithography followed by etching of gold with aqueous KI/I2 solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The spacing and the length of these electrodes were 6 mm and 4 mm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' 1 shows the schematic diagram of the fabricated IR sensor and photoresponse measurement circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The device was biased with a constant voltage (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='0 V) during collection of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='  To understand the reflection of light from graphene, reflectance from bi- layer substrate (SiO2/Si) and tri-layer substrate (graphene/SiO2/Si) were measured with a double beam UV/Visible spectrometer (Shimadzu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The reflectance spectra were investigated in the spectral range 300- 1100 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='  Au  A  V0  Graphene  SiO2  IR  Light  Si  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='1: Schematic of photoresponse measurement system (V0= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='0 V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Results and Discussions  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Surface Characterization The Raman spectroscopy has been used to characterize the quality of graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The Raman spectrum of Graphene gives for main bands corresponding to the vibration mode of graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' 2 shows the as- measured Raman spectra of graphene films produced on SiO2 surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The spectrum was normalized with respect to the intensity level of 2D band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The peak at around 1580 cm-1 and 2660 cm-1, respectively, indicate the G band and the 2D band, which are characteristics Raman peaks of graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' It has been reported that the defect free monolayer graphene can be identified with characteristic features of Raman band intensities [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The intensity of 2D band is ~2 times larger than the intensity of G band suggesting that the presence of less defective graphene on SiO2 surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' This fact is also supported by the weak intensity of D-band (1350 cm-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' 2:  Raman spectra of graphene transferred to silicon wafer (SiO2 + Si) scaled with respect to the maximum peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Photoconductivity  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Dynamic response Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' 3 shows the dynamic response of photoconductivity of graphene film for the NIR light at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' 3(a) shows the response and recovery of the device when the IR light was turned on and off, respectively, whereas Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' 3(b) indicates the same characteristic for one cycle only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The intensity of the IR light source used was 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='16 µW/mm2 at the device surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Although the intensity level was very low, a clear photoresponse of device was measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The photogeneration of carriers can be primarily attributed to the creation of bands at the defect of graphene sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' When graphene is deposited on a copper plate, defects are developed at the grain boundary of polycrystalline copper films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' We believe that these defects are responsible for the creation of localized photoactive regions, which contribute to the photogeneration of carriers [26,27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The photoresponse could be characterized with a time step function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' In both the photocurrent increase and drop cases, the experimental data were fitted well into the exponential form as [10],  \uf0f7\uf0f7  \uf0f8  \uf0f6  \uf0e7\uf0e7  \uf0e8  \uf0e6 −  +  =  \uf074  t  A  I  I  o  o  exp  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='  (1)  Here, I is the current, t is the response time and Io, \uf074 and A0 are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' 4(a) and 4(b) show the fit of the response in the form explained above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The data analysis indicated that the time constants were 10 ms and 31 ms for rise and fall of the photocurrent, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content="2 2D nsityRatio (ll) 1 8'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='6 G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='2 D G 人 0 1000 1500 2000 2500 3000 Raman Shift (anl) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' 3: The photoresponse of the device due to IR light for (a) different cycles and (b) for one cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' 4: The photoresponse of the device due to IR light for (a) response and (b) recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' 5: The photoresponse of the device due to IR light at (a) 50 0C and (b) 100 0C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The effect of temperature on photoconductivity Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' 6 shows the effect of temperature on the photoconductivity of graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The photoconductivity was tested at 50 0C and 100 0C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' During the experiment, the device was heated to the desired  (a)1,252 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='2515 b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='251 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='2505 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='2495 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='249 0 50 100 150 200 250 300 Time (ms):(a)(b)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='2888 (a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='2882 (vu) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='2864 20 40 60 80 100 120 140 Time (ms)(b)temperature for 30 minutes to ensure the thermal equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Transient responses of the device were 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='26 ms and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='57 ms and the transient recovery times were 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='55 ms and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='91 ms at 50 0C and 100 0C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' A significant difference in transient response of the device was not found when the device temperature was increased from room temperature to 50 0C and transient response time decreased by 40% when the temperature was changed from 50 0C to 100 0C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Similarly, the transient recovery time decreased by 60% when the temperature was changed from room temperature to 50 0C and it decreased by 50% when the temperature was changed from 50 0C to 100 0C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' On the other hand, the amplitude of the photocurrent didn’t show any significant difference when the temperature was changed from room temperature to 50 0C whereas it decreased by 50% when the temperature was changed from 50 0C to 100 0C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' A slight change in photocurrent at high temperature measurement (100 0C) from low temperature (50 0C) can be attributed to the career generation is influenced by thermal effect associated with defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Furthermore, the increase in current due to the thermal effect of IR light is less pronounced at elevated temperatures because the change in the temperature by IR heating is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='  Therefore, the total photocurrent generation can be attributed to the photo generation of carriers in the graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' 6: The photoresponse of the device in IR light due to hydrogenation at 100sccm of hydrogen flow for (a) difference cycle and (b) one cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='  On the other hand, the amplitude of the photocurrent didn’t show any significant difference when the temperature was changed from room temperature to 50 0C whereas it decreased by 50% when the temperature was changed from 50 0C to 100 0C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Smaller change in low temperature gradient can be attributed to the fact that small bandgap in graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Furthermore, the increase in current due to the thermal effect of IR light is less pronounced at elevated temperatures because the change in the temperature by IR heating is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Therefore, the total photocurrent generation can be attributed to the photo generation of carriers in the graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The effect of hydrogenation on photoconductivity The effect of hydrogenation on photoresponse of the device was tested at 100 0C for different concentrations of hydrogen flow rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The device was heated at 100 0C for 30 min to ensure the thermal equilibrium followed by the constant hydrogen flow for more than one hour until reach of the saturation of surface of graphene by hydrogen by adsorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The saturation was confirmed by monitoring resistance changes against time using two-point probe method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content="15026 LtzosT'o (a) 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='150234 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='150221 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='150208 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='150195 400 600 800 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='15026 (b) (mA) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='150221 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='150208 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='150195 460 500 Time (ms)Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' 7 shows the photoresponse of the device at different flow rates of hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Transient responses of the device were 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='05 ms and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='27 ms in 50 sccm and 100 sccm flow rate of hydrogen gas, respectively, and the corresponding values during recovery process were 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='1 ms and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='81 ms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The transient response of the device was found to differ by 17% in going from 50 to 100 sccm of hydrogen flow rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Table 1 lists the transient response and recovery times at different temperatures to compare the effect of hydrogenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' 7: The photoresponse of the device in IR light due to hydrogenation at (a) 50 sccm and (b) 100 sccm flow rate of hydrogen gas at 100 0C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='  Table 1: Transient response and recovery times at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Temperature (0C) Transient response (\uf0741) (ms) Transient recovery (\uf0742) (ms) In vacuum In hydrogen (100 sccm) In vacuum In hydrogen (100 sccm) Room Tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='04 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='90 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='26 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='29 100 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='57 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='24 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='91 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='81  The photoresponse of the device in hydrogen was also calculated and compared with that of the device in vacuum at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Response of the device was calculated using the formula given by [25],  %  100 2  2  1  \uf0f7\uf0f7  \uf0f8  \uf0f6  \uf0e7\uf0e7  \uf0e8  \uf0e6  −  =  I  I  I  S  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='  (2)  Where, I1 and I2 are the currents with and without IR light, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Generally, response is calculated in percentage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' 8 shows the comparison of the responses due to hydrogenation effect at 100 0C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The response was found to decrease by around 57% when the device was hydrogenated at 50 sccm flow rate of hydrogen gas while it decreased by around 68% when the flow rate was increased to 100 sccm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The effect of hydrogenation was even seen substantial at room temperature compared with hydrogenation at 100 0C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The flow of hydrogen was continued during cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The decrease in the response of the device due to hydrogenation effect was observed as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The semiconducting Behaviour of graphene is attributed  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='4933 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='4932 (a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='4931 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='493 mo 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='4929 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='4928 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='492 1L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='4926 0 10 28 30 40 Time (ws)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='5025 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='5024 (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='5021 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='502 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='5019 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='0 10 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' 30 40 50 Tine (ms)to the formation of bands at the defect sites [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' When hydronation is occurred, the conductivity can be reduced to a certain extent due to the passivation of defect sites with hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' 8: The photoresponse of the device in IR light at 100 0C in (a) hydrogenation at 100 sccm of hydrogen flow and (b) in ambient condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Conclusion In this paper, a graphene-based IR sensor was investigated in different conditions in terms of the photoresponse in the presence of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The device was fabricated between electrode materials and the presence of a monolayer of graphene was confirmed by Raman Spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The effect of temperature on photoconductivity was recorded at different temperature conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The photoconductivity of graphene films was interpreted as due to the creation of localized bands in defect sites at the gran boundaries of CVD graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The device exhibited a temperature-dependent effect on the photoresponse behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The transient response and recovery times were seen reduced in the high-temperature region, indicating that the thermal effect due to heating was more pronounced than the heating effect caused by the IR light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' It also revealed the fact that the net photocurrent change due to IR light decreases as the charge carriers responsible for conduction are already excited to the conduction band due to thermal heating before IR light was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The hydrogenation effect on photoconductivity was also studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' The hydrogenation caused a significant decrease in the photoresponse of the device at high temperature as expected because the hydrogen ions were believed to be adsorbed at the grain boundaries and passivate the defects that are responsible for photoconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='  As the device was illuminated with a low intensity (~ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='16 µW/mm2) of IR light, the net change in the photocurrent was not significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' However, the transient responses were observed around 100 times faster than the recently reported CNT-based IR sensor, which may lead to the application of graphene towards ultra-fast optical response devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='  Acknowledgements: This research was supported by a grant (Grant #: ECCS 0925783) from National Science Foundation (NSF) of USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='  References [1] S A McDonald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' 4 (2005) 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' [2]    B Pradhan, K Setyowati, H Liu, D H Waldeck and J Chen Nano Letters 8 (2008) 1142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' [3] M Acik, G Lee, C Mattevi, M Chhowalla, K Cho and Y J Chabal Nature Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' 9 (2010) 840.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' [4] M E Itkis, F Borondics, A Yu and R C Haddon Science 312  (2003) 413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' [5] K Yoshino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' 11 (1999) 1382.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' [6] C J Barber, C Winder, N S Sariciftci, J C Hummelen, A Dhanabalan and P A Hal Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' 12 (2002) 709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' [7] Y Yao, Y Liang, V Shrotriya, S Xiao, L Yu and Y Yang Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Mater 19  (2007) 3979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' [8] K Wai, C Lai, N Xi, K Carmen, F Fung, H Chen and T Tarn Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' 95  (2009) 221107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' [9] M Freitag, Y Martin, J A Misewich, R Martel and P Avouris Nano Letters 3  (2003) 1067.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='035 (t) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='005 0 20 40 60 80 Time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' (ms)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='06 (b) Response 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content='02 0 30 60 06 120 150 Time (ms)[10] S Lu and B Panchapakesan Nanotechnolgy 17 (2003) 1843.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' [11] X Qiu, M Freitag, V Perebeinos and P Avouris  Nano Letters 5 (2005) 749.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' [12] B Pradhan, R R Kohlmeyer, K Setyowati, H A Owen and J Chen Carbon 47  (2009) 1686.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' [13] E Song at al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Let.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' 96  (2001) 081911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' [14] Z H Ni at al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Nano Letters l  (2007) 2758.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' [15] M Gautam, Z Shi, and AH Jayatissa, Solar Energy Materials and Solar Cells 163 (2017) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' [16] L Xuesong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Science 324 (2009) 1312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' [17] O K Varghese et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Sensors and Actuators B 81  (2001) 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' [18] T Gupta and A H Jayatissa, Third IEEE Conference on Nanotechnology, IEEE-NANO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' 2 (2003) 469.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' [19] L Dong and Q Chen Front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' China 4  (2010) 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' [20] A H Neto, F Guinea, N M R Peres, K S Novoselov and A K Geim Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Modern Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' 81 (2009) 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' [21] A Reina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' C 112  (2008) 17741.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' [22] D Wei, Y Liu, Y Wang, H Zhang, L Huang and G Yu Nano Letters 9 (2009) 1752.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' [23] L Xuesong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Nano letters 9  (2009) 4359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' [24] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Ferrari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Let.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' 97 (2006) 187401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' [25] M Gautam, AH Jayatissa, Materials Science and Engineering: C 31 (2011) 1405 [26] L Liu, M Qing, Y Wang and S Chen J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' Science & Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' 31 (2015) 599 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' [27] J Sun, N Lin, Z Li, H Ren, C Tang and X Zhao Royal Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' of Chemistry Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}
+page_content=' 6 (2016) 1090.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'}

6dAyT4oBgHgl3EQfcvcQ/content/tmp_files/2301.00287v1.pdf.txt

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+Marked Graph Mosaics
+Seonmi Choi∗
+Sam Nelson†
+Abstract
+We consider the notion of mosaic diagrams for surface-links using marked graph diagrams. We estab-
+lish bounds, in some cases tight, on the mosaic numbers for the surface-links with ch-index up to 10. As
+an application, we use mosaic diagrams to enhance the kei counting invariant for unoriented surface-links
+as well as classical knots and links.
+Keywords: Mosaic knots, Surface-links, Marked graph diagrams, kei homset enhancements
+2020 MSC: 57K12
+1
+Introduction
+Surface-links are compact surfaces smoothly embedded in R4 or S4, i.e. surfaces which are knotted and
+linked in 4-space. Surface-links include many more distinct topological types of unknotted objects – spheres,
+tori, projective planes, Klein bottles, etc. – compared with classical knots, and additionally include both
+orientable and non-orientable cases.
+Introduced in [12], marked graph diagrams are knot diagrams with marked vertices representing saddle
+points of a surface-link. A marked graph diagram satisfying certain mild conditions determines a surface-
+link up to ambient isotopy in R4, and marked graph diagrams together with the Yoshikawa moves provide
+a convenient diagrammatic calculus for combinatorial computation with surface-links. Moreover, marked
+graph diagrams and their Yoshikawa equivalence classes provide a diagrammatic way to represent cobordisms
+between classical knots and links.
+A mosaic diagram for a classical knot K is a rectangular (usually square) arrangement of square tiles
+containing crossings, arcs or nothing such that the arcs join to form a diagram of K. Mosaics were used in
+[11] to define quantum knots, elements of Hilbert spaces generated by mosaic diagrams.
+In this paper we take the first steps toward extending these constructions to the case of surface-links by
+considering mosaic presentations for surface-links using marked graph diagrams. We establish a set of tiles
+and Yoshikawa moves for marked graph mosaics and provide mosaic diagrams for each of the surface-links in
+the Yoshikawa table of surface-links with up to ch-index 10, establishing an upper bound on mosaic number
+for these surface-links. As an application we use mosaic presentations to define a new enhancement of the kei
+counting invariant for classical knots and links as well as for surface-links. As with mosaic number, we can
+compute an upper bound with respect to a certain ordering on the new enhancement from a given diagram
+of a surface-link or classical knot or link.
+The paper is organized as follows. In Section 2 we review some preliminaries about knot mosaics and
+marked graph diagrams. In Section 3 we introduce marked graph mosaics and obtain some results including
+upper bounds, some tight, on the the mosaic numbers of both orientable and non-orientable surface-links
+with ch-index less than or equal to 10. In Section 5 we define kei-colored mosaics and use them to enhance
+the kei counting invariant for classical knots and links as well as surface-links. We conclude in Section 6 with
+some questions for future research.
+∗Email: [email protected]. Partially supported by Basic Science Research Program through the National Research Founda-
+tion of Korea(NRF) funded by the Ministry of Education(2021R1I1A1A01049100) and the National Research Foundation of
+Korea (NRF) grant funded by the Korean government (MSIT) (No. 2022R1A5A1033624).
+†Email: [email protected]. Partially supported by Simons Foundation Collaboration Grant 702597.
+1
+arXiv:2301.00287v1  [math.GT]  31 Dec 2022
+
+2
+Preliminaries
+We review knot mosaics and recall surface-links, marked graph diagrams and their relationships.
+2.1
+Surface-links and marked graph diagrams
+A surface-link is the image of a closed surface smoothly (piecewise linear and locally flatly) embedded in R4
+(or S4). If it is called a surface-knot, then the underlying surface is connected. A surface-link is orientable
+if the underlying surface is orientable; otherwise, it is nonorientable or unorientable. An unoriented surface-
+link is either an unorientable surface-link or an orientable surface link without a specified orientation. Two
+surface-links F and F ′ are equivalent if there exists an orientation-preserving homeomorphism h : R4 → R4
+such that h(F) = F ′. There are many useful schemes for describing for surface-links since it is difficult to
+directly deal with surface-links in 4-space for research. For example, broken surface diagrams, marked graph
+diagrams, motion pictures etc. See [2, 5, 6, 15] for more information.
+We use an effective tool for handling surface-links known as a marked graph diagram. A marked graph is
+a spatial graph embedded in R3 possibly with 4-valent vertices decorated by a line segment like
+. We call
+such a line segment a marker and a vertex with a marker a marked vertex.
+An orientation of edges incident with a marked vertex is one of two types of the orientation, such as
+or
+. A marked graph is said to be orientable if it admits an orientation. Otherwise, it is non-orientable. Two
+(oriented) marked graphs are said to be equivalent if they are ambient isotopic in R3 keeping the rectangular
+neighborhoods and markers (with orientation). In the same way as a link diagram, one can define a marked
+graph diagram which is a diagram in R2 with classical crossings and marked vertices.
+For each marked vertex
+of a marked graph diagram D, the local diagram obtained by splicing in
+a direction consistent with its marker (say + direction), looks like
+. By applying this in the opposite
+direction (called − direction), the resulting local diagram looks like
+. Therefore one can obtain two classical
+link diagrams, denoted by L+(D) and L−(D), from D by splicing every marked vertices in + direction and
+− direction, respectively. We call L+(D) and L−(D) the positive and negative resolutions of D, respectively.
+A marked graph diagram D is said to be admissible if both resolutions L−(D) and L+(D) are trivial. A
+marked graph is said to be admissible if it has an admissible marked graph diagram. For example, it is easy
+to check that a marked graph diagram D of the spun trefoil as follows is admissible.
+D
+L_(D)
+L+(D)
+Let D be a admissible marked graph diagram. Then a surface-link F(D) can be constructed and it is
+uniquely determined from D up to equivalence. Conversely, every surface-link F can be expressed by an
+admissible marked graph diagram D, that is, F(D) is equivalent to F. See [7, 12, 15] for more details.
+For example, the correspondence between the marked graph diagram and the standard projective plane are
+illustrated in the following figure.
+R3×{0}
+R3×{1}
+R3×{-1}
+R3×[1,∞)
+R3×[-1,∞)
+R4
+2
+
+The equivalence moves Γ1, · · · , Γ8 for marked graph diagrams is called Yoshikawa moves [15].
+Γ1
+Γ2
+Γ3
+Γ4
+Γ5
+Γ8
+Γ'4
+Γ6
+Γ7
+Γ'6
+Proposition 1 ([8, 14, 15]). Two marked graph diagrams D and D′ present equivalent oriented surface-
+links if and only if D can be obtained from D′ by a finite sequence of ambient isotopies in R2 and Yoshikawa
+moves.
+Definition 1. Let K be a marked graph diagram. The ch-index of K, denoted ch(K), is the total number
+of crossings and marked vertices in K.
+2.2
+Mosaic Knots
+A mosaic (unoriented) tile is one of rectangles with arcs and possibly with one crossing, depicted as follows.
+The set of mosaic tiles T0, T1, · · · , T10 is denoted by T(u) and there are exactly 5 tiles, up to rotation. The
+endpoints of an arc on a mosaic tile are called connection points of the tile and are also located the center
+of an edge. There are tiles with 0, 2 and 4 connection points in T(u).
+4 connection points
+0 connection points
+2 connection points
+An (m, n)-mosaic is an m × n matrix whose entries are mosaic tiles in T(u). If m = n, then it is simply
+called an n-mosaic. The sets of (m, n)-mosaics and n-mosaics are denoted by M(m,n) and M(n), respectively.
+Two tiles in a mosaic are said to be contiguous if they lie immediately next to each other in the same either
+row or column. A tile in a mosaic is said to be suitably connected if all its connection points touch the
+3
+
+connection points of contiguous tiles. all its connection points meet the connection points of contiguous tiles.
+Note that for 4-mosaic illustrated above, its (2, 2)-entry tile is suitably connected, but its (3, 3)-entry tile is
+not suitably connected.
+Definition 2. A knot (m, n)-mosaic is an (m, n)-mosaic in which all tiles are suitably connected. The set of
+all knot (m, n)-mosaic is the subset of M(m,n), denoted by K(m,n). If m = n, then it is called a knot n-mosaic
+and its set is denoted by K(n).
+Example 1. The trefoil 31 has a knot 5-mosaic and 4-mosaic, as follows.
+For the equivalence for mosaic knots, there are planar isotopy moves and Reidemeister moves by using
+mosaic tiles. The non-deterministic tiles are necessary to define the moves, as follows :
+Each non-deterministic tile means two types of tiles.
+or
+or
+Non-deterministic tiles labeled by the same letter A or B are synchronized.
+ A
+ A
+ A
+ B
+ B
+ B
+ B
+ A
+The equivalence of mosaic knots consists of 11 moves for planar isotopy, 2 moves for Reidemeister moves
+I, 4 moves for Reidemeister moves II and 6 moves for Reidemeister moves III.
+0. Planar isotopy moves : 11 types
+P1
+P4
+P2
+P3
+P7
+P5
+P6
+P10
+P11
+P8
+P9
+4
+
+1. Reidemeister moves I : 2 types
+2. Reidemeister moves II : 4 types
+3. Reidemeister moves III : 6 types
+ A
+ B
+ B
+ A
+ A
+ B
+ B
+ A
+ A
+ B
+ B
+ A
+ A
+ B
+ B
+ A
+ A
+ B
+ B
+ A
+ A
+ B
+ B
+ A
+All mosaic moves are permutations on the set M(n) of n-mosaics. Indeed, they are also in the group of
+all permutations of the set K(n) of knot n-mosaics.
+Definition 3. The ambient isotopy group A(n) is the subgroup of the group of all permutations of the set
+K(n) generated by all planar isotopy moves and all Reidemeister moves.
+Two n-mosaics M and M ′ are said to be of the same knot n-type, denoted by M
+n∼ M ′, if there exists an
+element of A(n) such that it transforms M into M ′. Two n-mosaics M and M ′ are said to be of the same
+knot type if there exists a non-negative integer k such that
+ikM
+n+k
+∼ ikM ′,
+where i : M(j) → M(j+1) is the mosaic injection by adding a row and a column consisting of only empty tiles.
+In [11], Lomonaco and Kauffman conjectured that tame knot theory is equivalent to knot mosaic theory
+and in [9], Kuriya and Shehab proved the conjecture.
+Proposition 2. Let K and K′ be two knot mosaics of two tame knots k and k′, respectively. Then K and
+K′ are of the same knot mosaic type if and only if k and k′ are of the same knot type.
+Definition 4. The mosaic number of a knot (or a link) K, denoted by m(K), is the smallest integer n for
+which K can be represented by a n-mosaic.
+It is obvious that the mosaic number is an invariant for knots and links. For example, the mosaic number
+of 31 is 4 and it is easy to show this. In the papers [13, 10], they calculated the mosaic number of knots up
+to 8 crossings.
+5
+
+3
+Marked Graph Mosaics
+Let T(u)
+M denote the set of 2 Symbols, called mosaic (unoriented) tiles with markers, as follows :
+Note that the two tiles are the same up to rotation and have 4 connection points. For constructing an
+n-mosaic for marked graph diagrams, consider all tiles of T(u) ∪ T(u)
+M as elementary tiles.
+Other definitions can be defined in a manner such as mosaic knots, for instance, connection points,
+contiguous, suitably connected. An (m, n)-mosaic is an m × n matrix M = (Mij) of tiles, with rows and
+columns indexed 0, 1, · · · , m − 1 where each (i, j)-entry Mij is an element of T(u) ∪ T(u)
+M . The set of (m, n)-
+mosaics is denoted by M(m,n)
+M
+. It m = n, then an (n, n)-mosaic is a n-mosaic and its set is denoted by
+M(n)
+M .
+Definition 5. A marked graph (m, n)-mosaic is a (m, n)-mosaic in which all tiles are suitably connected.
+The set of all marked graph (m, n)-mosaic is the subset of M(m,n)
+M
+, denoted by K(m,n)
+M
+. If m = n, then it is
+called a marked graph n-mosaic and its set is denoted by K(n)
+M .
+Example 2. The marked graph diagrams 01, 21
+1 and 60,1
+1
+have the marked graph mosaics as follows.
+21
+1
+01
+61
+0,1
+For the equivalence for marked graph mosaics, there are planar isotopy moves and Yoshikawa moves by
+using mosaic tiles in T(u) ∪ T(u)
+M . The mosaic moves for planar isotopy are the same P1, · · · , P11 with knot
+mosaic moves and 4 additional moves P ′
+8, P ′
+9, P ′
+10, P ′
+11 depicted as follows.
+P10'
+P11'
+P8'
+P9'
+6
+
+Yoshikawa moves Γ1, Γ2, Γ3 are the same with Reidemeister moves I, II, III. The mosaic moves for Yoshikawa
+moves Γ4, · · · , Γ8 are as follows.
+ A
+ B
+ B
+ A
+ A
+ B
+ B
+ A
+ A
+ B
+ B
+ A
+ A
+ B
+ B
+ A
+All marked graph mosaic moves are permutations on the set M(n)
+M of n-mosaics. Indeed, they are also in
+the group of all permutations of the set K(n)
+M of marked graph n-mosaics.
+Definition 6. The ambient isotopy group A(n)
+M is the subgroup of the group of all permutations of the set
+K(n)
+M generated by all planar isotopy moves and all Yoshikawa moves.
+Two marked graph n-mosaics M and M ′ are said to be of the same marked graph n-type, denoted by
+M
+n∼ M ′, if there exists an element of A(n)
+M such that it transforms M into M ′. Two marked graph n-mosaics
+M and M ′ are said to be of the same marked graph type if there exists a non-negative integer k such that
+ikM
+n+k
+∼ ikM ′,
+where i : M(j) → M(j+1) is the mosaic injection by adding a row and a column consisting of only empty
+tiles. Therefore, we can obtain the following result.
+Theorem 3. Let M and M ′ be two marked graph mosaics of two marked graphs K and K′, respectively.
+Then M and M ′ are of the same marked graph mosaic type if and only if K and K′ are equivalent.
+For oriented surface-links, consider original oriented mosaic tiles in T(o) (see in [11]) and add 4 oriented
+mosaic tiles with markers as follows. Then we can deal with oriented marked graph mosaics similar to oriented
+knot mosaics.
+7
+
+The definition of suitably connected when an orientation is given also considers only cases where the orien-
+tation is well matched. Therefore, the oriented marked graph mosaics can also follow the same flow.
+4
+Mosaic numbers
+The marked graph diagram 81 can reduce the size of its marked graph mosaic using mosaic moves.
+Definition 7. The mosaic number of a marked graph diagram K, denoted by m(K), is the smallest integer
+n for which K can be represented by a marked graph n-mosaic.
+It is obvious that the smallest number of the mosaic size of a marked graph diagram is an invariant for
+surface-links.
+Theorem 4. The mosaic number of a marked graph diagram is an invariant for surface-links.
+It is obvious that the mosaic number of the standard sphere 01 is 2 and the mosaic numbers of both 21
+1
+and 2−1
+1
+are 4.
+For finding the mosaic numbers, one can use twofold rule, introduced in [13]. For a given (m, n)-mosaic
+D, since there are exactly two ways to connect adjacent connection points in the boundary of D, one can
+obtain exactly two marked graph (m + 2, n + 2)-mosaics �D1 and �D2, where D is suitably connected except
+the connection point of its boundary. The entry tiles of D are called inner tiles of �D1 or �D2. It is obvious
+that a crossing and a marked vertex must be located in the position of inner tiles for the suitably connected
+condition.
+or
+It is clear that if one of four inner corners has a crossing or a marked vertex and if one of two mosaics
+by the twofold rule makes a kink, then the crossing or the marked vertex can be removed by Γ1 or Γ6, Γ′
+6,
+respectively.
+Γ'6
+Γ1
+Γ6
+Theorem 5. Let K be a marked graph K. If ch(K) ≥ 7, then m(K) ≥ 6 where ch(K) denotes the ch-index
+of K.
+8
+
+Proof. Let K be a marked graph whose ch-index is greater than or equal to 7. If ch(K) ≥ 10, then m(K) ≥ 6
+because the number of inner tiles of a 5-mosaic diagram is 9. Similarly, it is easy to check that m(K) ≥ 5 if
+ch(K) ≥ 7.
+In the case that ch(K) = 8, we will show that m(K) ̸= 5. Suppose that m(K) = 5, that is, there is a
+marked graph 5-mosaic diagram D of K such that the ch-index of D is 8. Since the number of inner tiles of
+D is 9, there are 9 types for inner tiles. All cases have at least 1 row in the boundary of inner tiles, whose
+all mosaic tiles are crossings or marked vertices, as follows up to rotation.
+  ?
+  ?
+  ?
+  ?
+  ?
+  ?
+  ?   ?
+  ?
+  ?
+  ?   ?
+  ?
+  ?
+  ?
+  ?
+  ?   ?
+or
+By applying the twofold rule, the resulting mosaics have always at least one kink. Therefore, one can
+remove the corresponding crossing or marked vertex. It contradicts that the ch-index is 8. Hence, m(K) ≥ 6.
+Similar that ch(K) = 7, suppose that m(K) = 5. Let D be a marked graph 5-mosaic diagram of K with
+ch-index 7. Then there are 36 cases of its inner tiles and they have at least 1 row as depicted above except 2
+cases. By applying the same argument of the case of ch(K) = 8, 34 cases are contradictory. In the remaining
+2 cases, both have exactly two corners with no crossings and no marked vertices. Then for each cases, there
+are 4 subcases as follows.
+  ?
+  ?
+By the twofold rule, for each subcase, there two marked graph mosaics; one of them has always at least one
+kink. Since we can reduce the ch-index of D, it contradicts that the ch-index is 7 and then m(K) ≥ 6.
+or
+or
+or
+or
+9
+
+The remaining diagrams of 4 subcase are the same shown as follows.
+It has exactly one component. It contradicts that the number of components of 70,−2
+1
+has two components.
+Hence, m(K) ≥ 6.
+The following diagrams are marked graph mosaics of surface-links with ch-index ≤ 10. The size of some
+mosaic diagrams are 6 as follows. By Theorem 5, we know that their mosaic numbers are exactly 6.
+101
+1
+101
+0,0,1
+101
+1,1
+101
+0,1
+102
+0,1
+91
+91
+0,1
+101
+103
+91
+1,-2
+102
+81
+21
+1
+01
+61
+0,1
+81
+1,1
+21
+-1
+81
+-1,-1
+71
+0,-2
+101
+-2,-2
+101
+-1,-1
+101
+0,-2
+102
+0,-2
+We conclude this section with a table of mosaic numbers for surface-links of small ch-index.
+10
+
+K
+m(K)
+01
+2
+21
+1, 2−1
+1
+4
+60,1
+1
+5, 6
+70,−2
+1
+, 81,1
+1 , 8−1,−1
+1
+, 100,1
+2
+6
+81, 91, 90,1
+1 , 91,−2
+1
+, 101, 102, 100,1
+1 , 101,1
+1 , 100,−2
+2
+, 10−1,−1
+1
+6, 7
+103, 101
+1, 100,0,1
+1
+, 100,−2
+1
+, 10−2,−2
+1
+6, 7, 8
+5
+Kei-Colored Mosaic Diagrams
+Recall that a kei is a set X with a binary operation ∗ satisfying the axioms
+(i) For all x ∈ X, x ∗ x = x,
+(ii) For all x, y ∈ X, we have (x ∗ y) ∗ y = x, and
+(iii) For all x, y, z ∈ X we have (x ∗ y) ∗ z = (x ∗ z) ∗ (y ∗ z).
+A map f : X → X′ between kei is a kei homomorphism if it satisfies
+f(x ∗ y) = f(x) ∗ f(y)
+for all x, y ∈ X. Kei are also called involutory quandles; see [3] for more.
+Example 3. Every group is a kei under the operation x ∗ y = yx−1y, called the core kei of the group.
+Every surface-link L (including classical knots and links, which can be regarded as trivial cobordisms) has
+a fundamental kei K(L) whose presentation can be read from a diagram of the surface-link. More precisely,
+the fundamental kei of a surface-link has generators corresponding to sheets, i.e., connected components of
+a marked graph diagram representing L where we divide at classical undercrossings, together with relations
+at the crossings as shown (suggestively as mosaic tiles)
+The elements of the fundamental kei are then equivalence classes of kei words in these generators modulo
+the equivalence relation generated by the crossing relations and the kei axioms. The isomorphism class of
+the fundamental kei is a well-known invariant of unoriented surface-links.
+Given a finite kei X, an assignment of elements of X to the sheets of an oriented marked graph diagram
+(i.e., segments ending at undercrossing points or marked vertices) is a kei coloring (also called an X-coloring)
+of the diagram if it satisfies the crossing condition pictured above at every crossing.
+An X-coloring of a diagram D of a surface-link L defines and is defined by a unique element of the set
+of kei homomorphisms Hom(K(L), X). This homset is an invariant of surface-links for every finite kei X,
+from which useful computable invariants can be extracted. The simplest example is the cardinality of the
+set, known as the kei counting invariant, denoted ΦZ
+X(L) = |Hom(K(L), X)|.
+Generally speaking, any invariant of kei-colored diagrams (or equivalently, homset elements) yields an
+invariant known as an enhancement of the kei counting invariant. Examples include the celebrated cocyle
+invariants studied in [1] and the more recent kei module invariants introduced in [4].
+We will use mosaic diagrams to enhance the kei counting invariant in the following way. Let L be a
+surface-link with mosaic diagram D and let X be a finite kei. Assigning elements of X (called “kei colors”)
+11
+
+y
+h*
+C
+yto each of the arcs on the tiles in D such that the colors match at connection points and satisfy the kei
+coloring conditions at the crossings and marked vertices, we obtain an X-colored mosaic diagram. If we let
+f ∈ Hom(K(L), X) be the homset element represented by this coloring, we may denote the colored diagram
+by Df.
+Definition 8. Let L be a surface-link represented by a marked graph diagram D and let X be a finite kei.
+For each kei coloring f ∈ Hom(K(L), X) let us define the kei deficiency of Df as the difference between the
+cardinality of the image subkei of f and the number of kei colors appearing in Df. Let φf be the minimal
+kei deficiency over the set of minimal mosaic number diagrams Df representing f. Then the multiset
+ΦMos,M
+X
+(L) = {φf | f ∈ Hom(K(L), X)}
+is the mosaic deficiency enhancement multiset of the kei homset invariant. For ease of comparison we may
+also convert this to polynomial form by summing over the multiset terms of the form uφf to define the
+mosaic deficiency enhancement polynomial
+ΦMos
+X
+(L) =
+�
+f∈Hom(K(L),X)
+uφf .
+Since there may be many distinct equivalent diagrams of L with minimal mosaic number, to get an
+invariant we take for each coloring the minimal kei deficiency over the (finite) set of all diagrams of L with
+minimal mosaic number. Then by construction, the multiset of φf-values forms an invariant of surface-links.
+From a given minimal-mosaic number diagram of L we can obtain an upper bound on each of the φf-values;
+to compute the invariant in general requires finding the complete set of minimal-mosaic number diagrams of
+L, which can be computationally difficult.
+Let us order the set of polynomials with nonnegative integer coefficients lexicographically by exponent.
+That is, to compare two polynomials we first compare their constant terms and in case of a tie, we use
+the linear term as a tiebreaker; if the constant and linear terms are equal, we use the quadratic term as
+a tiebreaker etc. Then finding a new diagram which reduces the deficiency moves a coloring representative
+from a higher exponent into a lower exponent, yielding a smaller lexicographical position; hence it follows
+that any particular diagram yields an upper bound on the invariant.
+To prove tightness of this bound, one can check exhaustively (which we have not done in the Example
+below) that all other mosaic diagrams with the same or lesser mosaic number of the link or surface-link in
+question have the same deficiencies for their colorings representing the nontrivial homset elements.
+Remark 1. We observe that we can similarly define deficiency enhancements using crossing number or
+ch-index in place of mosaic number. Generally speaking, on any diagram with nonzero deficiency we can
+perform Reidemeister II moves to reveal “missing” colors in the image subkei. Since these moves increase
+ch-index without changing the mosaic number, we expect that these should be different invariants.
+Example 4. Consider the surface-knot 101 and the kei Core(Z5). Our python computations show that 101
+has 25 colorings by the kei Core(Z5). These include five monochromatic colorings which have deficiency zero
+12
+
+and 20 nontrivial colorings, each of which is surjective with deficiency 1 on this diagram, e.g.
+.
+Then from this diagram we obtain an upper bound 5 + 20u on the kei deficiency polynomial.
+We end this section by defining another easy-to-define but difficult-to-compute invariant us surface-links
+using mosaics and kei.
+Definition 9. Let L be a surface-link and X a finite kei. For each f ∈ Hom(K(L), X) and each positive
+integer n ≥ 2, let ρn
+f be the minimal kei deficiency value over all n-mosaic diagrams of L. Then the sequence
+{ρn
+f }∞
+n=2 is the kei deficiency spectrum for f, and as before we have an invariant multiset of such spectra.
+Remark 2. We note that since classical knots can be understood as surface-links with an empty set of
+marked vertices (i.e., trivial cobordisms between two copies of the knot), the invariants defined in this
+section are also invariants of classical knots and links.
+6
+Questions
+There remains much to be done on the topic of mosaic surface-links. Finding efficient ways to prove tightness
+of bounds is of interest, as is extending the quantum knot constructions in [11].
+Say a surface-link L is X-deficiency heterogeneous if it has at least two homset elements which require
+different minimal-mosaic number diagrams to realize their minimal X-deficiencies. Is there any such surface-
+link? For a given kei X, which is the smallest ch-index of any link which is X-deficiency heterogeneous? For
+a fixed surface-link L, for which finite kei X, if any, is L X-deficiency heterogeneous?
+A question raised by Seiichi Kamada at a talk on this topic while this paper was in preparation is whether
+the ordering of surface-links by ch-number agrees with that induced by mosaic number – e.g., does there
+exist a surface-link whose minimal ch-diagram has greater mosaic number than its minimal mosaic diagram.
+As mentioned in Remark 1, since there are moves which change the ch-index without changing the mosaic
+number, it is not clear what is the relationship between these two notations of complexity of surface-links.
+References
+[1] J. S. Carter, D. Jelsovsky, S. Kamada, L. Langford, and M. Saito. State-sum invariants of knotted
+curves and surfaces from quandle cohomology. Electron. Res. Announc. Amer. Math. Soc., 5:146–156
+(electronic), 1999.
+13
+
+4
+5
+2
+101
+4
+5[2] S. Carter, S. Kamada, and M. Saito. Surfaces in 4-space. Encyclopaedia of Mathematical Sciences.
+Springer-Verlag, 2004.
+[3] M. Elhamdadi and S. Nelson. Quandles—an introduction to the algebra of knots, volume 74 of Student
+Mathematical Library. American Mathematical Society, Providence, RI, 2015.
+[4] Y. Joung and S. Nelson. Biquandle module invariants of oriented surface-links. Proc. Amer. Math. Soc.,
+148(7):3135–3148, 2020.
+[5] S. Kamada. Braid and knot theory in dimension four. Mathematical Surveys and Monographs. American
+Mathematical Society, 2002.
+[6] S. Kamada.
+Surface-knots in 4-space.
+Springer Monographs in Mathematics. Springer, 2017.
+An
+introduction.
+[7] A. Kawauchi, T. Shibuya, and S. Suzuki. Descriptions on surfaces in four-space. i. normal forms. Math.
+Sem. Notes Kobe Univ., 10:75–125, 1982.
+[8] C. Kearton and V. Kurlin.
+All 2-dimensional links in 4-space live inside a universal 3-dimensional
+polyhedron. Algebr. Geom. Topol., 8:1223–1247, 2008.
+[9] T. Kuriya and O. Shehab.
+The lomonaco-kauffman conjecture.
+J. Knot Theory Ramifications,
+23:1450003, 20 pp., 2014.
+[10] H. J. Lee, L. Ludwig, J. Paat, and A. Peiffer. Knot mosaic tabulation. Involve, 11:13–26, 2018.
+[11] S. J. Lomonaco and L. H. Kauffman. Quantum knots and mosaics. In Quantum information science
+and its contributions to mathematics, pages 177–208. American Mathematical Society, 2010.
+[12] S. J. Lomonaco, Jr. The homotopy groups of knots. I. How to compute the algebraic 2-type. Pacific J.
+Math., 95(2):349–390, 1981.
+[13] S. Oh, K. Hong, H. Lee, and H. J. Lee. Quantum knots and the number of knot mosaics. Quantum Inf.
+Process., 14:801–811, 2015.
+[14] F. J. Swenton.
+On a calculus for 2-knots and surfaces in 4-space.
+J. Knot Theory Ramifications,
+10:1133–1141, 2001.
+[15] K. Yoshikawa. An enumeration of surfaces in four-space. Osaka J. Math., 31:497–522, 1994.
+Nonlinear Dynamics and Mathematical Application Center
+Kyungpook National University
+Daegu, 41566, Republic of Korea
+Department of Mathematical Sciences
+Claremont McKenna College
+850 Columbia Ave.
+Claremont, CA 91711 USA
+14
+

6dAyT4oBgHgl3EQfcvcQ/content/tmp_files/load_file.txt

@@ -0,0 +1,440 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf,len=439
+page_content='Marked Graph Mosaics  Seonmi Choi∗  Sam Nelson†  Abstract  We consider the notion of mosaic diagrams for surface-links using marked graph diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' We estab-  lish bounds, in some cases tight, on the mosaic numbers for the surface-links with ch-index up to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' As  an application, we use mosaic diagrams to enhance the kei counting invariant for unoriented surface-links  as well as classical knots and links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Keywords: Mosaic knots, Surface-links, Marked graph diagrams, kei homset enhancements  2020 MSC: 57K12  1  Introduction  Surface-links are compact surfaces smoothly embedded in R4 or S4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' surfaces which are knotted and  linked in 4-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Surface-links include many more distinct topological types of unknotted objects – spheres,  tori, projective planes, Klein bottles, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' – compared with classical knots, and additionally include both  orientable and non-orientable cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Introduced in [12], marked graph diagrams are knot diagrams with marked vertices representing saddle  points of a surface-link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' A marked graph diagram satisfying certain mild conditions determines a surface-  link up to ambient isotopy in R4, and marked graph diagrams together with the Yoshikawa moves provide  a convenient diagrammatic calculus for combinatorial computation with surface-links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Moreover, marked  graph diagrams and their Yoshikawa equivalence classes provide a diagrammatic way to represent cobordisms  between classical knots and links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  A mosaic diagram for a classical knot K is a rectangular (usually square) arrangement of square tiles  containing crossings, arcs or nothing such that the arcs join to form a diagram of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Mosaics were used in  [11] to define quantum knots, elements of Hilbert spaces generated by mosaic diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  In this paper we take the first steps toward extending these constructions to the case of surface-links by  considering mosaic presentations for surface-links using marked graph diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' We establish a set of tiles  and Yoshikawa moves for marked graph mosaics and provide mosaic diagrams for each of the surface-links in  the Yoshikawa table of surface-links with up to ch-index 10, establishing an upper bound on mosaic number  for these surface-links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' As an application we use mosaic presentations to define a new enhancement of the kei  counting invariant for classical knots and links as well as for surface-links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' As with mosaic number, we can  compute an upper bound with respect to a certain ordering on the new enhancement from a given diagram  of a surface-link or classical knot or link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' In Section 2 we review some preliminaries about knot mosaics and  marked graph diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' In Section 3 we introduce marked graph mosaics and obtain some results including  upper bounds, some tight, on the the mosaic numbers of both orientable and non-orientable surface-links  with ch-index less than or equal to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' In Section 5 we define kei-colored mosaics and use them to enhance  the kei counting invariant for classical knots and links as well as surface-links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' We conclude in Section 6 with  some questions for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  ∗Email: smchoi@knu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='kr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Partially supported by Basic Science Research Program through the National Research Founda-  tion of Korea(NRF) funded by the Ministry of Education(2021R1I1A1A01049100) and the National Research Foundation of  Korea (NRF) grant funded by the Korean government (MSIT) (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' 2022R1A5A1033624).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  †Email: Sam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='Nelson@cmc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Partially supported by Simons Foundation Collaboration Grant 702597.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='00287v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='GT]  31 Dec 2022  2  Preliminaries  We review knot mosaics and recall surface-links, marked graph diagrams and their relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='1  Surface-links and marked graph diagrams  A surface-link is the image of a closed surface smoothly (piecewise linear and locally flatly) embedded in R4  (or S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' If it is called a surface-knot, then the underlying surface is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' A surface-link is orientable  if the underlying surface is orientable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' otherwise, it is nonorientable or unorientable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' An unoriented surface-  link is either an unorientable surface-link or an orientable surface link without a specified orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Two  surface-links F and F ′ are equivalent if there exists an orientation-preserving homeomorphism h : R4 → R4  such that h(F) = F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' There are many useful schemes for describing for surface-links since it is difficult to  directly deal with surface-links in 4-space for research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' For example, broken surface diagrams, marked graph  diagrams, motion pictures etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' See [2, 5, 6, 15] for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  We use an effective tool for handling surface-links known as a marked graph diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' A marked graph is  a spatial graph embedded in R3 possibly with 4-valent vertices decorated by a line segment like  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' We call  such a line segment a marker and a vertex with a marker a marked vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  An orientation of edges incident with a marked vertex is one of two types of the orientation, such as  or  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' A marked graph is said to be orientable if it admits an orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Otherwise, it is non-orientable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Two  (oriented) marked graphs are said to be equivalent if they are ambient isotopic in R3 keeping the rectangular  neighborhoods and markers (with orientation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' In the same way as a link diagram, one can define a marked  graph diagram which is a diagram in R2 with classical crossings and marked vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  For each marked vertex  of a marked graph diagram D, the local diagram obtained by splicing in  a direction consistent with its marker (say + direction), looks like  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' By applying this in the opposite  direction (called − direction), the resulting local diagram looks like  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Therefore one can obtain two classical  link diagrams, denoted by L+(D) and L−(D), from D by splicing every marked vertices in + direction and  − direction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' We call L+(D) and L−(D) the positive and negative resolutions of D, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  A marked graph diagram D is said to be admissible if both resolutions L−(D) and L+(D) are trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' A  marked graph is said to be admissible if it has an admissible marked graph diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' For example, it is easy  to check that a marked graph diagram D of the spun trefoil as follows is admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  D  L_(D)  L+(D)  Let D be a admissible marked graph diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Then a surface-link F(D) can be constructed and it is  uniquely determined from D up to equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Conversely, every surface-link F can be expressed by an  admissible marked graph diagram D, that is, F(D) is equivalent to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' See [7, 12, 15] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  For example, the correspondence between the marked graph diagram and the standard projective plane are  illustrated in the following figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  R3×{0}  R3×{1}  R3×{-1}  R3×[1,∞)  R3×[-1,∞)  R4  2  The equivalence moves Γ1, · · · , Γ8 for marked graph diagrams is called Yoshikawa moves [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content="  Γ1  Γ2  Γ3  Γ4  Γ5  Γ8  Γ'4  Γ6  Γ7  Γ'6  Proposition 1 ([8, 14, 15])." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Two marked graph diagrams D and D′ present equivalent oriented surface-  links if and only if D can be obtained from D′ by a finite sequence of ambient isotopies in R2 and Yoshikawa  moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Let K be a marked graph diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' The ch-index of K, denoted ch(K), is the total number  of crossings and marked vertices in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='2  Mosaic Knots  A mosaic (unoriented) tile is one of rectangles with arcs and possibly with one crossing, depicted as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  The set of mosaic tiles T0, T1, · · · , T10 is denoted by T(u) and there are exactly 5 tiles, up to rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' The  endpoints of an arc on a mosaic tile are called connection points of the tile and are also located the center  of an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' There are tiles with 0, 2 and 4 connection points in T(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  4 connection points  0 connection points  2 connection points  An (m, n)-mosaic is an m × n matrix whose entries are mosaic tiles in T(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' If m = n, then it is simply  called an n-mosaic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' The sets of (m, n)-mosaics and n-mosaics are denoted by M(m,n) and M(n), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Two tiles in a mosaic are said to be contiguous if they lie immediately next to each other in the same either  row or column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' A tile in a mosaic is said to be suitably connected if all its connection points touch the  3  connection points of contiguous tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' all its connection points meet the connection points of contiguous tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Note that for 4-mosaic illustrated above, its (2, 2)-entry tile is suitably connected, but its (3, 3)-entry tile is  not suitably connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' A knot (m, n)-mosaic is an (m, n)-mosaic in which all tiles are suitably connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' The set of  all knot (m, n)-mosaic is the subset of M(m,n), denoted by K(m,n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' If m = n, then it is called a knot n-mosaic  and its set is denoted by K(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' The trefoil 31 has a knot 5-mosaic and 4-mosaic, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  For the equivalence for mosaic knots, there are planar isotopy moves and Reidemeister moves by using  mosaic tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' The non-deterministic tiles are necessary to define the moves, as follows :  Each non-deterministic tile means two types of tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  or  or  Non-deterministic tiles labeled by the same letter A or B are synchronized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  A  A  A  B  B  B  B  A  The equivalence of mosaic knots consists of 11 moves for planar isotopy, 2 moves for Reidemeister moves  I, 4 moves for Reidemeister moves II and 6 moves for Reidemeister moves III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Planar isotopy moves : 11 types  P1  P4  P2  P3  P7  P5  P6  P10  P11  P8  P9  4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Reidemeister moves I : 2 types  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Reidemeister moves II : 4 types  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Reidemeister moves III : 6 types  A  B  B  A  A  B  B  A  A  B  B  A  A  B  B  A  A  B  B  A  A  B  B  A  All mosaic moves are permutations on the set M(n) of n-mosaics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Indeed, they are also in the group of  all permutations of the set K(n) of knot n-mosaics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' The ambient isotopy group A(n) is the subgroup of the group of all permutations of the set  K(n) generated by all planar isotopy moves and all Reidemeister moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Two n-mosaics M and M ′ are said to be of the same knot n-type, denoted by M  n∼ M ′, if there exists an  element of A(n) such that it transforms M into M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Two n-mosaics M and M ′ are said to be of the same  knot type if there exists a non-negative integer k such that  ikM  n+k  ∼ ikM ′,  where i : M(j) → M(j+1) is the mosaic injection by adding a row and a column consisting of only empty tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  In [11], Lomonaco and Kauffman conjectured that tame knot theory is equivalent to knot mosaic theory  and in [9], Kuriya and Shehab proved the conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Let K and K′ be two knot mosaics of two tame knots k and k′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Then K and  K′ are of the same knot mosaic type if and only if k and k′ are of the same knot type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' The mosaic number of a knot (or a link) K, denoted by m(K), is the smallest integer n for  which K can be represented by a n-mosaic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  It is obvious that the mosaic number is an invariant for knots and links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' For example, the mosaic number  of 31 is 4 and it is easy to show this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' In the papers [13, 10], they calculated the mosaic number of knots up  to 8 crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  5  3  Marked Graph Mosaics  Let T(u)  M denote the set of 2 Symbols, called mosaic (unoriented) tiles with markers, as follows :  Note that the two tiles are the same up to rotation and have 4 connection points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' For constructing an  n-mosaic for marked graph diagrams, consider all tiles of T(u) ∪ T(u)  M as elementary tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Other definitions can be defined in a manner such as mosaic knots, for instance, connection points,  contiguous, suitably connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' An (m, n)-mosaic is an m × n matrix M = (Mij) of tiles, with rows and  columns indexed 0, 1, · · · , m − 1 where each (i, j)-entry Mij is an element of T(u) ∪ T(u)  M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' The set of (m, n)-  mosaics is denoted by M(m,n)  M  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' It m = n, then an (n, n)-mosaic is a n-mosaic and its set is denoted by  M(n)  M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' A marked graph (m, n)-mosaic is a (m, n)-mosaic in which all tiles are suitably connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  The set of all marked graph (m, n)-mosaic is the subset of M(m,n)  M  , denoted by K(m,n)  M  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' If m = n, then it is  called a marked graph n-mosaic and its set is denoted by K(n)  M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' The marked graph diagrams 01, 21  1 and 60,1  1  have the marked graph mosaics as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  21  1  01  61  0,1  For the equivalence for marked graph mosaics, there are planar isotopy moves and Yoshikawa moves by  using mosaic tiles in T(u) ∪ T(u)  M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' The mosaic moves for planar isotopy are the same P1, · · · , P11 with knot  mosaic moves and 4 additional moves P ′  8, P ′  9, P ′  10, P ′  11 depicted as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content="  P10'  P11'  P8'  P9'  6  Yoshikawa moves Γ1, Γ2, Γ3 are the same with Reidemeister moves I, II, III." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' The mosaic moves for Yoshikawa  moves Γ4, · · · , Γ8 are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  A  B  B  A  A  B  B  A  A  B  B  A  A  B  B  A  All marked graph mosaic moves are permutations on the set M(n)  M of n-mosaics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Indeed, they are also in  the group of all permutations of the set K(n)  M of marked graph n-mosaics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' The ambient isotopy group A(n)  M is the subgroup of the group of all permutations of the set  K(n)  M generated by all planar isotopy moves and all Yoshikawa moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Two marked graph n-mosaics M and M ′ are said to be of the same marked graph n-type, denoted by  M  n∼ M ′, if there exists an element of A(n)  M such that it transforms M into M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Two marked graph n-mosaics  M and M ′ are said to be of the same marked graph type if there exists a non-negative integer k such that  ikM  n+k  ∼ ikM ′,  where i : M(j) → M(j+1) is the mosaic injection by adding a row and a column consisting of only empty  tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Therefore, we can obtain the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Let M and M ′ be two marked graph mosaics of two marked graphs K and K′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Then M and M ′ are of the same marked graph mosaic type if and only if K and K′ are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  For oriented surface-links, consider original oriented mosaic tiles in T(o) (see in [11]) and add 4 oriented  mosaic tiles with markers as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Then we can deal with oriented marked graph mosaics similar to oriented  knot mosaics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  7  The definition of suitably connected when an orientation is given also considers only cases where the orien-  tation is well matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Therefore, the oriented marked graph mosaics can also follow the same flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  4  Mosaic numbers  The marked graph diagram 81 can reduce the size of its marked graph mosaic using mosaic moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' The mosaic number of a marked graph diagram K, denoted by m(K), is the smallest integer  n for which K can be represented by a marked graph n-mosaic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  It is obvious that the smallest number of the mosaic size of a marked graph diagram is an invariant for  surface-links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' The mosaic number of a marked graph diagram is an invariant for surface-links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  It is obvious that the mosaic number of the standard sphere 01 is 2 and the mosaic numbers of both 21  1  and 2−1  1  are 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  For finding the mosaic numbers, one can use twofold rule, introduced in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' For a given (m, n)-mosaic  D, since there are exactly two ways to connect adjacent connection points in the boundary of D, one can  obtain exactly two marked graph (m + 2, n + 2)-mosaics �D1 and �D2, where D is suitably connected except  the connection point of its boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' The entry tiles of D are called inner tiles of �D1 or �D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' It is obvious  that a crossing and a marked vertex must be located in the position of inner tiles for the suitably connected  condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  or  It is clear that if one of four inner corners has a crossing or a marked vertex and if one of two mosaics  by the twofold rule makes a kink, then the crossing or the marked vertex can be removed by Γ1 or Γ6, Γ′  6,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content="  Γ'6  Γ1  Γ6  Theorem 5." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Let K be a marked graph K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' If ch(K) ≥ 7, then m(K) ≥ 6 where ch(K) denotes the ch-index  of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  8  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Let K be a marked graph whose ch-index is greater than or equal to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' If ch(K) ≥ 10, then m(K) ≥ 6  because the number of inner tiles of a 5-mosaic diagram is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Similarly, it is easy to check that m(K) ≥ 5 if  ch(K) ≥ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  In the case that ch(K) = 8, we will show that m(K) ̸= 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Suppose that m(K) = 5, that is, there is a  marked graph 5-mosaic diagram D of K such that the ch-index of D is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Since the number of inner tiles of  D is 9, there are 9 types for inner tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' All cases have at least 1 row in the boundary of inner tiles, whose  all mosaic tiles are crossings or marked vertices, as follows up to rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  or  By applying the twofold rule, the resulting mosaics have always at least one kink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Therefore, one can  remove the corresponding crossing or marked vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' It contradicts that the ch-index is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Hence, m(K) ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Similar that ch(K) = 7, suppose that m(K) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Let D be a marked graph 5-mosaic diagram of K with  ch-index 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Then there are 36 cases of its inner tiles and they have at least 1 row as depicted above except 2  cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' By applying the same argument of the case of ch(K) = 8, 34 cases are contradictory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' In the remaining  2 cases, both have exactly two corners with no crossings and no marked vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Then for each cases, there  are 4 subcases as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  By the twofold rule, for each subcase, there two marked graph mosaics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' one of them has always at least one  kink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Since we can reduce the ch-index of D, it contradicts that the ch-index is 7 and then m(K) ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  or  or  or  or  9  The remaining diagrams of 4 subcase are the same shown as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  It has exactly one component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' It contradicts that the number of components of 70,−2  1  has two components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Hence, m(K) ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  The following diagrams are marked graph mosaics of surface-links with ch-index ≤ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' The size of some  mosaic diagrams are 6 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' By Theorem 5, we know that their mosaic numbers are exactly 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  101  1  101  0,0,1  101  1,1  101  0,1  102  0,1  91  91  0,1  101  103  91  1,-2  102  81  21  1  01  61  0,1  81  1,1  21  1  81  1,-1  71  0,-2  101  2,-2  101  1,-1  101  0,-2  102  0,-2  We conclude this section with a table of mosaic numbers for surface-links of small ch-index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  10  K  m(K)  01  2  21  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' 2−1  1  4  60,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='1  1  5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' 6  70,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='−2  1  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' 81,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='1  1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' 8−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='−1  1  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' 100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='1  2  6  81,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' 91,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' 90,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='1  1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' 91,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='−2  1  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' 101,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' 102,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' 100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='1  1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' 101,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='1  1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' 100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='−2  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' 10−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='−1  1  6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' 7  103,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' 101  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' 100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='1  1  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' 100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='−2  1  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' 10−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='−2  1  6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' 8  5  Kei-Colored Mosaic Diagrams  Recall that a kei is a set X with a binary operation ∗ satisfying the axioms  (i) For all x ∈ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' x ∗ x = x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  (ii) For all x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' y ∈ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' we have (x ∗ y) ∗ y = x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' and  (iii) For all x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' z ∈ X we have (x ∗ y) ∗ z = (x ∗ z) ∗ (y ∗ z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  A map f : X → X′ between kei is a kei homomorphism if it satisfies  f(x ∗ y) = f(x) ∗ f(y)  for all x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Kei are also called involutory quandles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' see [3] for more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Every group is a kei under the operation x ∗ y = yx−1y, called the core kei of the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Every surface-link L (including classical knots and links, which can be regarded as trivial cobordisms) has  a fundamental kei K(L) whose presentation can be read from a diagram of the surface-link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' More precisely,  the fundamental kei of a surface-link has generators corresponding to sheets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=', connected components of  a marked graph diagram representing L where we divide at classical undercrossings, together with relations  at the crossings as shown (suggestively as mosaic tiles)  The elements of the fundamental kei are then equivalence classes of kei words in these generators modulo  the equivalence relation generated by the crossing relations and the kei axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' The isomorphism class of  the fundamental kei is a well-known invariant of unoriented surface-links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Given a finite kei X, an assignment of elements of X to the sheets of an oriented marked graph diagram  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=', segments ending at undercrossing points or marked vertices) is a kei coloring (also called an X-coloring)  of the diagram if it satisfies the crossing condition pictured above at every crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  An X-coloring of a diagram D of a surface-link L defines and is defined by a unique element of the set  of kei homomorphisms Hom(K(L), X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' This homset is an invariant of surface-links for every finite kei X,  from which useful computable invariants can be extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' The simplest example is the cardinality of the  set, known as the kei counting invariant, denoted ΦZ  X(L) = |Hom(K(L), X)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Generally speaking, any invariant of kei-colored diagrams (or equivalently, homset elements) yields an  invariant known as an enhancement of the kei counting invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Examples include the celebrated cocyle  invariants studied in [1] and the more recent kei module invariants introduced in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  We will use mosaic diagrams to enhance the kei counting invariant in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Let L be a  surface-link with mosaic diagram D and let X be a finite kei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Assigning elements of X (called “kei colors”)  11  y  h*  C  yto each of the arcs on the tiles in D such that the colors match at connection points and satisfy the kei  coloring conditions at the crossings and marked vertices, we obtain an X-colored mosaic diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' If we let  f ∈ Hom(K(L), X) be the homset element represented by this coloring, we may denote the colored diagram  by Df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Let L be a surface-link represented by a marked graph diagram D and let X be a finite kei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  For each kei coloring f ∈ Hom(K(L), X) let us define the kei deficiency of Df as the difference between the  cardinality of the image subkei of f and the number of kei colors appearing in Df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Let φf be the minimal  kei deficiency over the set of minimal mosaic number diagrams Df representing f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Then the multiset  ΦMos,M  X  (L) = {φf | f ∈ Hom(K(L), X)}  is the mosaic deficiency enhancement multiset of the kei homset invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' For ease of comparison we may  also convert this to polynomial form by summing over the multiset terms of the form uφf to define the  mosaic deficiency enhancement polynomial  ΦMos  X  (L) =  �  f∈Hom(K(L),X)  uφf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Since there may be many distinct equivalent diagrams of L with minimal mosaic number, to get an  invariant we take for each coloring the minimal kei deficiency over the (finite) set of all diagrams of L with  minimal mosaic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Then by construction, the multiset of φf-values forms an invariant of surface-links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  From a given minimal-mosaic number diagram of L we can obtain an upper bound on each of the φf-values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  to compute the invariant in general requires finding the complete set of minimal-mosaic number diagrams of  L, which can be computationally difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Let us order the set of polynomials with nonnegative integer coefficients lexicographically by exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  That is, to compare two polynomials we first compare their constant terms and in case of a tie, we use  the linear term as a tiebreaker;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' if the constant and linear terms are equal, we use the quadratic term as  a tiebreaker etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Then finding a new diagram which reduces the deficiency moves a coloring representative  from a higher exponent into a lower exponent, yielding a smaller lexicographical position;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' hence it follows  that any particular diagram yields an upper bound on the invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  To prove tightness of this bound, one can check exhaustively (which we have not done in the Example  below) that all other mosaic diagrams with the same or lesser mosaic number of the link or surface-link in  question have the same deficiencies for their colorings representing the nontrivial homset elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' We observe that we can similarly define deficiency enhancements using crossing number or  ch-index in place of mosaic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Generally speaking, on any diagram with nonzero deficiency we can  perform Reidemeister II moves to reveal “missing” colors in the image subkei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Since these moves increase  ch-index without changing the mosaic number, we expect that these should be different invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Consider the surface-knot 101 and the kei Core(Z5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Our python computations show that 101  has 25 colorings by the kei Core(Z5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' These include five monochromatic colorings which have deficiency zero  12  and 20 nontrivial colorings, each of which is surjective with deficiency 1 on this diagram, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Then from this diagram we obtain an upper bound 5 + 20u on the kei deficiency polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  We end this section by defining another easy-to-define but difficult-to-compute invariant us surface-links  using mosaics and kei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Let L be a surface-link and X a finite kei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' For each f ∈ Hom(K(L), X) and each positive  integer n ≥ 2, let ρn  f be the minimal kei deficiency value over all n-mosaic diagrams of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Then the sequence  {ρn  f }∞  n=2 is the kei deficiency spectrum for f, and as before we have an invariant multiset of such spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' We note that since classical knots can be understood as surface-links with an empty set of  marked vertices (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=', trivial cobordisms between two copies of the knot), the invariants defined in this  section are also invariants of classical knots and links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  6  Questions  There remains much to be done on the topic of mosaic surface-links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Finding efficient ways to prove tightness  of bounds is of interest, as is extending the quantum knot constructions in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Say a surface-link L is X-deficiency heterogeneous if it has at least two homset elements which require  different minimal-mosaic number diagrams to realize their minimal X-deficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Is there any such surface-  link?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' For a given kei X, which is the smallest ch-index of any link which is X-deficiency heterogeneous?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' For  a fixed surface-link L, for which finite kei X, if any, is L X-deficiency heterogeneous?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  A question raised by Seiichi Kamada at a talk on this topic while this paper was in preparation is whether  the ordering of surface-links by ch-number agrees with that induced by mosaic number – e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=', does there  exist a surface-link whose minimal ch-diagram has greater mosaic number than its minimal mosaic diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  As mentioned in Remark 1, since there are moves which change the ch-index without changing the mosaic  number, it is not clear what is the relationship between these two notations of complexity of surface-links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  References  [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Carter, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Jelsovsky, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Kamada, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Langford, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Saito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' State-sum invariants of knotted  curves and surfaces from quandle cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Announc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=', 5:146–156  (electronic), 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  13  4  5  2  101  4  5[2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Carter, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Kamada, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Saito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Surfaces in 4-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Encyclopaedia of Mathematical Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Springer-Verlag, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Elhamdadi and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Nelson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Quandles—an introduction to the algebra of knots, volume 74 of Student  Mathematical Library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' American Mathematical Society, Providence, RI, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  [4] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Joung and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Nelson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Biquandle module invariants of oriented surface-links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=',  148(7):3135–3148, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  [5] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Kamada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Braid and knot theory in dimension four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Mathematical Surveys and Monographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' American  Mathematical Society, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Kamada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Surface-knots in 4-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Springer Monographs in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Springer, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  An  introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Kawauchi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Shibuya, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Suzuki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Descriptions on surfaces in four-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' normal forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Sem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Notes Kobe Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=', 10:75–125, 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  [8] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Kearton and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Kurlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  All 2-dimensional links in 4-space live inside a universal 3-dimensional  polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Algebr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Topol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=', 8:1223–1247, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  [9] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Kuriya and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Shehab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  The lomonaco-kauffman conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Knot Theory Ramifications,  23:1450003, 20 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=', 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  [10] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Lee, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Ludwig, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Paat, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Peiffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Knot mosaic tabulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Involve, 11:13–26, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  [11] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Lomonaco and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Kauffman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Quantum knots and mosaics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' In Quantum information science  and its contributions to mathematics, pages 177–208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' American Mathematical Society, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  [12] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Lomonaco, Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' The homotopy groups of knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' How to compute the algebraic 2-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Pacific J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=', 95(2):349–390, 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  [13] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Oh, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Hong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Lee, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Quantum knots and the number of knot mosaics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Quantum Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=', 14:801–811, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  [14] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Swenton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  On a calculus for 2-knots and surfaces in 4-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Knot Theory Ramifications,  10:1133–1141, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  [15] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Yoshikawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' An enumeration of surfaces in four-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Osaka J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content=', 31:497–522, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Nonlinear Dynamics and Mathematical Application Center  Kyungpook National University  Daegu, 41566, Republic of Korea  Department of Mathematical Sciences  Claremont McKenna College  850 Columbia Ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}
+page_content='  Claremont, CA 91711 USA  14' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'}

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+Towards Dependable Autonomous Systems
+Based on Bayesian Deep Learning Components
+Fabio Arnez∗, Huascar Espinoza†, Ansgar Radermacher∗ and Franc¸ois Terrier∗
+∗Universit´e Paris-Saclay, CEA, List, F-91120, Palaiseau, France
+{name.lastname}@cea.fr
+†KDT JU, TO 56 05/16, B-1049 Brussels, Belgium
+[email protected]
+Abstract—As
+autonomous
+systems
+increasingly
+rely
+on
+Deep Neural Networks (DNN) to implement the navigation
+pipeline functions, uncertainty estimation methods have be-
+come paramount for estimating confidence in DNN predictions.
+Bayesian Deep Learning (BDL) offers a principled approach to
+model uncertainties in DNNs. However, in DNN-based systems,
+not all the components use uncertainty estimation methods
+and typically ignore the uncertainty propagation between them.
+This paper provides a method that considers the uncertainty
+and the interaction between BDL components to capture the
+overall system uncertainty. We study the effect of uncertainty
+propagation in a BDL-based system for autonomous aerial
+navigation. Experiments show that our approach allows us to
+capture useful uncertainty estimates while slightly improving the
+system’s performance in its final task. In addition, we discuss the
+benefits, challenges, and implications of adopting BDL to build
+dependable autonomous systems.
+Index Terms—Bayesian Deep Learning, Uncertainty Propaga-
+tion, Unmanned Aerial Vehicle, Navigation, Dynamic Depend-
+ability
+I. INTRODUCTION
+Navigation in complex environments still represents a big
+challenge for autonomous systems (AS). Particular instances
+of this problem are autonomous driving and autonomous aerial
+navigation in the context of self-driving cars and Unmanned
+Aerial Vehicles (UAVs), respectively. In both cases, the naviga-
+tion task is addressed by first acquiring rich and complex raw
+sensory information (e.g., from camera, radar, LiDAR, etc.),
+which is then processed to drive the autonomous agent towards
+its goal. Usually, this process is done in sequence, where tasks
+and specific software components are linked together in the so-
+called perception-planning-control software pipeline [1], [2].
+Over the last decade, Deep Neural Networks (DNNs) have
+become a popular choice to implement navigation pipeline
+components thanks to their effectiveness in processing com-
+plex sensory inputs, and their powerful representation learning
+that surpasses the performance of traditional methods. Cur-
+rently, three main paradigms exist to develop and train navi-
+gation components based on DNNs: Modular (isolated), End-
+to-End (E2E) learning, and mixed or hybrid approaches [2].
+Preprint version. Accepted and presented at the 18th European Depend-
+able Computing Conference (EDCC), Zaragoza, Spain, 2022. Digital Object
+Identifier (DOI) is available in the preprint description.
+Fig. 1. UAV BDL-based Aerial Navigation Pipeline: The downstream control
+component gets predictions of the previous perception component as input and
+must take their uncertainty into account.
+Despite the remarkable progress in representation learning,
+DNNs should also represent the confidence in their predictions
+to deploy them in safety-critical systems. McAllister et al. [2]
+proposed using Bayesian Deep Learning (BDL) to implement
+the components from navigation pipelines or stacks. Bayesian
+methods offer a principled framework to model and capture
+system uncertainty. However, if the Bayesian approach is
+followed, all the components in the system pipeline should
+use BDL to enable uncertainty propagation in the pipeline.
+Hence, BDL components should admit uncertainty information
+as an input to account for the uncertainty from the outputs of
+preceding BDL components (See Fig. 1).
+In recent years, a large body of literature has employed
+uncertainty estimation methods in robotic tasks thanks to
+its potential to improve the safety of automated functions
+[3], and the capacity to increase the task performance [4],
+[5]. However, uncertainty is captured partially in navigation
+pipelines that utilize DNNs. BDL methods are used mainly in
+perception tasks, and downstream components (e.g., planning
+and control) usually ignore the uncertainty from the preceding
+components or do not capture uncertainty in their predictions.
+Although some works propagate downstream perceptual
+uncertainty from intermediate representations [6]–[8], the
+overall system output does not take into account all the
+uncertainty sources from DNN components in the pipeline.
+Moreover, proposed frameworks for dynamic dependability
+management that use uncertainty information focus only on
+DNN-based perception tasks [9], [10], ignoring uncertainty
+propagation through the system pipeline, the interactions be-
+arXiv:2301.05297v1  [cs.RO]  12 Jan 2023
+
+Perception
+Controltween uncertainty-aware components, and the potential impact
+on system performance and safety.
+Quantifying uncertainty in a BDL-based system (i.e., a
+pipeline of BDL components) still remains a challenging task.
+Uncertainties from BDL components must be assembled in
+a principled way to provide a reliable measure of overall
+system uncertainty, based on which safe decisions can be
+made [2], [11]. In this paper, we propose to capture the
+uncertainty along a pipeline of BDL components and study
+the impact of uncertainty propagation on the aerial navigation
+task in a UAV. In addition, we propose an uncertainty-centric
+dynamic dependability management framework to cope with
+the challenges that arise from propagating uncertainty through
+BDL-based systems.
+II. RELATED WORK
+A. Neural Network Uncertainty Estimation
+Bayesian neural networks (BNN) have been widely used
+to represent the confidence in the predictions. A proper con-
+fidence representation in DNN predictions can be achieved
+by modeling two sources of uncertainty: aleatoric (data)
+and epistemic (model) uncertainty. For epistemic uncertainty,
+Bayesian inference is used to estimate the posterior predic-
+tive distribution. In practice, approximate Bayesian inference
+methods are often used [12]–[15] since the posterior on the
+model parameters p(θ | D) is intractable in DNNs.
+To model data uncertainty, [14], [16] propose to incorporate
+additional outputs to represent the parameters (mean and vari-
+ance) of a Gaussian distribution. Loquercio et al. [17] forward
+propagate sensor noise through the DNN. This approach does
+not require retraining, however, it assumes a fixed uncertainty
+value for the sensor noise at the input. Another family of
+methods aim to capture complex stochastic patterns such
+as multimodality or heteroscedasticity (aleatoric uncertainty)
+using latent variables (LV) as input. When BNNs are used with
+LV (BNN+LV), both types of uncertainty can be captured [18],
+[19]. In this approach, a BNN receives an input combined with
+a random disturbance coming from an LV (i.e., features are
+partially stochastic). In contrast, this paper considers that a
+BNN can receive a complete stochastic features at the input.
+B. Uncertainty in DNN-Based Navigation
+In an autonomous driving context, perception uncertainty is
+captured from implicit [8] and explicit representations [7] and
+used downstream for scene motion forecasting and trajectory
+planning respectively. In reinforcement learning, input uncer-
+tainty has been employed for model-based [20] and model-free
+control policies [21]. In the former case, a collision predictor
+uncertainty is passed to a model predictive controller. In the
+latter, perception uncertainty is mapped to the control policy
+uncertainty using heuristics. In the context of aerial navigation,
+a few works have considered uncertainty. [17] uses a fixed
+uncertainty value for sensors as an input to a control policy.
+[6] extends the work from [22] to use the uncertainty from
+perception noisy representations downstream in a BNN control
+policy. Although these approaches use perception uncertainty
+in downstream components, not all the DNN components in
+the pipeline employ uncertainty estimation methods.
+C. Uncertainty-based Dependability Frameworks
+For the deployment of dependable autonomous systems that
+use machine learning (ML) components, Trapp et al. [23] and
+Henne et al. [9] conceptualized the use and runtime monitoring
+of perception uncertainty to ensure safe behavior on AS. To
+model system behavior, probabilistic graphical models (PGMs)
+and, in particular, Bayesian Networks (BNs) have been used in
+dependability research for safety and reliability analyses and
+risk assessment applications [24]. BNs allow incorporating ex-
+pert domain knowledge, model complex relationships between
+components, and enable decision-making under uncertainty.
+In the context of autonomous aviation systems, [10] proposes
+a method for quantifying system assurance using perception
+component uncertainty and dynamic BNs. For autonomous
+vehicles, [25] offers a framework for dynamic risk assessment,
+using BNs to predict the behavior intents of other traffic par-
+ticipants. Unlike these works, this paper considers uncertainty
+from Bayesian deep learning components beyond perception.
+III. SYSTEM TASK FORMULATION
+In this paper, we address the problem of autonomous aerial
+navigation. The goal of the autonomous agent (i.e., UAV) is
+to navigate through a set of gates with unknown locations
+disposed in a circular track. Following prior work from [6],
+[22], the navigation architecture consists of two DNN-based
+components: one for perception and the other for control
+(see Fig. 2). Both DNNs are trained following the hybrid
+paradigm. To achieve the agent goal, the navigation task is
+formulated as a sequential-decision making problem, where
+a sequence of control actions are produced given environ-
+ment observations. In this regard, the simulation environment
+provides at each time step an observation comprised of an
+RGB image x acquired from a front-facing camera on the
+UAV. The perception component defines an encoder function
+qφ : X → Z that maps the input image x to a rich low
+dimensional representation z ��� R10. Next, a control policy
+πw : Z → Y maps the compact representation z to control
+commands y = [ ˙x, ˙y, ˙z, ˙ψ] ∈ R4, corresponding to linear and
+angular (yaw) velocities in the UAV body frame.
+In the perception component, a cross-modal variational
+autoencoder (CMVAE) [22], [26] is used to learn a rich and
+robust compact representation. A CMVAE is a variant of the
+traditional variational autoencoder (VAE) [27] that learns a
+single latent representation for multiple data modalities. In
+this case, the perception dataset Dp has two data modalities:
+the RGB images and the pose of the gate relative to the UAV
+body-frame. During training, the CMVAE encoder qφ maps an
+input image x to a noisy representation with mean µφ(x) and
+variance σ2
+φ(x) in the latent space, from where latent vectors
+z are sampled, z ∼ N(µφ, σ2
+φ). Next, a latent vector z is used
+to reconstruct the input image and estimate the gate pose (i.e.,
+recover the two data modalities) using two DNNs, a decoder
+and a feed-forward network. The CMVAE encoder qφ is based
+
+Fig. 2. System architecture for aerial navigation
+on the Dronet architecture [28], and additional constraints
+on the latent space are imposed through the loss function to
+promote the learning of robust disentangled representations.
+Once the perception component is trained, the downstream
+control task (control policy π) uses a feed-forward network to
+operate on the latent vectors z at the output of the CMVAE
+encoder qφ to predict UAV velocities. To this end, the control
+policy network is added at the output of the perception
+encoder qφ, forming the navigation pipeline DNN. The control
+component π uses a control imitation learning dataset (Dc).
+During training, we freeze the perception encoder qφ to update
+only the control policy network. For more information about
+the general architecture for aerial navigation, datasets, and
+training procedures, we refer the reader to [6], [22].
+IV. METHODOLOGY
+A. Uncertainty from Perception Representations
+Although the CMVAE encoder qφ employs Bayesian in-
+ference to obtain latent vectors z, CMVAE does not capture
+epistemic uncertainty since the encoder lacks a distribution
+over parameters φ. To capture uncertainty in the perception
+encoder we follow prior work from [29], [30] that attempts to
+capture epistemic uncertainty in VAEs. We adapt the CMVAE
+to capture the posterior qΦ(z | x, Dp) as shown in (1).
+qΦ(z | x, Dp) =
+�
+q(z | x, φ)p(φ | Dp)dφ
+(1)
+To approximate (1), we take a set Φ = {φm}M
+m of encoder
+parameters samples φm ∼ p(φ | Dp), to obtain a set of
+latent samples {zm}M
+m=1 ∼ qΦ(z | x, Dp) at the output
+of the encoder. In practice, we modify CMVAE by adding
+a dropout layer in the encoder. Then, we use Monte Carlo
+Dropout (MCD) [12] to approximate the posterior on the
+encoder weights p(φ | Dp). Finally, for a given input image x
+we perform M stochastic forward passes (with dropout “turned
+on”) to compute a set of M latent vector samples z at runtime.
+B. Input Uncertainty for Control
+In BDL, downstream uncertainty propagation assumes that
+a neural network component is able to handle or admit uncer-
+tainty at the input. In our navigation case, this implies that the
+DNN-based controller is able to handle the uncertainty coming
+from the perception encoder qΦ. To capture the navigation
+model uncertainty (overall system uncertainty at the output
+of the controller), we use the Bayesian approach to compute
+the posterior predictive distribution for target variable y∗
+associated with a new input image x∗, as shown in (2).
+p(y∗ | x∗, Dc, Dp) =
+��
+π(y | z, w)p(w | Dc)qΦ(z | x∗, Dp)dwdz
+(2)
+The integrals from (2) are intractable, and we rely on
+approximations to obtain an estimation of the predictive dis-
+tribution. The posterior p(w | Dc) is difficult to evaluate,
+thus we can approximate the inner integral using an ensemble
+of neural networks [15]. In practice, we train an ensemble
+of N probabilistic control policies {πn(y | z, wn)}N
+n=1,
+with weights {wn}N
+n=1 ∼ p(w|D). Each control policy πn
+in the ensemble predicts the mean µ and variance σ2 for
+each velocity command, i.e., yµ
+=
+[µ ˙x, µ ˙y, µ ˙z, µ ˙ψ] and
+yσ2 = [σ2
+˙x, σ2
+˙y, σ2
+˙z, σ2
+˙ψ]. For training the control policy we
+use imitation learning and the heteroscedastic loss function,
+as suggested by [14], [16].
+The outer integral is approximated by taking a set of
+samples from the perception component latent space. In
+[6] latent samples are drawn using the encoder mean and
+variance z ∼ N(µφ, σ2
+φ). For the sake of simplicity, we
+directly use the samples obtained in the perception component
+{zm}M
+m ∼ qΦ(z | x, Dp) to take into account the epistemic
+uncertainty from the previous stage. Finally, the predictions
+that we get from passing each latent vector z through each
+ensemble member are used to estimate the posterior predictive
+distribution in (2). From the control policy perspective, using
+multiple latent samples z can be seen as taking a better
+“picture” of the latent space (perception representation) to
+gather more information about the environment.
+V. EXPERIMENTS & DISCUSSION
+For our experiments, we seek to study the impact of
+uncertainty propagation in the navigation pipeline. In par-
+ticular, we seek to answer the following research questions:
+RQ1. How does uncertainty from perception representations
+affect downstream component uncertainty estimation quality?
+RQ2. Can uncertainty propagation improve system perfor-
+mance? RQ3. Could uncertainty-aware components in the
+pipeline help detect challenging scenes that can threaten the
+system mission? To answer these questions we perform a
+quantitative and qualitative comparison between uncertainty-
+aware aerial navigation models.
+A. Experimental setup
+1) Navigation Model Baselines: All the navigation archi-
+tectures are based on [22] and are implemented using PyTorch.
+Table I shows the uncertainty-aware navigation architectures
+used in our experiments, detailing the type of perception
+component, the number of latent variable samples (LVS), the
+type of control policy, and the number of control prediction
+
+TrainingOnly
+Ensemble
+Probabilistic NeuralNetworks
+2
+CMVAE
+Yμl
+2
+元1
+Yol
+Z1
+Yμ3
+Z2
+:
+元3
+Yo3
+p(y* I x*, Dc, Dp)
+2
+Yμ5
+Yo5
+元5
+q(z / x, Dp)
+[Tn(y I z, Wn))N
+Perception
+ControlTABLE I
+UNCERTAINTY-AWARE NAVIGATION MODELS IN THE EXPERIMENTS
+Model
+Perception (qφ)
+LVS
+Control Policy (π)
+CPS
+M0
+MCD-CMVAE
+32
+Ensemble (N = 5) Prob.
+160
+M1
+CMVAE
+32
+Ensemble (N = 5) Prob.
+160
+M2
+CMVAE
+1
+Ensemble (N = 5) Prob.
+5
+M3
+CMVAE
+32
+Deterministic
+32
+M4
+CMVAE
+1
+Prob.
+1
+samples (CPS) at the output of the system. For instance, M0
+represents our Bayesian navigation pipeline. M0 perception
+component captures epistemic uncertainty using MCD with
+32 forward passes for each input to get 32 latent variable pre-
+dictions. For the sake of simplicity, perception predictions are
+directly used as latent variable samples in downstream control.
+The control component uses an ensemble of 5 probabilistic
+control policies obtaining 160 control prediction samples. M1
+to M4 partially capture uncertainty in the pipeline since
+they use a deterministic perception component (CMVAE). For
+the control component, M1 and M2 take 32 and 1 latent
+variable samples (LVS) respectively, and use the samples later
+with an ensemble of 5 probabilistic control policies capturing
+epistemic and aleatoric uncertainty; M3 uses 32 LVS, and
+the control component is completely deterministic; M4 uses
+1 LVS with a probabilistic control policy to capture aleatoric
+uncertainty. For UAV control, we use the expected value of
+the predicted velocities means at the output of the control
+component [14], i.e., ˆyµ = E([µ ˙x, µ ˙y, µ ˙z, µ ˙ψ]).
+2) Datasets: We use two independent datasets for each
+component in the navigation pipeline. The perception CMVAE
+uses a dataset (Dp) of 300k images where a gate is visible and
+gate-pose annotations area available. The control component
+uses a dataset (Dc) of 17k images with UAV velocity anno-
+tations. Dc is collected by flying the UAV in a circular track
+with gates, using traditional methods for trajectory planning
+and control (see [22] for more details). The perception dataset
+is divided into 80% for training, and the remaining 20% for
+validation and testing. The control dataset uses a split of 90%
+for training and the remaining for validation and testing. In
+both cases the image size is 64x64 pixels. In addition, using the
+validation data from Dp and Dc, we generate refined validation
+sub-datasets with images that have: exactly one visible gate
+(ideal situation), no visible gate in front, and multiple gates
+visible. The last two types of images represent situations that
+can pose a risk to the system task. Each sub-dataset contains
+200 images.
+B. Experiments
+In the context of RQ2, we use the validation dataset from the
+control component to measure the regression Expected Cali-
+bration Error (ECE) [31] to compare the quality of uncertainty
+estimates from navigation models at the output of the system,
+(i.e., the control component output).
+In order to answer RQ2, we evaluate our navigation archi-
+tecture under controlled simulations using the AirSim simu-
+(a) Circular track view without noise (left) and with noise (right).
+(b) UAV mission scenes
+Fig. 3. UAV Mission: Navigation tracks and scenes from birds-eye view, and
+view from UAV perspective
+lation environment. The UAV mission resembles the scenario
+and the conditions observed in the training dataset. Therefore,
+we use a circular track with eight equally spaced gates posi-
+tioned initially in a radius of 8m and constant height. To assess
+the system performance to perturbations in the environment,
+we generate new tracks adding random noise to each gate
+radius and height.
+In the context of the AirSim [32] simulation environment, a
+track is entirely defined by a set of gates, their poses in three-
+dimensional space, and the expected navigation direction of
+the agent. For perception-based navigation, the complexity of
+a track resides in the “gate-visibility” difficulty [33], [34], i.e.,
+how well the camera Field-of-View (FoV) captures the gate. A
+natural way to increase track complexity is by adding a random
+displacement to the position of each gate. A track without
+random displacement in the gates has a circular fashion. Gate
+position randomness alters the shape of the track, affecting the
+gate visibility, i.e., gates are: not visible, partially visible, or
+multiple gates can be captured in the UAV FoV as presented
+in Fig. 3. The images from these scenarios are challenging
+given its potential impact on system performance.
+To measure the system performance we take the average
+number of gates passed in all generated tracks. For track
+generation we use a random seed to produce circular tracks
+with two levels of noise in the gates offset, i.e., each random
+seed generates two (reproducible) noisy tracks. In total, we use
+6 random seeds to produce 12 tracks, 6 tracks per noise level.
+The two noise levels are a combination of Gate Radius Noise
+(GRN) and Gate Height Noise (GHN). Finally, all navigation
+models are tested in the same generated tracks for a fair
+comparison, and each model has 3 trials per track.
+To address RQ3, we perform a qualitative comparison of
+the component predicted densities using scenes (images) from
+challenging situations during the UAV mission. To this end,
+we first use the images from the generated sub-datasets. Next,
+we use the scenes from Fig. 3b as an input to the Bayesian
+navigation model M0 to analyze the effect uncertainty prop-
+agation under specific situations.
+
+口口
+口
+口
+口TABLE II
+UNCERTAINTY-AWARE NAVIGATION MODELS:
+ECE & AVG. NUMBER OF GATES PASSED
+Model
+ECE (↓)
+Performance with Track Gate Noise (↑)
+GRN ∼ U[−1.0, 1.0)
+GHN ∼ U[0, 2.0)
+GRN ∼ U[−1.5, 1.5)
+GHN ∼ U[0, 3.0)
+M0
+0.00700
+19.77
+9.22
+M1
+0.00129
+17.67
+6.0
+M2
+0.00136
+17.33
+4.0
+M3
+0.05709
+8.33
+5.0
+M4
+0.00050
+15.16
+4.38
+C. Results
+Table II summarizes the ECE for all the navigation models
+using the validation dataset from the control component. M4
+has the best uncertainty quality since the model learned to
+predict the noise from the data using the heteroscedastic
+loss function. On the contrary, M2 has the worst calibration
+results caused by the deterministic control choice and its
+inability to learn the data uncertainty. M1 and M2 have
+similar values since both receive the one noisy encoding from
+perception. However, M1 takes multiple samples from the
+noisy perception encoding which causes a reduction of the
+ECE value. Finally, M0 shows a higher ECE value compared
+to the previous models. This is caused by applying MCD in
+the perception CMVAE and the dispersion of the latent codes
+at the output of the perception encoder qΦ. The uncertainty
+quality of the downstream control is slightly affected because
+the control component did not see the same perception encod-
+ing dispersion (uncertainty) during training.
+For RQ2, Table II presents the navigation performance
+results for all the navigation models. In general, learning to
+predict uncertainty in the control component can boost the
+performance significantly. However, for M3, sampling from a
+noisy perception representation adds sufficient diversity to the
+downstream control predictions, resulting in better decisions
+than M2 in tracks with higher noise levels. In M4, the good
+performance suggests that the track noise observed at test time
+(lower noise level), resembles the data noise observed during
+the training of the single probabilistic model.
+In case of M0, the diversity from perception prediction
+samples improves the performance. Interestingly, the perfor-
+mance difference with other models is not significant. This
+situation can make us wonder if an uncertainty estimation is
+needed along the whole pipeline. Nonetheless, we believe that
+performance similarity is rooted in how we use our model
+predictions and uncertainties. The control output is computed
+by taking the mean and variance of the policy ensemble
+mixture, and only the mean values are passed to the UAV
+control. However, the multimodal predictions in Fig. 5 show
+that admitting perception uncertainty (samples) at the input of
+the control component permits the representation of ambiguity
+in the predictions. Hence, a proper use of predictions and
+associated uncertainties is needed. For example, in a bi-modal
+predictive distribution at the output, we can use the modes
+(a) Visible gate sub-dataset
+(b) No visible gate sub-dataset
+(c) Multiple gates visible sub-dataset
+Fig. 4. Navigation model standard deviation (ˆσ) prediction comparison
+(i.e., distribution peaks) instead of the expected value to avoid
+sub-optimal control decisions (e.g., near distribution valleys).
+In the context of RQ3, Fig. 4 shows the estimated uncer-
+tainty densities (ˆσ) for each velocity command at the output
+of the system, using the images from the generated datasets.
+In this case, M0 allows higher uncertainty estimates while
+reducing the dispersion in the sub-datasets from each situation.
+Fig. 5 shows M0 predictions at the output of the perception
+(z) and control (ˆµ, ˆσ) components. Predictions are made using
+the three sample images from Fig. 3b, using the LVS and CPS
+to estimate the densities.
+M0 perception and control outputs show high uncertainty
+(dispersion) values when a gate is not visible (mid-right). The
+ˆµ ˙y density suggests that the UAV control predictions will
+follow the training dataset (Dc) bias, rotating clockwise and
+moving to the right when no gate is in-front. Interestingly,
+the predicted densities in the bottom plots show that M0 is
+able to represent the ambiguity in the input, i.e. sample image
+
+DoubleorMultipleVisibleGatesSubdataset:PredictedStandardDeviationDensities
+1.2
+Navigation Model
+Velocity (m/s) or (deg/s)
+Mo
+1.0
+Mi
+M2
+0.8
+0.2
+0.0
+ox
+ModelPredictionVisibleGate Subdataset:Predicted Standard DeviationDensities
+1.2
+Navigation Model
+Mo
+1.0
+Mi
+M2
+0.8
+0.2
+0.0
+ox
+Mode/PredictionNoVisibleGate Subdataset:Predicted Standard DeviationDensities
+1.2
+Navigation Model
+Velocity (m/s) or (deg/s)
+Mo
+1.0
+M1
+M2
+0.8
+0.6
+0.4
+0.2
+0.0
+x
+ModelPrediction(a) Single gate prediction densities
+(b) No visible gate prediction densities
+(c) Double gate prediction densities
+Fig. 5. Bayesian navigation model M0: Perception qΦ predictions z density (left); Predicted velocity ˆµ density (mid); Predicted velocity ˆσ (right)
+.
+with two gates. The predicted densities have a multimodal
+distribution (two peaks) for ˆµ ˙y and ˆσ ˙y commands. Further,
+the predicted densities for the latent vector z show that
+the uncertainty from perception outputs is different for each
+type of sample, which is suitable for the early detection
+of anomalies based on uncertainty information. In addition,
+detecting multi-modality in prediction distributions can help
+expressing situations where decisions must be made.
+D. Dynamic Dependability Management using Uncertainty
+from DNN-Based Systems
+Based on the results and observations in the previous
+sub-sections, uncertainty propagation through a DNN-based
+can impact downstream component predictions and their per-
+formance. Thus, using uncertainty information to improve
+system dependability or safety can be a challenging task. For
+example, building monitoring functions based on uncertainty
+information is no simple task. The uncertainty intervals we ob-
+served for different situations present overlaps that can lead to
+false-positive or false-negative verdicts. Moreover, the multi-
+modal nature of some predictions under specific conditions or
+scenes demands knowledge of multiple intervals around the
+monitored uncertainty value. Therefore dependable and safe
+automated systems require more than a simple composition of
+predicates around some confidence measures.
+Towards building dependable autonomous systems, we pro-
+pose to align with previous frameworks that leverage percep-
+tion uncertainty (cf. subsection II-C). However existing frame-
+works for system dependability do not consider the impact
+of uncertainty propagation in uncertainty-aware systems. To
+overcome these new challenges, we propose to capture and
+use uncertainty beyond perception and consider as well the
+uncertainty from downstream components along the navigation
+pipeline, as presented in Fig.6 1 . Our approach for dynamic
+dependability management takes inspiration from [35] and
+focuses on safety. Therefore, we propose an architecture for
+dynamic risk assessment and management where we devise
+three functional blocks, as shown in Fig. 6 2 : Monitoring
+functions, risk estimation and behavior arbitration modules.
+1) Monitoring Functions: Monitoring is a widely-known
+dependability technique for runtime verification intended to
+track system variables (e.g. component inputs and outputs).
+In the automotive domain, SOTIF and ISO26262 suggest the
+use of monitoring functions as a solution for error detection in
+hardware and software components [36]. Monitoring functions
+are designed using a set of rules, based on a model of the
+system and its environment, and the properties they should
+guarantee. Hence, monitors basically perform a binary classi-
+fication task to check if a property holds or not.
+Designing monitoring functions for ML components is
+different given the probabilistic nature of the outputs and
+the difficulty in specifying the component behavior at design
+time. For ML-based components in general, typical monitoring
+
+PerceptionCMVAEEncodergoPredictionDensities
+value
+iable
+vari
+.atent
+Zo
+Z1
+Z2
+Z3
+Z4
+Z5
+Z6
+Z7
+Z8
+Z9
+LatentvectorzvariablesControl EnsembleMixtureVelocity μPredictionDensities
+Mo Prediction
+1.75
+μx
+1.50
+py
+1.25
+1.00
+0.75
+0.50
+0.25 
+0.00
+-1.0
+0.5
+0.0 
+0.5 
+1.0 
+1.5 
+2.0 
+2.5
+3.0
+3.5 
+Predictedvelocityμ(m/s)or(deg/s)Control Ensemble Mixture Velocity  Prediction Densities
+3.5 
+Mo Prediction
+3.0 
+ox
+oy
+2.5 
+Density
+02
+2.0
+1.5 
+1.0
+0.5 
+0.0
+0.0 
+0.2 
+0.4 
+0.6
+0.8 
+1.0
+Predicted velocity (m/s)or (deg/s)Perception CMVAE Encoder go Prediction Densities
+value
+variable
+.atent
+-3
+Zo
+Z1
+Z2
+Z3
+Z4
+Z5
+Z6
+Z7
+Z8
+Zg
+LatentvectorzvariablesControl Ensemble MixtureVelocity μPredictionDensities
+2.00
+Mo Prediction
+μx
+1.75
+ily
+1.50
+1.25
+1.00
+0.75 
+0.50 
+0.25 
+0.00
+1.0
+0.5
+ 0.0 
+0.5 
+1.0
+1.5
+2.0
+2.5 
+3.0 
+3.5 
+Predictedvelocityμ (m/s)or(deg/s)ControlEnsembleMixtureVelocityoPredictionDensities
+Mo Prediction
+3.0 
+0x
+2.5
+oy
+02
+1.5
+1.0
+0.5 
+0.0 
+0.0 
+0.2
+0.4
+0.6
+0.8 
+1.0
+Predicted velocity  (m/s)or (deg/s)Perception CMVAE Encoder go Prediction Densities
+value
+variable
+.atent
+Zo
+Z1
+Z2
+Z3
+Z4
+Z5
+Z6
+Z7
+Z8
+Zg
+LatentvectorzvariablesControl Ensemble Mixture Velocityμ Prediction Densities
+3.5 
+Mo Prediction
+3.0 
+px
+2.5 
+Density
+2.0
+1.5
+1.0
+0.5 
+0.0 
+0.5
+0.0 
+0.5 
+1.0 
+1.5
+2.0
+2.5 
+3.0
+Predicted velocity μ(m/s)or(deg/s)Control Ensemble Mixture Velocity  Prediction Densities
+4.0
+Mo Prediction
+ox
+3.5 
+<
+3.0 
+02
+2.0
+1.5
+1.0 
+ 0.5 
+0.0 
+0.0 
+0.2 
+0.4 
+0.6 
+0.8 
+1.0 
+Predicted velocity (m/s)or (deg/s)Fig. 6. Runtime risk assessment & management framework
+function tasks include Out-of-Distribution (OoD) detection or
+Out-of-Boundary (OoB) detection and can be implemented
+with rules, data-driven methods or a mix of both.
+2) Probabilistic Inference for Risk Assessment: To enable
+dynamic uncertainty-aware reasoning and provide context to
+risk estimates, we propose to use Bayesian networks. Fol-
+lowing the methodology described in [24], BNs for risk
+assessment and safety can be constructed using a combination
+of expert domain knowledge and data. The experts provide a
+model of causal relations and can have support from traditional
+dependability analysis (e.g., fault tree analysis) to build the BN
+structure while system data is used to provide the conditional
+probabilities between random variables.
+In our framework, the BN of the system can receive
+the predictions from components in the pipeline (probability
+distributions) and the verdicts from monitoring functions ap-
+plied to system sensors, component predictions, and relevant
+environmental variables. The output of the BN is represented
+by all the critical events identified by experts. Hence, during
+inference, the BN estimates the probability of a critical event,
+which is used along with its severity to compute the system’s
+risk at runtime [37]. Though we focus on risk assessment, in a
+general way the output of BNs can be any assurance measure
+variables linked to dependability attributes [10]. Further, the
+BN should handle uncertain evidence [38] to preserve the
+probabilistic nature of component and monitor predictions.
+3) Behavior Arbitration: The last building block in our
+framework aims at keeping the system in a safe state by
+taking or discarding navigation pipeline predictions. Safe
+decisions must be made in the presence of high-risk values in
+a given context caused by erroneous component predictions or
+associated uncertainties and external environmental variables.
+To this end, we propose using Behavior Trees (BTs) to adopt
+different system behaviors while facing high-risk situations.
+BTs are sophisticated modular decision-making engines for
+reactive and fault-tolerant task execution [39]. Compositions
+of BTs can preserve safety and robustness properties [40] and
+are widely adopted tools in robotics. In the context of our
+system, we can have a dedicated behavior to search for a gate
+when we detect that there are no gates in the UAV FoV. This
+behavior will put the system back into a state where the levels
+of uncertainty do not represent a risk.
+VI. CONCLUSION
+We presented a method to capture and propagate uncertainty
+along a navigation pipeline implemented with Bayesian deep
+learning components for UAV aerial navigation. We analyzed
+the effect of uncertainty propagation regarding system com-
+ponent predictions and performance. Our experiments show
+that our approach to capturing and propagating uncertainty
+along the system can provide valuable predictions for decision-
+making and identifying situations that are critical for the
+system. However, proper use and management of component
+predictions and uncertainty estimates are needed to create
+dependable and highly-performant systems. In this sense and
+based on our observations, we also proposed a framework for
+system dependability management using system uncertainty
+and focused on safety. In future work, we aim to implement
+our proposed dependability framework and explore sampling-
+free methods [41] for uncertainty estimation to reduce the
+computational budget and memory footprint in our approach.
+ACKNOWLEDGMENT
+This work has received funding from the COMP4DRONES
+project, under ECSEL Joint Undertaking (JU) grant agreement
+N°826610. The ECSEL JU receives support from the European
+Union’s Horizon 2020 research and innovation programme and
+from Spain, Austria, Belgium, Czech Republic, France, Italy,
+Latvia, Netherlands.
+REFERENCES
+[1] S. Grigorescu, B. Trasnea, T. Cocias, and G. Macesanu, “A survey
+of deep learning techniques for autonomous driving,” Journal of Field
+Robotics, vol. 37, no. 3, pp. 362–386, 2020.
+[2] R. McAllister, Y. Gal, A. Kendall, M. Van Der Wilk, A. Shah, R. Cipolla,
+and A. Weller, “Concrete problems for autonomous vehicle safety: Ad-
+vantages of bayesian deep learning,” in Proceedings of the Twenty-Sixth
+International Joint Conference on Artificial Intelligence.
+International
+Joint Conferences on Artificial Intelligence, Inc., 2017.
+[3] R. Michelmore, M. Kwiatkowska, and Y. Gal, “Evaluating uncertainty
+quantification in end-to-end autonomous driving control,” arXiv preprint
+arXiv:1811.06817, 2018.
+[4] F. Nozarian, C. M¨uller, and P. Slusallek, “Uncertainty quantification
+and calibration of imitation learning policy in autonomous driving.” in
+TAILOR, 2020, pp. 146–162.
+[5] E. Ohn-Bar, A. Prakash, A. Behl, K. Chitta, and A. Geiger, “Learning
+situational driving,” in Proceedings of the IEEE/CVF Conference on
+Computer Vision and Pattern Recognition, 2020, pp. 11 296–11 305.
+[6] F. Arnez, H. Espinoza, A. Radermacher, and F. Terrier, “Improving
+robustness of deep neural networks for aerial navigation by incorporating
+input uncertainty,” in International Conference on Computer Safety,
+Reliability, and Security.
+Springer, 2021, pp. 219–225.
+[7] B. Ivanovic, K.-H. Lee, P. Tokmakov, B. Wulfe, R. McAllister,
+A. Gaidon, and M. Pavone, “Heterogeneous-agent trajectory forecasting
+incorporating class uncertainty,” arXiv preprint arXiv:2104.12446, 2021.
+[8] S. Casas, C. Gulino, S. Suo, K. Luo, R. Liao, and R. Urtasun,
+“Implicit latent variable model for scene-consistent motion forecasting,”
+in Computer Vision–ECCV 2020: 16th European Conference, Glasgow,
+UK, August 23–28, 2020, Proceedings, Part XXIII 16.
+Springer, 2020,
+pp. 624–641.
+[9] M. Henne, A. Schwaiger, and G. Weiss, “Managing uncertainty of ai-
+based perception for autonomous systems.” in AISafety@ IJCAI, 2019.
+
+Bayesian Deep Learning-Based Navigation Pipeline
+(Uncertainty Propagation Between Components)
+Sensors
+Perception
+Planning
+Control
+Monitoring Functions
+(Data-Driven / STL / Mixed)
+Environment
+Graphical Model
+Probabilistic
+Behavior
+Arbitration
+个
+个
+Expertknowledge&Data
+Risk Estimation
+2
+Risk Assessment & Management[10] E. Asaadi, E. Denney, and G. Pai, “Quantifying assurance in learning-
+enabled systems,” in International Conference on Computer Safety,
+Reliability, and Security.
+Springer, 2020, pp. 270–286.
+[11] A. Lavin, C. M. Gilligan-Lee, A. Visnjic, S. Ganju, D. Newman,
+S. Ganguly, D. Lange, A. G. Baydin, A. Sharma, A. Gibson et al.,
+“Technology readiness levels for machine learning systems,” arXiv
+preprint arXiv:2101.03989, 2021.
+[12] Y. Gal and Z. Ghahramani, “Dropout as a bayesian approximation:
+Representing model uncertainty in deep learning,” in international
+conference on machine learning, 2016, pp. 1050–1059.
+[13] Y. Gal, J. Hron, and A. Kendall, “Concrete dropout,” in Advances in
+neural information processing systems, 2017, pp. 3581–3590.
+[14] B. Lakshminarayanan, A. Pritzel, and C. Blundell, “Simple and scalable
+predictive uncertainty estimation using deep ensembles,” in Advances in
+neural information processing systems, 2017, pp. 6402–6413.
+[15] F. K. Gustafsson, M. Danelljan, and T. B. Sch¨on, “Evaluating scalable
+bayesian deep learning methods for robust computer vision,” arXiv
+preprint arXiv:1906.01620, 2019.
+[16] A. Kendall and Y. Gal, “What uncertainties do we need in bayesian
+deep learning for computer vision?” in Advances in neural information
+processing systems, 2017, pp. 5574–5584.
+[17] A. Loquercio, M. Segu, and D. Scaramuzza, “A general framework for
+uncertainty estimation in deep learning,” IEEE Robotics and Automation
+Letters, vol. 5, no. 2, pp. 3153–3160, 2020.
+[18] S. Depeweg, J. Hern´andez-Lobato, F. Doshi-Velez, and S. Udluft,
+“Learning and policy search in stochastic dynamical systems with
+bayesian neural networks,” in 5th International Conference on Learning
+Representations, ICLR 2017-Conference Track Proceedings, 2017.
+[19] S. Depeweg, J.-M. Hernandez-Lobato, F. Doshi-Velez, and S. Udluft,
+“Decomposition of uncertainty in bayesian deep learning for efficient
+and risk-sensitive learning,” in International Conference on Machine
+Learning.
+PMLR, 2018, pp. 1184–1193.
+[20] B. L¨utjens, M. Everett, and J. P. How, “Safe reinforcement learning
+with model uncertainty estimates,” in 2019 International Conference on
+Robotics and Automation (ICRA).
+IEEE, 2019, pp. 8662–8668.
+[21] T. Fan, P. Long, W. Liu, J. Pan, R. Yang, and D. Manocha, “Learning
+resilient behaviors for navigation under uncertainty,” in 2020 IEEE
+International Conference on Robotics and Automation (ICRA).
+IEEE,
+2020, pp. 5299–5305.
+[22] R. Bonatti, R. Madaan, V. Vineet, S. Scherer, and A. Kapoor, “Learning
+visuomotor policies for aerial navigation using cross-modal representa-
+tions,” arXiv preprint arXiv:1909.06993, 2019.
+[23] M. Trapp, D. Schneider, and G. Weiss, “Towards safety-awareness
+and dynamic safety management,” in 2018 14th European Dependable
+Computing Conference (EDCC).
+IEEE, 2018, pp. 107–111.
+[24] S. Kabir and Y. Papadopoulos, “Applications of bayesian networks and
+petri nets in safety, reliability, and risk assessments: A review,” Safety
+science, vol. 115, pp. 154–175, 2019.
+[25] J. Reich, M. Wellstein, I. Sorokos, F. Oboril, and K.-U. Scholl, “Towards
+a software component to perform situation-aware dynamic risk assess-
+ment for autonomous vehicles,” in Dependable Computing–EDCC 2021
+Workshops: DREAMS, DSOGRI, SERENE 2021, Munich, Germany,
+September 13, 2021, Proceedings.
+Springer Nature, 2021, p. 3.
+[26] A. Spurr, J. Song, S. Park, and O. Hilliges, “Cross-modal deep varia-
+tional hand pose estimation,” in Proceedings of the IEEE Conference on
+Computer Vision and Pattern Recognition, 2018, pp. 89–98.
+[27] D. P. Kingma and M. Welling, “Auto-encoding variational bayes,” arXiv
+preprint arXiv:1312.6114, 2013.
+[28] A. Loquercio, A. I. Maqueda, C. R. Del-Blanco, and D. Scaramuzza,
+“Dronet: Learning to fly by driving,” IEEE Robotics and Automation
+Letters, vol. 3, no. 2, pp. 1088–1095, 2018.
+[29] E. Daxberger and J. M. Hern´andez-Lobato, “Bayesian variational
+autoencoders for unsupervised out-of-distribution detection,” arXiv
+preprint arXiv:1912.05651, 2019.
+[30] A. Jesson, S. Mindermann, U. Shalit, and Y. Gal, “Identifying causal-
+effect inference failure with uncertainty-aware models,” Advances in
+Neural Information Processing Systems, vol. 33, pp. 11 637–11 649,
+2020.
+[31] V. Kuleshov, N. Fenner, and S. Ermon, “Accurate uncertainties for deep
+learning using calibrated regression,” arXiv preprint arXiv:1807.00263,
+2018.
+[32] S. Shah, D. Dey, C. Lovett, and A. Kapoor, “Airsim: High-fidelity visual
+and physical simulation for autonomous vehicles,” in Field and service
+robotics.
+Springer, 2018, pp. 621–635.
+[33] R. Madaan, N. Gyde, S. Vemprala, M. Brown, K. Nagami, T. Taubner,
+E. Cristofalo, D. Scaramuzza, M. Schwager, and A. Kapoor, “Airsim
+drone racing lab,” arXiv preprint arXiv:2003.05654, 2020.
+[34] Y. Song, M. Steinweg, E. Kaufmann, and D. Scaramuzza, “Autonomous
+drone racing with deep reinforcement learning,” in 2021 IEEE/RSJ
+International Conference on Intelligent Robots and Systems (IROS).
+IEEE, 2021, pp. 1205–1212.
+[35] P. Moosbrugger, K. Y. Rozier, and J. Schumann, “R2u2: monitoring
+and diagnosis of security threats for unmanned aerial systems,” Formal
+Methods in System Design, vol. 51, no. 1, pp. 31–61, 2017.
+[36] S. Mohseni, M. Pitale, V. Singh, and Z. Wang, “Practical solutions
+for machine learning safety in autonomous vehicles,” arXiv preprint
+arXiv:1912.09630, 2019.
+[37] J. Eggert, “Risk estimation for driving support and behavior planning
+in intelligent vehicles,” at-Automatisierungstechnik, vol. 66, no. 2, pp.
+119–131, 2018.
+[38] A. B. Mrad, V. Delcroix, S. Piechowiak, P. Leicester, and M. Abid,
+“An explication of uncertain evidence in bayesian networks: likelihood
+evidence and probabilistic evidence,” Applied Intelligence, vol. 43, no. 4,
+pp. 802–824, 2015.
+[39] M. Colledanchise and L. Natale, “On the implementation of behavior
+trees in robotics,” IEEE Robotics and Automation Letters, vol. 6, no. 3,
+pp. 5929–5936, 2021.
+[40] M. Colledanchise and P. ¨Ogren, “How behavior trees modularize ro-
+bustness and safety in hybrid systems,” in 2014 IEEE/RSJ International
+Conference on Intelligent Robots and Systems.
+IEEE, 2014, pp. 1482–
+1488.
+[41] B. Charpentier, O. Borchert, D. Z¨ugner, S. Geisler, and S. G¨unnemann,
+“Natural posterior network: Deep bayesian predictive uncertainty for ex-
+ponential family distributions,” arXiv preprint arXiv:2105.04471, 2021.
+

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+DEEP REINFORCEMENT LEARNING FOR ASSET ALLOCATION:
+REWARD CLIPPING
+Jiwon Kim
+SK Inc.(SK C&C)
+[email protected]
+MOON-JU KANG
+[email protected]
+KangHun Lee
+SK Inc.(SK C&C)
+[email protected]
+HyungJun Moon
+SK Inc.(SK C&C)
+[email protected]
+BO-KWAN JEON
+SK Inc.(SK C&C)
+[email protected]
+ABSTRACT
+Recently, there are many trials to apply reinforcement learning in asset allocation for earning
+more stable profits. In this paper, we compare performance between several reinforcement learning
+algorithms - actor-only, actor-critic and PPO models. Furthermore, we analyze each models’ character
+and then introduce the advanced algorithm, so called Reward clipping model. It seems that the
+Reward Clipping model is better than other existing models in finance domain, especially portfolio
+optimization - it has strength both in bull and bear markets. Finally, we compare the performance for
+these models with traditional investment strategies during decreasing and increasing markets.
+Keywords DEEP REINFORCEMENT LEARNING · PORTFOLIO MANAGEMENT · POLICY GRADIENT ·
+PROXIMAL POLICY OPTIMIZATION · REWARD CLIPPING
+1
+Introduction
+In recent years, AI algorithms-deep or machine learnings are used in financial market for various fields like stock
+prediction, auto trading, deep hedging, [1]. At the same time, passing through the bull market right after COVID-19, we
+are currently experiencing difficulties in dealing with market by inflation and the rising interest rate. In this situation, to
+get more return with less risk, asset allocation(portfolio optimization) using Robo-Advisor with reinforcement learning
+is in the spotlight.
+Zhipeng Liang et al. implement three reinforcement learning algorithm - DDPG, PPO and Adversarial PG in portfolio
+management in [2]. They showed PG algorithm outperforms URCP in China stock market. Also, in [3], Farzan
+Soleymani and Eric Paquet present a deep reinforcement learning combined with a restricted stacked autoencoder and a
+convolutional neural network in portfolio management. Here they apply SARSA algorithm which is enforced with a
+CNN. Jung hoon Kim proposed reinforcement learning to make a short position especially in downward trends of stock
+markets ([4]).
+This paper is composed of three parts. Firstly, we conduct existing three reinforcement algorithms: actor-only, actor-
+critic and PPO. In several research, it is shown that PPO has potential in portfolio management, [5], [6]. Especially,
+Amine Mohamed Aboussalah et al. provide the stability of several RL models including PPO with a cross-sectional
+analysis ([7]).
+Here, our all three models are based on policy gradient method. Note that from [8] chapter13, in a policy gradient
+method, the reward function is defined by
+J(θ) = Σs∈Sµπ(s)Σa∈Aπθ(a|s)qπ(s, a)
+(1)
+where µπ(s) = limt→∞P(st = s|s0, πθ) the stationary distribution for the policy πθ.
+Furthermore (see [8])
+arXiv:2301.05300v1  [q-fin.CP]  2 Jan 2023
+
+Reinforcement Learning in Asset Allocation
+∇θJ(θ) ∝ µπ(s)Σa∈Aqπ(s, a)∇θπθ(a|s)
+(2)
+Our actor-critic model contains this policy gradient method, and actor-only model has the only actor part of the
+actor-critic model.
+Also, from [9], for PPO algorithm we use
+LCLIP (θ) = ˆEt[min(rt(θ) ˆAt, clip(rt(θ), 1 − ϵ, 1 + ϵ) ˆAt)]
+(3)
+where ˆEt indicates the empirical average over a finite batch of samples, in an algorithm that alternates between sampling
+and optimization and ˆAt is an estimator of the advantage function at timestep t. And
+LCLIP ′(θ) = ˆEt[LCLIP (θ) − c1(Vθ(s) − Vtarget)2 + c2H(s, πθ)]
+(4)
+where c1 and c2 are hyperparameter constants. Here, the equation(4) means that when applying PPO for policy (actor)
+as well as value (critic) functions, besides the clipped reward, the objective function is strengthened with an error term
+on the value estimation and an entropy term to incentivize sufficient exploration [6].
+Secondly, after checking the performance of three existing RL models and analyzing the characteristics of them, we
+introduce the modified new model which is called Reward Clipping model. When we test three RL models, actor-only
+and actor-critic models show high-risk and high-return. They gain high profitability in a bull market, but also have a big
+loss rate during a bear market. On the other hand, PPO model moves opposite way. It shows good defensive movement
+when a stock market is decreasing, but it cannot get enough return when a stock market is growing. So we combine
+these models to get advantages only - the result model gets high return during bull market but also good defense in a
+bear market. For this, we use modified PPO algorithm:
+LCLIP (θ)NEW = ˆEt[min( ˆAt, clip( ˆAt, 1 − ϵ1, 1 + ϵ2)]
+(5)
+In original PPO algorithm (equation (3)), clipping is given for the probability ratio rt(θ) =
+πθ(at|st)
+πθold(at|st), i.e. in our case
+for the proportion of each asset(product). But in financial market stability is needed for advantages- return, MDD and
+so on, not for portions of portfolio. Furthermore, bigger return and sharpe ratio are better, we set different values ϵ1, ϵ2
+saying lower and upper bounds. So we modify the clipping logic in PPO to equation (5). This is more intuitive since by
+controlling the advantage function directly, we can get immediate effects in our rewards. And it looks that this is more
+fittable model in finance area.
+Finally, we compare performance of RL models with traditional quant investment strategies - All Weather Portfolio, 6:4
+(equity:bond) and equal weight rules. These results can suggest us the direction of our RL models and give necessity of
+use of AI models in financial portfolio optimization.
+In the following experiments, we use two sets of products. The first set is composed of 68 products, 22 in Europe,
+Korea, US bond, 44 in US, Europe, Korea, Japan equity and 2 in gold. From this we can see that the RL models give us
+not only an optimal asset allocation but also a product selection. For the second set, we use 25 products, 16 in US and
+KOREA stocks, 4 in intermediate-term treasuries, 2 in long-term treasuries, 2 in commodities including REITs and
+gold. With the second product set, we compare the performance of RL models to ALL Weather Portfolio strategy.
+2
+Existing Models
+In this section, we implement three different existing methodologies, actor-only, actor-critic and PPO in asset opti-
+mization. We show how each models work, especially in the view of returns, sharpe ratio, standard deviation and
+MDD.
+2.1
+Construction and Experiments
+In our experiments, one state includes previous closing price, volume or some other financial indices in a fixed window.
+And an action is the desired allocating weights.
+The actor-only model is the actor part of the actor-critic model. And PPO model is the model constructed from the
+actor-critic model by replacing actor part to PPO algorithm. Hence all three models have the same architecture for the
+actor part. The following Figure 1 is the common architecture of three models and the output after doing softmax is the
+proportion for each asset.
+2
+
+Reinforcement Learning in Asset Allocation
+Figure 1: Architecture
+Note that Q-value function is estimated using a function approximator with weight vector θ : Q(s, a; θ) for action
+values. And DQN iteratively improves an estimate of Q∗ by minimizing the sequence of loss functions:
+Li(θi) = Es,a,r,s′[(yDQN
+i
+− Q(s, a; θi))2],
+(6)
+with
+yDQN
+i
+= r + γmaxa′Q(s′, a′; θi−1)
+(7)
+Harm van Seijen et al. proposed in [10] to decompose the reward function Renv into n reward functions (see Figure 1
+in [10]):
+Renv(s, a, s′) =
+n
+�
+k=1
+Rk(s, a, s′),
+(8)
+for all s, a, s′, and to train a separate reinforcement-learning agent on each of these reward functions. Hence the
+associated sequence of loss function is:
+Li(θi)′ = Es,a,r,s′[
+n
+�
+k=1
+(yk,i − Qk(s, a; θi))2],
+(9)
+with
+yk,i = Rk(s, a, s′) + γ
+�
+a′∈A
+1
+|A|Qk(s′, a′; θi−1).
+(10)
+(See [10]) Here, we use return, sharpe ratio and antibias as our rewards.
+2.2
+Experimental Results
+Following is the result for three RL models: actor-only, actor-critic(AC) and PPO. We train the models from 2010-01-01
+to 2019-06-10, and test them from 2019-07-18 to 2021-06-16. We want to see the movement of models for sharp
+drawing down and increasing stock market during the COVID-19. For this experiment, the first data set is used (a
+product selection of 68 products is also reflected).
+From the Table 1 and Figure 2, we can see that Actor-only and AC models have big draw down(MDD) but AC has good
+return. On the other hand, PPO model has less MDD than other two models, but small return too. Also, Figure 2 shows
+3
+
+CNN
+CNN
+CNN
+Conv1
+Canv1
+Conv1
+Conv2
+Cov2
+Conv2
+BatchNormallzation
+BatchNormalization
+BatchNormallzation
+Max Pooling
+Max Pooling
+Max Pooling
+Q
+Dropout
+Dropout
+Dropout
+R
+.
+.
+Conv
+Con
+Conv
+Q:
+BatchNormalization
+BathNomalization
+BahNgwalanon
+Fc1
+Fc1
+Fc1
+DNN
+Fc2 (Dropout)
+Fc3 (Dropout)
+10 1m =I=0
+Softmax
+0.1
+0.3
+0.4
+0.2Reinforcement Learning in Asset Allocation
+Model
+Annual Return
+Sharpe Ratio
+Standard Deviation
+MDD
+Sortino
+Actor-only
+13.61
+0.8068
+0.1670
+-24.65
+1.1432
+Actor-critic
+18.64
+1.0635
+0.1616
+-27.12
+1.6766
+PPO
+10.25
+1.0160
+0.0966
+-18.36
+1.4575
+Table 1: Performance of Actor-only vs AC vs PPO
+Figure 2: Actor-only vs AC vs PPO
+some patterns for these three models as well. As we can see in the graph, general policy gradient actor(for AC and
+actor-only model) makes big drop. But if we compare AC and actor-only model, critic part makes the model get more
+returns although it cannot defense the crash. On the other hand, PPO model has smooth movement in its returns. From
+this, we infer that AC model works for bull-market and PPO model is good for bear market.
+3
+Reward Clipping Model
+As we can see in the previous section, Actor-critic and PPO algorithm have its own characteristic. The previous one
+has a strength for increasing market but failing to defense in the depressed stock market. On the other hand, the PPO
+algorithm operates the other way around. So here, we introduce the new algorithm so called Reward Clipping model
+which is strong both in increasing and decreasing stock markets.
+3.1
+Idea
+Note that from the PPO equation([9]), it ensures that the update is not too large, that is the old policy is not too different
+from the new policy. We guess this logic makes PPO move smoothly by giving clipping on the main object, in our
+case proportion for each asset in a portfolio. But in a financial market, especially in an asset allocation, big changes in
+proportions of assets between old and new portfolio is not a problem if we get enough benefit to the point where we can
+ignore a turnover. Since our main purpose is return or sharp ratio even though our output is the portfolio, we apply
+clipping logic to our rewards, not to the main object-asset portfolio.
+With simple experiment(not with full products) , we can see the effect of upper and lower bound in reward clipping (see
+Figure 3). Here, RC_-0.4_0.4 model is the reward clipping model with both upper and lower bounds which are -0.4 and
+0.4. The model RC_0.4 is the reward clipping model with upper bound only which is 0.4. It says RC_0.4 model has no
+restriction moving downward on its rewards. Similarly, RC_-0.4 means the reward clipping has lower bound -0.4 only
+(no upper bound so it can move upward freely).
+As you can see the result in Figure 3, if models have the upper clipping bound on their rewards, it seems that they have a
+limitation to go up, so they cannot get enough return. On the other hands, if a model has no upper clipping bound (lower
+4
+
+2019-07-18-2021-06-16
+140000
+Actor-only
+Actor-critic
+130000
+PPO
+120000
+110000
+100000
+90000
+80000
+2019-07-18
+2019-12-05
+2020-04-23
+2020-09-10
+2021-01-28Reinforcement Learning in Asset Allocation
+Figure 3: Reward Clipping Upper and Lower bound effects
+bound only, RC_-0.4), it moves go up (gets more profit) more than other models, but less down (than other models
+move). Hence we can conclude that if we don’t set a upper reward clipping, we can get more rewards and prevent loss
+of reward by giving a lower reward clipping.
+Especially, we can find if the model has an upper bound on its reward clipping, it cannot be converged. It is because
+that since the model was constructed to purchase more rewards (higher reward is better), if we set up the upper bound,
+it seems to make confliction with the model’s pursuit. The followings are the figures of the convergence for the three
+models. The leftmost is the convergence of RC_-0.4_0.4, the middle is for RC_-0.4 and the rightmost is for RC_0.4.
+Figure 4: Convergence for RC_-0.4_0.4, RC_-0.4, RC_0.4
+3.2
+Construction
+The basic construction for the Reward Clipping model is same with Figure 1. The only different part is that we apply
+clipping logic in PPO onto reward parts in Actor-Only model. For example, for the return reward, our formula is
+max(avg(Σn
+i=1(Wi · daily returni)))
+where Wi is the weight and daily returni = (Ai,t, ...Ai,t+T ) and Ai,t is the daily return at time t of i asset.
+The following pseudo-codes show that which parts are modified from original PPO algorithm to Reward Clipping one.
+5
+
+2019-06-10-2021-07-30
+160000
+RC_-0.4_0.4
+RC_-0.4
+150000
+wy
+RC_0.4
+140000
+130000
+120000
+110000
+100000
+90000
+2019-06-10
+2019-10-28
+2020-03-16
+2020-08-03
+2020-12-21
+2021-05-101.26
+1.24
+1.22
+1.20
+118
+1.16
+114 
+1.12
+0
+2500
+5000
+7500
+10000
+12500
+15000175001.40
+135
+1.30
+1.25
+1.20
+0
+2500
+5000
+7500
+1000012500 15000175001.24
+1.23
+1.22
+121
+1.20
+119
+2500
+5000
+7500
+10000
+12500
+15000
+17500Reinforcement Learning in Asset Allocation
+Algorithm 1 PPO-Clip
+1: for iteration = 1, 2, . . . do
+2:
+for actor = 1, 2, . . . , N do
+3:
+Run policy πθold in environment for T time steps
+4:
+Compute advantage estimates ˆA1, . . . , ˆAT where ˆAt = Wi · Ai,t
+5:
+end for
+6:
+Update the policy by maximizing the PPO-Clip objective:
+θk+1 = argmaxθ
+1
+T
+T
+�
+t=0
+min( πθ(at|st)
+πθk(at|st)Aπθk (st, at), g(ϵ, Aπθk (st, at)))
+7:
+Optimize surrogate L wrt. θ, with K epochs and minibatch size M ≤ NT
+8:
+θold ← θ
+9: end for
+Algorithm 2 Reward-Clip
+1: for iteration = 1, 2, . . . do
+2:
+for actor = 1, 2, . . . , N do
+3:
+Run policy πθold in environment for T time steps
+4:
+Compute advantage estimates ˆA1, . . . , ˆAT where ˆAt = Wi · Ai,t
+5:
+end for
+6:
+Update the policy by maximizing the Reward-Clip objective:
+θk+1 = argmaxθ
+1
+T
+T
+�
+t=0
+min( At
+At−1
+, ϵ1, ϵ2)
+(11)
+where ϵ1, ϵ2 are lower and upper bounds.
+7:
+Optimize surrogate L wrt. θ, with K epochs and minibatch size M ≤ NT
+8:
+θold ← θ
+9: end for
+The Equation 11 in 2 is the biggest changed part in our new model.
+Note that in PPO algorithm, the clipping object is the result of the action-portfolio, but the Reward-Clip object is the
+reward. Since we apply reward clipping to actor-only, in the above pseudo-code, the critic part is excluded.
+With the simple experiment introduced in the previous section, we only consider the model with a lower clipping bound
+in its rewards.
+3.3
+Experimental Results and Model Comparisons
+In the next two subsections, we give two experimental results and comparisons. To see that the reward clipping model
+has strength in a bear market but has enough profit in a bull market, we conduct two experiments during two period-
+falling and increasing markets and compare its performance with other models.
+3.3.1
+Reward Clipping in a falling market
+Here, we check the effect of the Reward Clipping model in a falling market. We train the model from 2010-01-01 to
+2021-06-10 and test it from 2021-07-26 to 2022-07-22. We pick this test period to see how the reward clipping model
+with lower bound work in current market situation. Since we apply the reward clipping logic on actor-only model, to
+see the effect of lower bounded reward clipping, we compare performance with actor-only model. As you can see in
+Figure 5, reward clipping with lower bound (and no upper bound) is effective for a falling market but the same return
+with actor-only model when a market is increasing. The detail is given in Table 2.
+To see the market trend like the degree of decline, we put KOSPI and S&P500 indices too.
+6
+
+Reinforcement Learning in Asset Allocation
+Model
+Annual Return
+Sharpe Ratio
+Standard Deviation
+MDD
+Sortino
+Actor-only
+-6.95
+-0.3616
+0.1633
+-20.86
+-0.5544
+Reward Clipping
+-4.21
+-0.2809
+0.1256
+-14.54
+-0.4422
+KOSPI
+-24.80
+-1.6585
+0.1653
+-30.13
+-2.5470
+S&P500
+-10.02
+-0.4394
+0.1972
+-23.39
+-0.6689
+Table 2: Table for Reward Clipping effect in a falling market
+Figure 5: Reward Clipping effect in a falling market
+In this test(in a falling market), to compare the results with All weather portfolio (in the next section), we use the second
+set of products (16 products in equity, 6 in bond, 2 in commodities and 1 gold). With the above MDD and sortino (and
+Annual Return) in Table 2, we can see that the reward clipping model with a lower bound has a good defense in a falling
+market situation.
+The following Figure 6 is shown the proportion of asset classes for Actor-only and RC models.
+Figure 6: Proportion of asset classes in bear market
+Here, we can see that RC model defense the bear market better than the Actor-only model (especially after April, 2022)
+by increasing the portion of Intermediate-term bond (ITBOND).
+7
+
+2021-07-26 -2022-07-22
+110000
+Actor-Only
+105000
+KOSPI
+S&P500
+100000
+Reward Clipping
+95000
+90000
+85000
+80000
+75000
+70000
+2021-07-26
+2021-10-04
+2021-12-13
+2022-02-21
+2022-05-02
+2022-07-11100
+Actor-only : 2021-07-26 - 2022-07-22
+08
+60
+40 
+20 
+0
+2021-07-26
+2021-08-23
+2021-09-20
+2021-10-18
+2021-11-15
+2021-12-13
+2022-01-10
+2022-03-07
+2022-04-04,
+2022-05-02
+2022-05-30
+2022-06-27Reward-Clipping : 2021-07-26 - 2022-07-22
+100
+COMMODITIES_MT
+COMMODITIES_REITS
+80 
+EQUITY-KR
+EQUITY-US
+GOLD
+60
+ITBOND
+LTBOND
+40 
+0 -
+2021-07-26
+2021-08-23
+2021-09-20
+2021-10-18
+2021-11-15
+2021-12-13
+2022-01-10
+2022-02-07
+L0-E0-7
+2022-04-04
+2022-05-30
+2022-06-27
+2022-Reinforcement Learning in Asset Allocation
+3.3.2
+Model Comparison for four models
+The below Table 3 and Figure 7 show comparison of four models- Actor-only, AC, PPO and Reward Clipping. In
+section 2.2 we’ve already seen the result for existing three models, so we just add the performance of Reward Clipping
+model.
+Model
+Annual Return
+Sharpe Ratio
+Standard Deviation
+MDD
+Sortino
+Actor-only
+13.61
+0.8068
+0.1670
+-24.65
+1.1432
+Actor-critic
+18.64
+1.0635
+0.1616
+-27.12
+1.6766
+PPO
+10.25
+1.0160
+0.0966
+-18.36
+1.4575
+Reward Clipping
+18.45
+1.2746
+0.1301
+-21.45
+2.0391
+Table 3: Comparison four models
+Figure 7: comparison four models
+If you compare Actor-only and Reward clipping models in Figure 7, we can see that Reward clipping has less draw
+down but more benefits in increasing situation. You can check this in Table 3 by comparing MDD, sortino and Annual
+Return - Reward clipping model has less MDD but bigger Annual Return and sortino than Actor-only model. It has
+the almost same bottom point to PPO but the same top point to AC at the end. It has the same increasing strength
+with Actor-critic but also the same defensive power with the PPO algorithm. This means by clipping onto reward in
+Actor-only model, we can get advantages of both Actor-critic and PPO algorithms - strength both in increasing and
+decreasing stock markets. Furthermore, as you can see in Figure 4, reward clipping model with lower bound doesn’t
+much resource (actually it turns out that the reward clipping model requires less resources than PPO model) so we have
+benefit in the point of view of resources and time saving.
+The following Figure 8 shows the change of proportion of asset classes. Note that we apply rebalancing every month
+regularly. As we can see in Figure 8 the existing three models - Actor only, Actor critic and PPO have stable movement.
+Especially, PPO shows almost constant movement - it is almost the same with equal weight. On the other hand, Reward
+Clipping model moves actively that is supposed the basis why the model has good performance in both bull and bear
+markets.
+Furthermore, since PPO model needs more resources - for example, time for convergence, in the above result we can
+see not only the goodness of the performance but also resource effectiveness (Figure 4) of the reward clipping model.
+8
+
+2019-07-18-2021-06-16
+140000
+Actor-only
+Actor-critic
+130000
+PPO
+Reward Clipping
+120000
+110000
+100000
+90000
+80000
+2019-07-18
+2019-12-05
+2020-04-23
+2020-09-10
+2021-01-28Reinforcement Learning in Asset Allocation
+Figure 8: Proportion of asset classes in bull market
+4
+Further work
+There are still many interesting further work using deep reinforcement learning in asset allocation. Firstly, we deal with
+ETF(Exchange Traded Fund)s only since each of them has representative index so AI models can train the indices - and
+so we can also contain ETF’s which are launched recently although there are not enough time to train a model. But
+many financial corporation or customers require to expand products to several financial products - stocks, funds and so
+on. Secondly, in this paper we have applied reward clipping algorithm to actor-only model. So, the next step is to apply
+reward clipping algorithm to actor-critic model. Since actor-critic model has higher return than actor-only model and
+similar draw down, by defending the fall of actor-critic model, we expect that actor-critic model with lower bounded
+reward clipping has better performance. Thirdly, in our tests, we execute rebalancing every month regularly, but in a
+real situation, risk management system is also a necessary requisite. Actually there are several trials to apply AI to
+detect and react risks. In [11] Yang-Yu Liu et al. show a phenomenon called "loss aversion" which says that people
+are much more sensitive to losses than to gains of the same magnitude. And it will affect individual decision-makings
+and portfolio asset prices in financial markets ([12], [13]). With this prior research outcomes, Qing Yang Eddy Lim et
+al. provide an alternative view in maximising portfolio returns using RL by considering dynamic risks appropriate to
+market conditions through dynamic portfolio rebalancing ([14]). Finally, we can still try other RL algorithms. Although
+we select Actor-critic and PPO models by limitations of resources in this paper, there are many other trials to apply RL
+algorithms in asset allocation ([15]). In [16], Ricard Durall conduct 9 different algorithms including A2C, PPO, DDPG,
+SAC and TD3.
+5
+Conclusion
+In this paper, we apply deep reinforcement learning algorithms to portfolio optimization. At first, we compare the
+performances of existing models- Actor-only, Actor-critic and PPO. And then analyze the characteristics of three models.
+Finally, we introduce a new model which has strengths only of each models - the new model, Reward Clipping model
+has out-performed return in a bull market but also a good defense in a bear market. To see the model’s performance, we
+compare them with the traditional approaches - Equal Weight, 6:4(equity:bond) and All-Weather portfolio ([17]). Here
+we apply All-Weather portfolio only to the second product set (in a bear market) because the second set is consisted of
+proper asset classes for All-Weather method.
+In Table 4, Figure 9 and Table 5, we can see that Equal weight has less MDD than other models, but small return in a
+9
+
+Actor-only : 2019-07-18 - 2021-06-16
+100
+08
+60
+40
+20
+2019-07-18
+2019-08-15
+2019-09-12
+2019-10-10
+2019-11-07
+-05
+2020-01-30
+2020-02-27
+2020-03-26
+020-04-23
+0-05-21
+020-06-18
+020-07-16
+020-08-13
+2020-09-10
+2020-10-08
+-05
+2020-12-03
+LE*
+019-12-
+0-11-
+020-12-
+TO-
+021-02-
+2020
+021Actor-critic : 2019-07-18 - 2021-06-16
+100
+USA_BOND
+USA_EQUIT
+80 
+GOLD
+KOR_EQUIT
+60
+KOR_BOND
+JP_EQUIT
+UK_EQUIT
+40 
+UK_BOND
+DX_EQUIT
+DX_BOND
+20 
+0
+2019-07-18
+019-08-15
+2019-09-12
+2019-10-10
+2019-11-07
+020-01-30
+Z~
+-06-18
+09-10
+10-08
+-05
+12-03
+LE*
+5
+-25
+2021-04-22
+2021-05-20
+2019-12-
+020-02-
+LO
+20-11-
+0-12-
+1-01-
+2021-03-
+020-
+-0z
+020-
+021
+2021PPO : 2019-07-18 - 2021-06-16
+100
+08
+60
+40
+20
+2019-07-18
+2019-08-15
+2019-09-12
+2019-10-10
+2019-11-07
+2020-01-02
+0E-T0-0Z0
+020-02-27
+2020-03-26
+2020-04-23
+2020-05-21
+2020-06-18
+2020-07-16
+020-08-13
+020-09-1(
+2020-10-08
+2020-11-05
+020.12-03
+2021-01-28
+2021-02-25
+2021-03-25
+021-04-
+N
+-05Reward Clipping : 2019-07-18 - 2021-06-16
+100
+USA_BOND
+USA_EQUIT
+08
+GOLD
+KOR_EQUIT
+60
+KOR_BOND
+JP_EQUIT
+UK_EQUIT
+40 
+UK_BOND
+DX_EQUIT
+DX_BOND
+20
+2019-07-18
+2019-08-15
+2019-09-12
+2019-10-10
+2019-11-07
+2019-12-05
+Z0-T0-0Z02
+2020-01-30
+2020-02-27
+2020-03-26
+2020-04-23
+2020-05-21
+2020-06-18
+2020-07-16
+2020-08-13
+2020-09-10
+2020-10-08
+2
+2021-05-20
+020-11
+-t0-Reinforcement Learning in Asset Allocation
+bull market. When we consider Return, MDD, and sortino rate, Reward Clipping model works best for both bull and
+bear markets.
+Model
+Annual Return
+Sharpe Ratio
+Standard Deviation
+MDD
+Sortino
+Actor-only
+13.61
+0.8068
+0.1670
+-24.65
+1.1432
+Actor-critic
+18.64
+1.0635
+0.1616
+-27.12
+1.6766
+PPO
+10.25
+1.0160
+0.0966
+-18.36
+1.4575
+Reward Clipping
+18.45
+1.2746
+0.1301
+-21.45
+2.0391
+Equal weight
+10.10
+1.0012
+0.0968
+-18.44
+1.4386
+6:4
+10.70
+0.9588
+0.1074
+-20.36
+1.3811
+Table 4: models vs traditional approaches during COVID-19
+Model
+Annual Return
+Sharpe Ratio
+Standard Deviation
+MDD
+Sortino
+Actor-only
+-6.95
+-0.3616
+0.1633
+-20.86
+-0.5544
+Reward Clipping
+-4.21
+-0.2809
+0.1256
+-14.54
+-0.4422
+Equal weight
+-10.23
+-1.0956
+0.0950
+-15.27
+-1.5984
+6:4
+-13.53
+-1.5420
+0.0921
+-17.43
+-2.2477
+All-Weather
+-14.73
+-1.7794
+0.0880
+-19.39
+-2.4975
+KOSPI
+-24.80
+-1.6585
+0.1653
+-30.13
+-2.5470
+S&P500
+-10.02
+-0.4394
+0.1972
+-23.39
+-0.6689
+Table 5: models vs traditional approaches in a bear market
+Figure 9: models vs traditional approaches during COVID-19(bull market) and a bear market
+In our experiments, the existing models have its own characteristics - some have an advantage in defense for drawing
+down and others have a strength for profits but not in both. And also depending on the direction of each model, it seems
+that each model selects two or three main products/asset classes to achieve their purpose. But Reward Clipping model
+which the model has advantages of existing models we introduced has a strong strength for both in two opposite market
+situations (Table 4, Table 5, and Figure 9). And it turns out that the Reward Clipping model has dynamics to select
+products/asset classes to pursue more profits managing a draw-down.
+6
+Acknowledgment
+We appreciate Seongjae Huh for his advice about traditional investment strategies. We also would like to say thanks to
+Yong Qu Lee, team leader of SK C&C for his support.
+10
+
+2019-07-18-2021-06-16
+140000
+Actor-critic
+Reward Clipping
+130000
+Actor-only
+PPO
+120000
+60:40
+Equal weight
+110000
+100000
+90000
+80000
+2019-07-18
+2019-12-05
+2020-04-23
+2020-09-10
+2021-01-282021-07-26-2022-07-22
+Actor-only
+105000
+Reward-Clipping
+60:40
+All whether
+100000
+Equal weight
+95000
+90000
+85000
+2021-07-26
+2021-10-04
+2021-12-13
+2022-02-21
+2022-05-02
+2022-07-11Reinforcement Learning in Asset Allocation
+References
+[1] Thomas G. Fischer. Reinforcement learning in financial markets - a survey. FAU Discussion Papers in Economics,
+(12/2018), October 2018.
+[2] Zipeng Liang, Hao Chen, Junhao Zhu, Kangkang Jiang, and Yanran Li. Adversarial deep reinforcement learning
+in portfolio management. arXiv:1808.09940v3 [q-fin.PM], November 2018.
+[3] Farzan Soleymani and Eric Paquet. Financial portfolio optimization with online deep reinforcement learning and
+restricted stacked autoencoder-deepbreath. Expert Systems with Applications, 156(113456), October 2020.
+[4] Jung hoon Kim. Efficient portfolio management using deep reinforcement learning. Seoul National University,
+December 2020.
+[5] Andres Heurtas. A reinforcement learning application for portfolio optimization in the stock market. UNIVERSITY
+OF HELSINKI, June 2020.
+[6] Amine Mohamed Aboussalah, Ziyun Xu, and Chi-Guhn Lee. What is the value of the cross-sectional approach to
+deep reinforcement learning? Quantitative Finance, 22(Issue 6):1091–1111, 2022.
+[7] Jeffrey M. Wooldridge. Part 1: Regression analysis with cross sectional data. Introductory Econometrics A
+Modern Approach. 4th edition, 2009.
+[8] Richard S. Sutton and Andrew G. Barto. Reinforcement Learning: An Introduction. The MIT Press, 2nd edition,
+2018.
+[9] John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization
+algorithms. arXiv:1707.06347v2 [cs.LG], August 2017.
+[10] Harm van Seijen, Mehdi Fatemi, Joshua Romoff, Romain Laroche, Tavian Barnes, and Jeffrey Tsang. Hybrid
+reward architecture for reinforcement learning. arxiv:1706.04208v2 [cs.LG], November 2017.
+[11] Yang-Yu Liu, Jose C. Nacher, Tomoshiro Ochiai, Mauro Martino, and Yaniv Altshuler. Prospect theory for online
+financial trading. PLOS ONE, 9(Issue 10, e109458), 2014.
+[12] Donghyun Cheong, Young Min Kim, Hyun Woo Byun, Kyong Joo Oh, and Tae Yoon Kim. Using genetic
+algorithm to support clustering-based portfolio optimization by investor information. Applied Soft Computing,
+61:593–602, December 2017.
+[13] Liyan Yang. Loss aversion in financial markets. Journal of Mechanism and Institution Design, 4(1):119–137,
+2019.
+[14] Qing Yang Eddy Lim, Qi Cao, and Chai Quek. Dynamic portfolio rebalancing through reinforcement learning.
+Neural Computing and Applications, 34:7125–7139, 2022.
+[15] Miquel Noguer i Alonso and Sonam Srivastava. Deep reinforcement learning for asset allocation in us equities.
+arXiv:2010.04404v1 [q-fin.PM], October 2020.
+[16] Ricard Durall. Asset allocation: From markowitz to deep reinforcement learning. arXiv:2208.07158v1 [q-fin.PM],
+July 2022.
+[17] Youssef Louraoui. The all-weather portfolio approach: The holy grail of portfolio management. SSRN, (4021133),
+2022.
+11
+

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+arXiv:2301.00541v1  [math.AP]  2 Jan 2023
+BOUNDARY REGULARITY FOR AN EVEN ORDER ELLIPTIC SYSTEM IN
+THE CRITICAL DIMENSION
+MING-LUN LIU AND YAO-LAN TIAN*
+Abstract. In this short note, we consider the Dirichlet problem associated to an even
+order elliptic system with antisymmetric first order potential.
+Given any continuous
+boundary data, we show that weak solutions are continuous up to boundary.
+Keywords: Polyharmonic maps, higher order elliptic system, Boudary continuity, Dirichlet prob-
+lem
+2010 Mathematics Subject Classification: 35J48, 35B65, 35G35
+1. Introduction
+In this paper, we consider the Dirichlet problem for the following even order elliptic
+system for u ∈ W k,2(Ω, Rm):
+(1.1)
+∆ku =
+k−1
+�
+l=0
+∆l ⟨Vl, du⟩ +
+k−2
+�
+l=0
+∆lδ(wldu)
+in Ω ⊂ R2k
+with the following regularity assumptions on the coefficients:
+(1.2)
+wi ∈ W 2i+2−k,2 �
+Ω, Rm×m�
+for i ∈ {0, . . . , k − 2},
+Vi ∈ W 2i+1−k,2 �
+Ω, Rm×m ⊗ ∧1R2k�
+for i ∈ {1, . . . , k − 1},
+and
+V0 = dη + F
+with
+(1.3)
+η ∈ W 2−k,2 (Ω, so(m))
+and
+F ∈ W 2−k, 2k
+k+1,1 �
+Ω, Rm×m ⊗ ∧1R2k�
+.
+This system was initially introduced by de Longueville and Gastel [2], aiming at
+a further extesion of the second order theory by Rivi`ere [11] (corresponding to the case
+k = 1) and the fourth order theory by Lamm-Rivi`ere [7] (corresponding to the case k = 2),
+addressing an open problem of Rivi`ere. It includes the Euler-Lagrange equations of many
+interesting classes of geometric mappings such as the harmonic mappings, biharmonic
+mappings, polyharmonic mappings and so on; see [1, 12, 11, 7, 3, 5, 6].
+A distinguished feature of this system is the criticality. To see it, we consider the
+simpler case k = 1. Then system (1.1) reduces to the second order Rivi`ere system
+(1.4)
+∆u = Ω′ · ∇u,
+Corresponding author: Yao-Lan Tian.
+Both authors are partially supported by the Young Scientist Program of the Ministry of Sci-
+ence and Technology of China (No. 2021YFA1002200), the National Natural Science Foundation of
+China (No. 12101362) and the Natural Science Foundation of Shandong Province (No. ZR2022YQ01,
+ZR2021QA003).
+1
+
+2
+M.-L. LIU AND Y.-L. TIAN
+where u ∈ W 1,2(Ω, Rm) and Ω′ ∈ L2(Ω, so(m) ⊗ Λ1R2). The right hand side of (1.4)
+is merely in L1 by H¨older’s inequality and so standard Lp regularity theory for elliptic
+equations fails to apply here. In the celebrated work [11], Rivi`ere succeeded in rewriting
+(1.4) into an equivalent conservation law, from which the continuity of weak solutions
+follows. The techniques were further extended to fourth order system in [7] and finally to
+general even order systems in [2].
+In this paper, we shall consider the Dirichlet boundary value problem for (1.1). Recall
+that we say that u ∈ W k,2(Ω, Rm) has Dirichlet boundary value g ∈ Ck−1(Ω, Rm) if
+∇αu = ∇αg
+on ∂Ω
+holds in the sense of traces for all 2k-dimensional multi-indices α with |α| ≤ k − 1.
+Similarly, we say that u has Navier boundary value hi ∈ C(Ω, Rm), i = 0, · · · , k − 1, if for
+all i ∈ {0, · · · , k − 1}
+∆iu = hi
+on ∂Ω.
+Now, we can state our main theorem.
+Theorem 1.1. Fix k ∈ N and Ω ⊂ R2k a bounded smooth domain.
+Suppose u ∈
+W k,2(Ω, Rm) is a solution of (1.1) with (1.2) and (1.3). If either the Dirichlet bound-
+ary value g ∈ Ck−1(Ω, Rm) or the Navier boundary value hi for i = 0, · · · , k − 1, then
+u ∈ C(Ω, Rm).
+Theorem 1.1 can be viewed as a natural extension of the corresponding boundary
+continuity results of M¨uller-Schikorra [9] for second order system, and Guo-Xiang [4] for
+fourth order system. As a special case of Theorem 1.1, we infer that every (extrinsic or
+intrinsic) polyharmonic mapping from the unit ball B2k ⊂ R2k into a closed manifold
+N ֒→ Rm is continuous up the boundary, under the Dirichlet boundary value condition.
+This partially extends the corrosponding boundary continuity reuslt of Lamm-Wang [8].
+The approach to Theorem 1.1 is similar to that of Lamm and Wang [8], relying on
+interior H¨older regularity and a boundary maximal principle.
+As noticed in [4], this
+approach only requies zero order boundary assumption “u = g” or “u = h0” on ∂Ω for
+some continuous function g or h0. This observation extends to the general even order
+system (1.1).
+Our natations are rather standard. We write Br(x) for a ball centred at x with radius
+r in R2k. The notation C denotes various constants that may be different from line to line.
+We sometimes write A ≲ B meaning that A ≤ CB for some constant C > 0 depending
+only on the quantitative data.
+2. The proof of main result
+2.1. Interior regularity. Continuity of weak solutions for (1.1) was first obtained in [2].
+But for boundary continuity, we need the stronger interior H¨older regularity. We first
+recall the following interior H¨older regularity result for (1.1) from [5, Theorem 1.3] or [6,
+Theorem 1.1].
+Theorem 2.1 (Interior Regularity). Suppose u ∈ W k,2(B2k, Rm) is a solution of (1.1) with
+(1.2) and (1.3). Then there exist α ∈ (0, 1), C > 0 and r0, depending only on k, m and
+the data from (1.2) and (1.3), such that u is locally α-H¨older continuous and
+oscBr(x)u ≤ Crα∥u∥W k,2(B2k)
+
+BOUNDARY REGULARITY
+3
+for all x ∈ B 1
+4(0) and all 0 < r < r0.
+We shall use the following version of Theorem 2.1 in our later proofs. There exists
+R0 > 0 sufficiently small, such that, if x ∈ Ω and 0 < r < min{R0, dist(x, ∂Ω)/4}, then
+u ∈ C0,α(Br(x0), Rm) for some α ∈ (0, 1) and
+(2.1)
+oscBτr(x)u ≲ τ α∥u∥W k,2(B4r(x0))
+for 0 < τ ≤ 1.
+By [6, Theorem 1.1], one can indeed infer that (2.1) holds for all α ∈ (0, 1). But for our
+purpose, the current estimate is sufficient.
+2.2. Boundary Maximum Principle. Another ingredient for boundary regularity is a
+boundary maximum principle, originally discovered by Qing [10] in his proof of boundary
+regularity for weakly harmonic maps and was later adapted to the polyharmonic case in
+Lamm-Wang [8].
+For x ∈ Ω and R > 0, denote by ΩR(x) = Ω ∩ BR(x). We shall prove the following
+boundary maximal principle for solutions of (1.1).
+Proposition 2.2 (Boundary Maximum Principle). There exists a constant C > 0 such that
+for any x ∈ Ω and 0 < R < R0/4, for any q ∈ Rm, there holds
+(2.2)
+max
+ΩR(x) |u − q| ≤ C
+�
+max
+∂ΩR(x) |u − q| + ∥u∥W k,2(Ω4R(x))
+�
+.
+To prove Proposition 2.2, we need the following version of Courant-Lebesgue lemma,
+which was essentially established in [8]. Since the formulation is slightly different from
+there, for the convenience of the readers, we recall the proofs here.
+Lemma 2.3. There exists C > 0 such that for any R > 0 and any x0 ∈ Ω, there exists
+R1 ∈ (R, 2R) so that
+osc∂BR1(x0)∩Ωu ≤ C∥u∥W k,2(Ω4R(x0)).
+Proof. For x ∈ B2R(x0), set r = |x − x0| ∈ [0, 2R]. By Fubini’s theorem, we have
+�
+Ω2R(x0)
+|∇u|2kdx ≥
+� 2R
+R
+dr
+�
+∂Br(x0)∩Ω
+|∇Tu|2kdH2k−1
+≥
+�� 2R
+R
+1
+rdr
+�
+inf
+R≤r≤2R
+�
+r
+�
+∂Br(x0)∩Ω
+|∇Tu|2kdH2k−1
+�
+= ln 2
+inf
+R≤r≤2R
+�
+r
+�
+∂Br(x0)∩Ω
+|∇Tu|2kdH2k−1
+�
+,
+where ∇T denotes the gradient operator on ∂Br(x0) and dH2k−1 is the volume element
+on ∂Br(x0). Then there exists R1 ∈ (R, 2R) such that
+R1
+�
+∂BR1(x0)∩Ω
+|∇Tu|2kdH2k−1 ≤
+1
+ln 2
+�
+Ω2R(x0)
+|∇u|2kdx.
+Hence u(R1, ·) ∈ W 1,2k(∂BR1(x0) ∩ Ω, Rm) and the Sobolev embedding theorem implies
+that u(R1, ·) ∈ C
+1
+2k (∂BR1(x0) ∩ Ω, Rm) and
+osc∂BR1(x0)∩Ωu ≲ R1
+�
+∂BR1(x0)∩Ω
+|∇Tu|2kdH2k−1 ≲ ∥u∥W k,2(Ω4R(x0)).
+□
+
+4
+M.-L. LIU AND Y.-L. TIAN
+Proof of Proposition 2.2. Denote M = max
+ΩR(x) |u−q|, here q ∈ Rm is fixed. We may assume
+that
+(2.3)
+M ≥ ∥u∥W k,2(ΩR(x)).
+Choose x0 ∈ ΩR(x) such that
+(2.4)
+|u(x0) − q| ≥ 3
+4M.
+Let r0 = dist(x0, ∂ΩR(x0)). Note that r0 ≤ R ≤ R0. Thus (2.1) implies that for any
+r ∈ (0, r0
+4 ), we have
+(2.5)
+oscBr(x0)u ≤ C
+� r
+r0
+�α0
+∥u∥W k,2(ΩR(x)) ≤ CM
+� r
+r0
+�α0
+.
+Pick r1 = r0/(4C)1/α in the above, and we obtain
+oscBr1(x0)u ≤ 1
+4M
+This together with (2.4) yields
+(2.6)
+inf
+Br1(x0) |u − q| ≥ |u(x0) − q| − oscBr1(x0)u ≥ 1
+2M.
+By Lemma 2.3, there exists r2 ∈ (r0, 2r0) such that
+(2.7)
+osc∂Br2(x0)∩ΩR(x)u ≤ C∥u∥W k,2(Ω4R(x)).
+Note that ∂Br2(x0)∩∂ΩR(x) ̸= ∅. Using polar coordinates centered at x0, we estimate
+inf
+�
+|u(r1, θ)−u(r2, θ)| : (ri, θ) ∈ ∂Bri(x0) ∩ ΩR(x), i = 1, 2
+�
+≤C
+�
+S2k−1 dθ
+� r2
+r1
+|ur|χ[r1,r2]×S2k−1(r, θ)dr
+≤ C
+r2k−1
+1
+�
+S2k−1 dθ
+� r2
+r1
+|ur|χ[r1,r2]×S2k−1(r, θ)r2k−1dr
+≤ C
+r2k−1
+1
+�
+B2r0(x)∩ΩR(x)
+|ur|dx
+≤ C
+r2k−1
+1
+|B2r0(x)|
+2k−1
+2k
+��
+ΩR(x)
+|∇u|2kdx
+� 1
+2k
+≤C r2k−1
+0
+r2k−1
+1
+∥u∥W k,2(Ω4R(x)) ≤ C∥u∥W k,2(Ω4R(x)).
+This implies that there exists θ0 ∈ ∂B1(x0) such that
+(2.8)
+|u(r1, θ0) − u(r2, θ0)| ≤ C∥u∥W k,2(Ω4R(x)).
+
+BOUNDARY REGULARITY
+5
+Hence, by choosing an arbitrary x∗ ∈ ∂Br2(x0) ∩ ∂ΩR(x), we obtain from (2.6), (2.7) and
+(2.8) that
+M
+2 ≤
+inf
+Br1(x0) |u − q| ≤ |u(r1, θ0) − q|
+≤|u(r1, θ0) − u(r2, θ0)| + |u(r2, θ0) − u(x∗)| + |u(x∗) − q|
+≤C∥u∥W k,2(Ω4R(x)) + osc∂Br2(x0)∩ΩR(x)u + sup
+∂ΩR(x)
+|u − q|
+≤C
+�
+sup
+∂ΩR(x)
+|u − q| + ∥u∥W k,2(Ω4R(x))
+�
+.
+The proof is complete.
+□
+2.3. Proof of Theorem 1.1. Now we are ready to prove Theorem 1.1.
+Proof of Theorem 1.1. Let x0 ∈ ∂Ω and take q = g(x0) = u(x0) in Proposition 2.2. Note
+that
+max
+∂ΩR(x0) |u − u(x0)| ≤
+max
+∂ΩR(x0)∩∂Ω |u − u(x0)| + osc∂ΩR(x0)∩Ωu
+=
+max
+∂ΩR(x0)∩∂Ω |g − g(x0)| + osc∂ΩR(x0)∩Ωu.
+The first term tends to 0 as R → 0 since g ∈ C(∂Ω). The second term tends to 0 as
+R → 0 by Lemma 2.3. This implies the continuity of u as desired.
+□
+Remark 2.4. The proof above extends to solutions to the following inhomogeneous elliptic
+system
+(2.9)
+∆ku =
+k−1
+�
+l=0
+∆l ⟨Vl, du⟩ +
+k−2
+�
+l=0
+∆lδ(wldu) + f
+in Ω ⊂ R2k
+with f ∈ Lp for some p > 1 and (1.2), (1.3). Indeed, by [6, Theorem 1.1], in this case, the
+interior regularity estimate (2.1) becomes
+(2.10)
+oscBτr(x)u ≲ τ α �
+∥u∥W k,2(B4r(x0)) + ∥f∥Lp(B4r(x0))
+�
+for 0 < τ ≤ 1.
+With this, the buondary maximal principle (2.2) remains valid with an extra term ∥f∥Lp(Ω4R(x))
+on the right hand side. The proof of Theorem 1.1 then works with obvious modifications.
+Acknowledgements. The authors would like to Prof. Chang-Lin Xiang and Chang-Yu
+Guo for posing this question to them and for many useful conservations.
+References
+[1] S.-Y.A. Chang, L. Wang and P.C. Yang, A regularity theory of biharmonic maps. Commun.
+Pure Appl. Math. 52(9) (1999), 1113-1137.
+[2] F.L. de Longueville and A. Gastel, Conservation laws for even order systems of polyharmonic
+map type. Calc. Var. Partial Differential Equations 60, 138 (2021).
+[3] A. Gastel and C. Scheven, Regularity of polyharmonic maps in the critical dimension. Comm.
+Anal. Geom. 17 (2009), no. 2, 185-226.
+[4] C.-Y. Guo and C.-L. Xiang, Regularity of solutions for a fourth order linear system via conser-
+vation law. J. Lond. Math. Soc. (2) 101 (2020), no. 3, 907-922.
+[5] C.-Y. Guo and C.-L. Xiang, Regularity of weak solutions to higher order elliptic systems in critical
+dimensions. Tran. Amer. Math. Soc. 374 (2021), no. 5, 3579-3602.
+
+6
+M.-L. LIU AND Y.-L. TIAN
+[6] C.-Y. Guo, C.-L. Xiang and G.-F. Zheng, Lp regularity theory for even order elliptic systems
+with antisymmetric first order potentials., J. Math. Pures Appl. 165 (2022) 286-324.
+[7] T. Lamm and T. Rivi`ere, Conservation laws for fourth order systems in four dimensions. Comm.
+Partial Differential Equations 33 (2008), 245-262.
+[8] T. Lamm and C.Y. Wang, Boundary regularity for polyharmonic maps in the critical dimension.
+Adv. Calc. Var. 2 (2009), 1-16.
+[9] F. M¨uller and A. Schikorra, Boundary regularity via Uhlenbeck-Rivi`ere decomposition. Analysis
+(Munich) 29 (2009), 199-220.
+[10] J. Qing, Boundary regularity of weakly harmonic maps from surfaces, J. Funct. Anal. 114 (1993)
+458-466.
+[11] T. Rivi`ere, Conservation laws for conformally invariant variational problems. Invent. Math. 168
+(2007), 1-22.
+[12] C.Y. Wang, Stationary biharmonic maps from Rm into a Riemannian manifold. Comm. Pure Appl.
+Math. 57 (2004), 419-444.
+(Ming-Lun Liu) Research Center for Mathematics and Interdisciplinary Sciences, Shan-
+dong University, Qingdao 266237, P. R. China and Frontiers Science Center for Nonlin-
+ear Expectations, Ministry of Education, Qingdao, P. R. China
+Email address: [email protected]
+(Yao-Lan Tian) Center for Optics Research and Engineering, Shandong University,
+Qingdao 266237, P. R. China
+Email address: [email protected]
+

9tAyT4oBgHgl3EQfqPjX/content/tmp_files/load_file.txt

@@ -0,0 +1,285 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf,len=284
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='00541v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='AP]  2 Jan 2023  BOUNDARY REGULARITY FOR AN EVEN ORDER ELLIPTIC SYSTEM IN  THE CRITICAL DIMENSION  MING-LUN LIU AND YAO-LAN TIAN*  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' In this short note, we consider the Dirichlet problem associated to an even  order elliptic system with antisymmetric first order potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  Given any continuous  boundary data, we show that weak solutions are continuous up to boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  Keywords: Polyharmonic maps, higher order elliptic system, Boudary continuity, Dirichlet prob-  lem  2010 Mathematics Subject Classification: 35J48, 35B65, 35G35  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Introduction  In this paper, we consider the Dirichlet problem for the following even order elliptic  system for u ∈ W k,2(Ω, Rm):  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='1)  ∆ku =  k−1  �  l=0  ∆l ⟨Vl, du⟩ +  k−2  �  l=0  ∆lδ(wldu)  in Ω ⊂ R2k  with the following regularity assumptions on the coefficients:  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='2)  wi ∈ W 2i+2−k,2 �  Ω, Rm×m�  for i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' , k − 2},  Vi ∈ W 2i+1−k,2 �  Ω, Rm×m ⊗ ∧1R2k�  for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' , k − 1},  and  V0 = dη + F  with  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='3)  η ∈ W 2−k,2 (Ω, so(m))  and  F ∈ W 2−k, 2k  k+1,1 �  Ω, Rm×m ⊗ ∧1R2k�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  This system was initially introduced by de Longueville and Gastel [2], aiming at  a further extesion of the second order theory by Rivi`ere [11] (corresponding to the case  k = 1) and the fourth order theory by Lamm-Rivi`ere [7] (corresponding to the case k = 2),  addressing an open problem of Rivi`ere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' It includes the Euler-Lagrange equations of many  interesting classes of geometric mappings such as the harmonic mappings, biharmonic  mappings, polyharmonic mappings and so on;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' see [1, 12, 11, 7, 3, 5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  A distinguished feature of this system is the criticality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' To see it, we consider the  simpler case k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Then system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='1) reduces to the second order Rivi`ere system  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='4)  ∆u = Ω′ · ∇u,  Corresponding author: Yao-Lan Tian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  Both authors are partially supported by the Young Scientist Program of the Ministry of Sci-  ence and Technology of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' 2021YFA1002200), the National Natural Science Foundation of  China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' 12101362) and the Natural Science Foundation of Shandong Province (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' ZR2022YQ01,  ZR2021QA003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  1  2  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' LIU AND Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' TIAN  where u ∈ W 1,2(Ω, Rm) and Ω′ ∈ L2(Ω, so(m) ⊗ Λ1R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' The right hand side of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='4)  is merely in L1 by H¨older’s inequality and so standard Lp regularity theory for elliptic  equations fails to apply here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' In the celebrated work [11], Rivi`ere succeeded in rewriting  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='4) into an equivalent conservation law, from which the continuity of weak solutions  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' The techniques were further extended to fourth order system in [7] and finally to  general even order systems in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  In this paper, we shall consider the Dirichlet boundary value problem for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Recall  that we say that u ∈ W k,2(Ω, Rm) has Dirichlet boundary value g ∈ Ck−1(Ω, Rm) if  ∇αu = ∇αg  on ∂Ω  holds in the sense of traces for all 2k-dimensional multi-indices α with |α| ≤ k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  Similarly, we say that u has Navier boundary value hi ∈ C(Ω, Rm), i = 0, · · · , k − 1, if for  all i ∈ {0, · · · , k − 1}  ∆iu = hi  on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  Now, we can state our main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Fix k ∈ N and Ω ⊂ R2k a bounded smooth domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  Suppose u ∈  W k,2(Ω, Rm) is a solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='1) with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' If either the Dirichlet bound-  ary value g ∈ Ck−1(Ω, Rm) or the Navier boundary value hi for i = 0, · · · , k − 1, then  u ∈ C(Ω, Rm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='1 can be viewed as a natural extension of the corresponding boundary  continuity results of M¨uller-Schikorra [9] for second order system, and Guo-Xiang [4] for  fourth order system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' As a special case of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='1, we infer that every (extrinsic or  intrinsic) polyharmonic mapping from the unit ball B2k ⊂ R2k into a closed manifold  N ֒→ Rm is continuous up the boundary, under the Dirichlet boundary value condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  This partially extends the corrosponding boundary continuity reuslt of Lamm-Wang [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  The approach to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='1 is similar to that of Lamm and Wang [8], relying on  interior H¨older regularity and a boundary maximal principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  As noticed in [4], this  approach only requies zero order boundary assumption “u = g” or “u = h0” on ∂Ω for  some continuous function g or h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' This observation extends to the general even order  system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  Our natations are rather standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' We write Br(x) for a ball centred at x with radius  r in R2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' The notation C denotes various constants that may be different from line to line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  We sometimes write A ≲ B meaning that A ≤ CB for some constant C > 0 depending  only on the quantitative data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' The proof of main result  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Interior regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Continuity of weak solutions for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='1) was first obtained in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  But for boundary continuity, we need the stronger interior H¨older regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' We first  recall the following interior H¨older regularity result for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='1) from [5, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='3] or [6,  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='1 (Interior Regularity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Suppose u ∈ W k,2(B2k, Rm) is a solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='1) with  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Then there exist α ∈ (0, 1), C > 0 and r0, depending only on k, m and  the data from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='3), such that u is locally α-H¨older continuous and  oscBr(x)u ≤ Crα∥u∥W k,2(B2k)  BOUNDARY REGULARITY  3  for all x ∈ B 1  4(0) and all 0 < r < r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  We shall use the following version of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='1 in our later proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' There exists  R0 > 0 sufficiently small, such that, if x ∈ Ω and 0 < r < min{R0, dist(x, ∂Ω)/4}, then  u ∈ C0,α(Br(x0), Rm) for some α ∈ (0, 1) and  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='1)  oscBτr(x)u ≲ τ α∥u∥W k,2(B4r(x0))  for 0 < τ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  By [6, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='1], one can indeed infer that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='1) holds for all α ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' But for our  purpose, the current estimate is sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Boundary Maximum Principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Another ingredient for boundary regularity is a  boundary maximum principle, originally discovered by Qing [10] in his proof of boundary  regularity for weakly harmonic maps and was later adapted to the polyharmonic case in  Lamm-Wang [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  For x ∈ Ω and R > 0, denote by ΩR(x) = Ω ∩ BR(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' We shall prove the following  boundary maximal principle for solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='2 (Boundary Maximum Principle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' There exists a constant C > 0 such that  for any x ∈ Ω and 0 < R < R0/4, for any q ∈ Rm, there holds  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='2)  max  ΩR(x) |u − q| ≤ C  �  max  ∂ΩR(x) |u − q| + ∥u∥W k,2(Ω4R(x))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  To prove Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='2, we need the following version of Courant-Lebesgue lemma,  which was essentially established in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Since the formulation is slightly different from  there, for the convenience of the readers, we recall the proofs here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' There exists C > 0 such that for any R > 0 and any x0 ∈ Ω, there exists  R1 ∈ (R, 2R) so that  osc∂BR1(x0)∩Ωu ≤ C∥u∥W k,2(Ω4R(x0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' For x ∈ B2R(x0), set r = |x − x0| ∈ [0, 2R].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' By Fubini’s theorem, we have  �  Ω2R(x0)  |∇u|2kdx ≥  � 2R  R  dr  �  ∂Br(x0)∩Ω  |∇Tu|2kdH2k−1  ≥  �� 2R  R  1  rdr  �  inf  R≤r≤2R  �  r  �  ∂Br(x0)∩Ω  |∇Tu|2kdH2k−1  �  = ln 2  inf  R≤r≤2R  �  r  �  ∂Br(x0)∩Ω  |∇Tu|2kdH2k−1  �  ,  where ∇T denotes the gradient operator on ∂Br(x0) and dH2k−1 is the volume element  on ∂Br(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Then there exists R1 ∈ (R, 2R) such that  R1  �  ∂BR1(x0)∩Ω  |∇Tu|2kdH2k−1 ≤  1  ln 2  �  Ω2R(x0)  |∇u|2kdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  Hence u(R1, ·) ∈ W 1,2k(∂BR1(x0) ∩ Ω, Rm) and the Sobolev embedding theorem implies  that u(R1, ·) ∈ C  1  2k (∂BR1(x0) ∩ Ω, Rm) and  osc∂BR1(x0)∩Ωu ≲ R1  �  ∂BR1(x0)∩Ω  |∇Tu|2kdH2k−1 ≲ ∥u∥W k,2(Ω4R(x0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  □  4  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' LIU AND Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' TIAN  Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Denote M = max  ΩR(x) |u−q|, here q ∈ Rm is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' We may assume  that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='3)  M ≥ ∥u∥W k,2(ΩR(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  Choose x0 ∈ ΩR(x) such that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='4)  |u(x0) − q| ≥ 3  4M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  Let r0 = dist(x0, ∂ΩR(x0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Note that r0 ≤ R ≤ R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Thus (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='1) implies that for any  r ∈ (0, r0  4 ), we have  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='5)  oscBr(x0)u ≤ C  � r  r0  �α0  ∥u∥W k,2(ΩR(x)) ≤ CM  � r  r0  �α0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  Pick r1 = r0/(4C)1/α in the above, and we obtain  oscBr1(x0)u ≤ 1  4M  This together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='4) yields  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='6)  inf  Br1(x0) |u − q| ≥ |u(x0) − q| − oscBr1(x0)u ≥ 1  2M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='3, there exists r2 ∈ (r0, 2r0) such that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='7)  osc∂Br2(x0)∩ΩR(x)u ≤ C∥u∥W k,2(Ω4R(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  Note that ∂Br2(x0)∩∂ΩR(x) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Using polar coordinates centered at x0, we estimate  inf  �  |u(r1, θ)−u(r2, θ)| : (ri, θ) ∈ ∂Bri(x0) ∩ ΩR(x), i = 1, 2  �  ≤C  �  S2k−1 dθ  � r2  r1  |ur|χ[r1,r2]×S2k−1(r, θ)dr  ≤ C  r2k−1  1  �  S2k−1 dθ  � r2  r1  |ur|χ[r1,r2]×S2k−1(r, θ)r2k−1dr  ≤ C  r2k−1  1  �  B2r0(x)∩ΩR(x)  |ur|dx  ≤ C  r2k−1  1  |B2r0(x)|  2k−1  2k  ��  ΩR(x)  |∇u|2kdx  � 1  2k  ≤C r2k−1  0  r2k−1  1  ∥u∥W k,2(Ω4R(x)) ≤ C∥u∥W k,2(Ω4R(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  This implies that there exists θ0 ∈ ∂B1(x0) such that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='8)  |u(r1, θ0) − u(r2, θ0)| ≤ C∥u∥W k,2(Ω4R(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  BOUNDARY REGULARITY  5  Hence, by choosing an arbitrary x∗ ∈ ∂Br2(x0) ∩ ∂ΩR(x), we obtain from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='6), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='7) and  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='8) that  M  2 ≤  inf  Br1(x0) |u − q| ≤ |u(r1, θ0) − q|  ≤|u(r1, θ0) − u(r2, θ0)| + |u(r2, θ0) − u(x∗)| + |u(x∗) − q|  ≤C∥u∥W k,2(Ω4R(x)) + osc∂Br2(x0)∩ΩR(x)u + sup  ∂ΩR(x)  |u − q|  ≤C  �  sup  ∂ΩR(x)  |u − q| + ∥u∥W k,2(Ω4R(x))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  □  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Now we are ready to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Let x0 ∈ ∂Ω and take q = g(x0) = u(x0) in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Note  that  max  ∂ΩR(x0) |u − u(x0)| ≤  max  ∂ΩR(x0)∩∂Ω |u − u(x0)| + osc∂ΩR(x0)∩Ωu  =  max  ∂ΩR(x0)∩∂Ω |g − g(x0)| + osc∂ΩR(x0)∩Ωu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  The first term tends to 0 as R → 0 since g ∈ C(∂Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' The second term tends to 0 as  R → 0 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' This implies the continuity of u as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  □  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' The proof above extends to solutions to the following inhomogeneous elliptic  system  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='9)  ∆ku =  k−1  �  l=0  ∆l ⟨Vl, du⟩ +  k−2  �  l=0  ∆lδ(wldu) + f  in Ω ⊂ R2k  with f ∈ Lp for some p > 1 and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='2), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Indeed, by [6, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='1], in this case, the  interior regularity estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='1) becomes  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='10)  oscBτr(x)u ≲ τ α �  ∥u∥W k,2(B4r(x0)) + ∥f∥Lp(B4r(x0))  �  for 0 < τ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  With this, the buondary maximal principle (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='2) remains valid with an extra term ∥f∥Lp(Ω4R(x))  on the right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='1 then works with obvious modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' The authors would like to Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Chang-Lin Xiang and Chang-Yu  Guo for posing this question to them and for many useful conservations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  References  [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Chang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Wang and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Yang, A regularity theory of biharmonic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' 52(9) (1999), 1113-1137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  [2] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' de Longueville and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Gastel, Conservation laws for even order systems of polyharmonic  map type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Partial Differential Equations 60, 138 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Gastel and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Scheven, Regularity of polyharmonic maps in the critical dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' 17 (2009), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' 2, 185-226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  [4] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Guo and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Xiang, Regularity of solutions for a fourth order linear system via conser-  vation law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Lond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' (2) 101 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' 3, 907-922.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  [5] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Guo and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Xiang, Regularity of weak solutions to higher order elliptic systems in critical  dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Tran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' 374 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' 5, 3579-3602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  6  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' LIU AND Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' TIAN  [6] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Guo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Xiang and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Zheng, Lp regularity theory for even order elliptic systems  with antisymmetric first order potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Pures Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' 165 (2022) 286-324.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  [7] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Lamm and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Rivi`ere, Conservation laws for fourth order systems in four dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  Partial Differential Equations 33 (2008), 245-262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  [8] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Lamm and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Wang, Boundary regularity for polyharmonic maps in the critical dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' 2 (2009), 1-16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  [9] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' M¨uller and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Schikorra, Boundary regularity via Uhlenbeck-Rivi`ere decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Analysis  (Munich) 29 (2009), 199-220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  [10] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Qing, Boundary regularity of weakly harmonic maps from surfaces, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' 114 (1993)  458-466.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  [11] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Rivi`ere, Conservation laws for conformally invariant variational problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' 168  (2007), 1-22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  [12] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Wang, Stationary biharmonic maps from Rm into a Riemannian manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' 57 (2004), 419-444.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='  (Ming-Lun Liu) Research Center for Mathematics and Interdisciplinary Sciences, Shan-  dong University, Qingdao 266237, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' China and Frontiers Science Center for Nonlin-  ear Expectations, Ministry of Education, Qingdao, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content=' China  Email address: minglunliu2021@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
+page_content='com  (Yao-Lan Tian) Center for Optics Research and Engineering, Shandong University,  Qingdao 266237, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
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+page_content=' China  Email address: tianylbnu@126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'}
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BtE4T4oBgHgl3EQfeA0g/content/tmp_files/2301.05095v1.pdf.txt

@@ -0,0 +1,1577 @@
+Electron cooling in graphene enhanced by plasmon-hydron resonance
+Xiaoqing Yu1, Alessandro Principi2, Klaas-Jan Tielrooij3,4, Mischa Bonn1 and Nikita Kavokine1,5
+1Max Planck Institute for Polymer Research,
+Ackermannweg 10, Mainz 55128, Germany
+2School of Physics and Astronomy, University of Manchester, M13 9PL Manchester, U.K.
+3Catalan Institute of Nanoscience and Nanotechnology (ICN2),
+BIST and CSIC, Campus UAB, Bellaterra, Barcelona, 08193, Spain
+4Department of Applied Physics, TU Eindhoven,
+Den Dolech 2, 5612 AZ, Eindhoven, The Netherlands and
+5Center for Computational Quantum Physics,
+Flatiron Institute, 162 5th Avenue, New York, NY 10010, USA
+Evidence is accumulating for the crucial role of a solid’s free electrons in the
+dynamics of solid-liquid interfaces.
+Liquids induce electronic polarization and
+drive electric currents as they flow; electronic excitations, in turn, participate in
+hydrodynamic friction. Yet, the underlying solid-liquid interactions have been
+lacking a direct experimental probe. Here, we study the energy transfer across
+liquid-graphene interfaces using ultrafast spectroscopy. The graphene electrons
+are heated up quasi-instantaneously by a visible excitation pulse, and the time
+evolution of the electronic temperature is then monitored with a terahertz pulse.
+We observe that water accelerates the cooling of the graphene electrons, whereas
+other polar liquids leave the cooling dynamics largely unaffected. A quantum
+theory of solid-liquid heat transfer accounts for the water-specific cooling en-
+hancement through a resonance between the graphene surface plasmon mode
+and the so-called hydrons – water charge fluctuations –, particularly the water
+libration modes, that allows for efficient energy transfer.
+Our results provide
+direct experimental evidence of a solid-liquid interaction mediated by collective
+modes and support the theoretically proposed mechanism for quantum friction.
+They further reveal a particularly large thermal boundary conductance for the
+water-graphene interface and suggest strategies for enhancing the thermal con-
+ductivity in graphene-based nanostructures.
+arXiv:2301.05095v1  [cond-mat.mes-hall]  12 Jan 2023
+
+2
+Free electrons in graphene exhibit rather unique dynamics in the terahertz (THz) frequency
+range, including a highly non-linear response to photoexcitation by THz pulses [1, 2]. Graphene’s
+distinctive dynamical properties on picosecond timescales have found several applications in, e.g.,
+ultrafast photodetectors, modulators, and receivers [3–5]. The THz frequency range acquires par-
+ticular importance at room temperature T, where it corresponds to the typical frequency of thermal
+fluctuations: kBT/ℏ ∼ 6 THz, with kB Boltzmann’s constant and ℏ Planck’s constant. One may
+therefore expect non-trivial couplings between the graphene electrons and the thermal fluctuations
+of their environment. These couplings have been intensively studied in the case of a solid environ-
+ment: for instance, non-adiabatic effects have been shown to arise in the graphene electron-phonon
+interaction [6], and plasmon-phonon coupling between graphene and a polar substrate has been
+demonstrated [7–9]. More recently, it has been theoretically proposed that similar effects are at
+play when graphene has a liquid environment: then, the interaction between the liquid’s charge
+fluctuations – dubbed hydrons – and graphene’s electronic excitations tunes the hydrodynamic
+friction at the carbon surface [10, 11]. This "quantum friction" mechanism holds the potential of
+entirely new strategies for controlling liquid flows at nanometer scales [12, 13].
+Obtaining an experimental signature of the quantum friction mechanism would involve directly
+visualizing momentum transfer between a solid and a liquid: that is, measuring a force. Force
+measurements at solid-liquid interfaces suffer from a strong sensitivity to the surface state, coupled
+with enormous technical challenges [14–16]. In this Article, we overcome this obstacle by measuring
+energy transfer as a proxy for momentum transfer. Specifically, we use a femtosecond visible pulse to
+introduce a quasi-instantaneous temperature difference between the graphene electrons and their
+environment. The cooling rate of the electronic system is followed in real-time using terahertz
+pulses. Such Optical Pump - Terahertz Probe (OPTP) spectroscopy is a well-established tool for
+probing electron relaxation in 2D materials [17–21]. In high-quality graphene, it has been used to
+identify the interaction of hot carriers with optical phonons [19, 20] and with substrate phonons
+as the main electron cooling mechanisms [22].
+Here, we measure the electron relaxation time in the presence of different polar liquids to probe
+the electron-liquid interaction, which we find to be significant compared to the electron - optical
+phonon interaction only when the liquid is water. A complete theoretical analysis shows that this
+specificity of water is explained by the strong coupling of its THz (libration) modes to the graphene
+surface plasmon, with the electron-electron interactions in graphene playing a crucial role.
+
+3
+a
+b
+Liquid
+Solid
+Quantum friction
+Liquid
+Solid
+Quantum heat transfer
+c
+FIG. 1. From friction to heat transfer. a. Artist’s view of the system under study: the interface
+between a liquid and a graphene sheet. The liquid, at temperature T, may flow with an interfacial velocity
+v, while the graphene electrons (depicted by the orange cloud) may be heated up to a temperature T + ∆T.
+b. Schematic of the solid-liquid quantum friction mechanism: momentum is transferred directly through
+quasiparticle tunneling at a rate γ between surface modes of the solid and the liquid (depicted by the blue
+parabolas), at wavevector q and frequency ωq. The Bose distribution nB predicts a higher occupation of the
+liquid mode (filling of the blue parabola) due to a flow-induced Doppler shift. c. Schematic, with the same
+notations as in b, of solid-liquid quantum heat transfer. Here, the solid’s mode has a higher occupation
+due to a higher temperature than the liquid. Energy and momentum transfer involve the same interaction
+between surface modes.
+Solid-liquid heat transfer
+The energy transfer between a solid and a liquid is usually considered to be mediated by
+molecular vibrations at the interface, as most of a solid’s heat capacity is contained in its phonon
+modes [23]. Even if an optical excitation of the solid’s electrons is used to create the temperature
+difference, the electrons are typically assumed to thermalize with phonons on a very short time
+scale, so that the solid’s phonons ultimately mediate the energy transfer to the liquid’s vibrational
+modes [24, 25]. However, if the electrons were to transfer energy to the liquid faster than to the
+phonons, the interfacial thermal conductivity would contain a non-negligible contribution from
+near-field radiative heat transfer [26, 27]. Such an electronic or "quantum" contribution to heat
+transfer is in close analogy with the quantum contribution to hydrodynamic friction (Fig.
+1).
+Quantum hydrodynamic friction relies on momentum being transferred directly between the solid’s
+and the liquid’s charge fluctuation modes.
+In a simplified Fermi golden rule picture [10], the
+corresponding friction force can be written as
+FQ =
+�
+dqdω ℏq ∆γq(ω).
+(1)
+
+4
+SiO2 cell
+THz probe
+Liquid
+Graphene
+Optical pump
+∆E(t)
+a
+Te = 623 K
+Pump-probe delay (ps)
+Normalized ∆T
+b
+N2
+H2O
+D2O
+Methanol
+Ethanol
+1.4
+1.6
+1.8
+2.0
+2.2
+2.4
+ 
+ 
+Te= 1241 K
+Te = 1023 K
+Te = 770 K
+Te = 623 K
+Cooling time (ps)
+c
+0
+1
+2
+3
+4
+5
+6
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+N2
+H2O
+Methanol
+Ethanol
+ D2O
+FIG. 2. Measurement of picosecond hot electron relaxation in graphene. a. Schematic of the
+experimental setup. A graphene sample is placed in contact with a liquid inside a fused silica flow cell. An
+optical excitation pulse impulsively heats up the graphene electrons, and the electron temperature dynamics
+are then monitored with a THz probe. b. Normalized electron temperature as a function of time after
+photoexcitation. The dotted lines represent raw data and the full lines are exponential fits. c. Electron
+cooling time obtained through exponential fitting (see b) for the different liquids that have been placed in
+the flow cell and different initial electron temperatures, set by the excitation laser fluence. Faster cooling
+is observed in the presence of water and heavy water. Error bars represent 95% confidence intervals of the
+exponential fits.
+It is a sum over all the in-plane wavevectors q and frequencies ω of the elementary momentum
+ℏq, multiplied by the net quasiparticle tunneling rate ∆γq(ω) between the solid’s and the liquid’s
+modes at wavevector q and frequency ω. The quantum contribution to the solid-liquid energy
+transfer rate then reads
+QQ =
+�
+dqdω ℏω ∆γq(ω),
+(2)
+with the momentum quantum ℏq being replaced by the energy quantum ℏω.
+Thus, quantum
+friction and quantum energy transfer rely on the same solid-liquid interactions, contained in the
+tunneling rates ∆γq(ω). In the same way that probing quantum friction requires it to dominate
+over the surface roughness contribution, the quantum energy transfer needs to exceed the classical
+phonon-based energy transfer in order to become measurable. We now show that this condition is
+met upon optically exciting of a graphene-water interface, owing, in particular, to graphene’s weak
+electron-phonon coupling [28].
+Time-resolved electron cooling
+Our experimental setup is schematically represented in Fig. 2a. A monolayer graphene sample
+
+5
+was transferred onto a fused silica flow cell, filled with either nitrogen gas or a liquid of our choice
+(SI Sec.
+1.1).
+In a typical experiment, the graphene electrons were excited using a ∼ 50 fs
+laser pulse with 800 nm central wavelength. Then, the attenuation of a ∼ 1 ps THz probe pulse
+(precisely, the modulation of the peak electric field) was monitored as a function of the pump-
+probe delay (SI Sec.
+1.2).
+After absorption of the exciting pump pulse, the non-equilibrium
+electron distribution typically thermalizes over a sub-100 fs timescale through electron-electron
+scattering [29]: it can then be described as a Fermi-Dirac distribution at a given temperature. A
+hotter electron distribution results in a lower THz photoconductivity, since hotter electrons are less
+efficient at screening charged impurities [30, 31]. The pump-probe measurement thus gives access
+to the electron temperature dynamics after photoexcitation (Fig. 2b).
+Regardless of the medium that the graphene is in contact with, the electronic temperature T(t)
+exhibits a relaxation that can be approximated by an exponential function : ∆T(t) = T(t) − T0 =
+∆T0e−t/τ. This allows us to extract the cooling times τ for the different liquids and different initial
+electronic temperatures (determined by the excitation laser fluence), displayed in Fig. 2c. We
+observe that the cooling time is longer for an initially hotter electron distribution, in agreement
+with previous reports [20]. Now, for all initial temperatures, we consistently observe the same
+dependence of the cooling time on the sample’s liquid environment.
+In the presence of water
+(H2O) and heavy water (D2O), the graphene electrons cool faster than they do intrinsically, in an
+inert nitrogen atmosphere. Conversely, methanol and ethanol have almost no effect on the electron
+cooling time. Interestingly, we observe an isotope effect in the electron cooling process: there is
+a difference in the cooling times in the presence of H2O and D2O that well exceeds experimental
+uncertainties.
+We are thus led to hypothesize, as anticipated above, that the liquid provides the electrons with
+a supplementary cooling pathway, which, in the case of water, has an efficiency comparable to the
+intrinsic cooling pathway. We then interpret the faster cooling as a signature of "quantum" electron-
+liquid energy transfer. We assess the pertinence of this hypothesis by developing a complete theory
+of quantum energy transfer at the solid-liquid interface.
+Theoretical framework
+In order to tackle the interaction between a classical liquid and an electronic system whose
+behavior is intrinsically quantum, we describe the liquid in a formally quantum way. Following
+ref. [10], we represent the liquid’s charge density as a free fluctuating field with prescribed corre-
+
+6
+lation functions. This naturally leads to a Fourier-space description of the solid-liquid interface
+in terms of its collective modes, rather than the usual molecular scale interactions. Within this
+description, the quantum solid-liquid energy transfer amounts to electron relaxation upon coupling
+to a bosonic bath, a problem that has been extensively studied in condensed matter systems [32].
+Interestingly, in the case of graphene, many of these studies are carried out within a single-particle
+Boltzmann formalism, which may incorporate multiple screening effects only in an ad hoc fash-
+ion [20, 28, 33]. These effects turn out to be crucial for the solid-liquid system under consideration:
+we have therefore developed an ab initio theory of solid-liquid heat transfer based on the non-
+equilibrium Keldysh formalism [34], which has only very recently been considered for problems of
+interfacial heat transfer [35]. Our computation, detailed in the SI Sec. 2.2, is closely analogous to
+the one carried out for quantum friction in ref. [10]. The theoretical framework can formally apply
+to fully non-equilibrium situations and take interactions into account to arbitrary order. However,
+to obtain a closed-form result, we restrict ourselves to a two-temperature model, where the liquid
+and the solid are assumed to be internally equilibrated at temperatures Tℓ and Te respectively. Fur-
+thermore, we take electron-electron and electron-liquid Coulomb interactions into account at the
+Random Phase Approximation (RPA) level. We these assumptions, we obtain the electron-liquid
+energy transfer rate as
+QQ =
+1
+2π3
+�
+dq
+� +∞
+0
+dω ℏω[nB(ω, Te) − nB(ω, Tℓ)]Im [ge(q, ω)]Im [gℓ(q, ω)]
+|1 − ge(q, ω)gℓ(q, ω)|2 ,
+(3)
+consistently with the general form anticipated in Eq. (2). Here, nB(ω, T) = 1/(eℏω/T − 1) is the
+Bose distribution and the ge,ℓ are surface response functions of the solid and the liquid, respectively.
+These are analogues of the dielectric function for semi-infinite media, whose precise definition is
+given in the SI, Sec.
+2.3.
+For the liquids under consideration, it will be sufficient to use the
+long-wavelength-limit expression of the surface response function:
+gℓ(q → 0, ω) = ϵℓ(ω) − 1
+ϵℓ(ω) + 1,
+(4)
+where ϵℓ(ω) is the liquid’s bulk dielectric function. For two-dimensional graphene, we show in the
+SI (Sec. 2.3) that the surface response function can be expressed as
+ge(q, ω) = − e2
+2ϵ0qχ(q, ω),
+(5)
+where χ(q, ω) is graphene’s charge susceptibility. We note that the result in Eq. (3) has been derived
+for two solids separated by a vacuum gap in the framework of fluctuation-induced electromagnetic
+phenomena [26, 36, 37]; we believe, however, that the framework we provide is better suited to the
+solid-liquid system under consideration.
+
+7
+Coulomb
+interaction
+Surface
+plasmon
+Electron
+cloud
+Hindered rotation
+(libration)
+a
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+Wavevector (nm-1)
+0
+0.05
+0.1
+0.15
+0.2
+Frequency (eV)
+Dirac cone
+Plasmon-hydron
+resonance
+Te = 623 K
+Cooling rate (1/ps)
+Total rate (experiment)
+Liquid contribution (theory)
+N2
+H2O
+D2O
+Methanol
+Ethanol
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+Wavevector (nm-1)
+0
+0.05
+0.1
+0.15
+0.2
+Frequency (eV)
+Te = 623 K
+Dirac cone
+0.05
+0.1
+0.15
+0.2
+Frequency (eV)
+0
+0.1
+0.2
+0.3
+0.4
+Surface excitation spectrum
+H2O
+D2O
+Methanol
+Ethanol
+b
+c
+0.5
+0.6
+0.7
+0.8
+0
+0.2
+0.4
+0.6
+0.8
+1
+Cooling rate (1/ps)
+N2
+H2O
+D2O
+Methanol
+Ethanol
+d
+e
+f
+Plasmon
+10
+20
+30
+40
+50
+Frequency (THz)
+FIG. 3. Mechanism of electron-liquid heat transfer. a. Surface excitation spectra Im [gℓ(ω)] of the
+different liquids under study obtained according to Eq. (4) from the experimentally-measured bulk dielectric
+permittivities. The arrows indicate the libration modes of H2O and D2O. b. Graphene surface excitation
+spectrum Im [ge(q, ω)], calculated at a chemical potential µ = 100 meV and temperature Te = 623 K.
+The main feature is the collective plasmon mode. c. Theoretical prediction for the graphene-water energy
+transfer rate resolved in frequency-wavevector space. The main contribution originates from a resonance
+between the graphene plasmon mode and the water libration mode. d. Experimentally-measured electron
+cooling rate in the presence of the various liquids. e. Theoretical prediction for the liquid contribution
+to the electron cooling rate, reproducing the experimentally-observed trend in terms of the nature of the
+liquid. The symbol size in the vertical direction represents the variation in the theoretical prediction when
+the graphene chemical potential spans the range [100 meV, 180 meV]. f. Schematic of the water-mediated
+electron cooling mechanism inferred from the combination of theoretical and experimental results.
+The
+cooling proceeds through the Coulomb interaction between the graphene plasmon mode and the hindered
+molecular rotations (librations) in water.
+Plasmon-hydron resonance
+If the interaction with the liquid is the only mechanism for electron relaxation, our result in
+Eq. (3) determines the time evolution of the electron temperature according to
+C(Te)dTe(t)
+dt
+= −QQ(Te, Tℓ),
+(6)
+
+8
+where C(Te) is the graphene electronic heat capacity at temperature Te. This allows us to de-
+fine the liquid contribution to the electron cooling rate as 1/τ = QQ(Te, Tℓ)/(C(Te) × (Te − Tℓ)),
+which may be compared with the experimental results. The quantitative evaluation of τ requires
+the surface response functions of graphene and of the various liquids. We compute the graphene
+surface response function according to Eq. (5) by numerical integration [38], at the chemical po-
+tential determined for our samples by Raman spectroscopy (SI Sec.
+1.4).
+For the liquids, we
+use the expression in Eq. (4), with the bulk dielectric function determined by infrared absorption
+spectroscopy (Fig. 3a and SI Sec. 1.3).
+Our theoretical prediction for the various liquids’ contribution to the electron cooling rate is
+shown in Fig. 3e. Quantitatively, we obtain cooling rates of the order of 1 ps−1, in excellent
+agreement with the experimentally observed range (Fig.
+3d) : our theory indicates that the
+quantum electron-liquid cooling is a sufficiently efficient process to compete with intrinsic electron
+relaxation mechanisms. Moreover, our theory reproduces the experimentally observed trend in
+cooling rates, with a significant liquid contribution arising only for water and heavy water; the
+dependence of the cooling rate on initial electron temperature is also well-reproduced (Fig. S7).
+Finally, the theory reproduces the isotope effect, that is, the slightly slower cooling observed with
+D2O as compared to H2O.
+We may now exploit the theory to gain insight into the microscopic mechanism of the liquid-
+mediated cooling process. In Eq. (3), the difference of Bose distributions decreases exponentially
+at frequencies above Te/ℏ ∼ 100 meV. At frequencies below 100 meV, the graphene spectrum is
+dominated by a plasmon mode, that corresponds to the collective oscillation of electrons in the
+plane of the graphene layer [38] (Fig. 3b). In this same frequency range, water and heavy water
+have a high spectral density due to their libration mode, that corresponds to hindered molecular
+rotations [39] (Fig. 3a). As a result, the energy transfer rate resolved in frequency-momentum
+space (the integrand in Eq. (3), plotted in Fig. 3c) has its main contribution from the spectral
+region where the two modes overlap. We conclude that the particularly efficient electron-water
+cooling is due to a resonance between the graphene plasmon mode and the water libration mode.
+This conclusion is further supported by the isotope effect. Indeed, the libration of the heavier D2O
+is at slightly lower frequency than that of the lighter H2O, and a higher frequency mode makes a
+larger contribution to the cooling rate due to the factor ℏω in Eq. (3). Physically, the quasiparticle
+tunneling rates are almost the same for the graphene-H2O and graphene-D2O systems, but in the
+case of H2O each quasiparticle carries more energy. Overall, our experiments evidence a direct
+interaction between the graphene plasmon and water libration, as shown schematically in Fig. 3f.
+
+9
+Cooling rate (1/ps)
+Renormalized
+No e-e interactions
+Bare
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+Wavevector (nm-1)
+0
+0.05
+0.1
+0.15
+0.2
+Frequency (eV)
+Te = 623 K
+b
+Full theory
+First order
+a
+0
+0.1
+0.2
+0
+20
+40
+60
+Energy transfer rate (meV·Å2·s-1)
+First order
+Full theory
+Frequency (eV)
+c
+H2O
+D2O
+Methanol
+Ethanol
+10-1
+100
+101
+0
+0.1
+0.2
+Frequency (eV)
+FIG. 4. Strong plasmon-hydron coupling. a. Theoretical prediction for the graphene electron cooling
+rate in contact with different liquids, within different treatments of interactions. The cooling rate is strongly
+overestimated if no electron-electron interactions are taken into account (blue symbols), and underestimated
+if the electron-liquid interactions are considered only to first order (orange symbols). b. Graphene surface
+excitation spectrum Im [ge(q, ω)], calculated at a chemical potential µ = 180 meV and temperature Te =
+623 K, renormalized by the presence of water according to Eq. (7). The white dashed lines are guides to
+the eye showing the strongly-coupled plasmon-hydron mode. Inset: bare and renormalized graphene spectra
+at fixed wavevector q0 = 0.15 nm−1. c. Comparison between the spectrally resolved energy transfer rates
+obtained to first order and to arbitrary order in the solid-liquid interaction. Higher-order effects enhance
+the energy transfer rate at low frequencies.
+Interactions and strong coupling
+The combination of theory and experiment allows us to identify the key physical ingredients
+that are required to account for energy transfer at the water-graphene interface. First, our results
+reveal that electron-electron interactions are crucial, since they produce the plasmon mode that
+is instrumental to the energy transfer mechanism. Indeed, applying our theory to non-interacting
+graphene would result in a strongly overestimated liquid contribution to the cooling rate (Fig. 4a).
+This precludes single-particle Boltzmann approaches – such as those that have been used for the
+electron-phonon interaction in graphene [20, 28] – for accurately describing the water-graphene
+interaction.
+Furthermore, the detailed examination of our theoretical result reveals that the efficiency of the
+electron-water cooling is enhanced by the formation of a strongly-coupled plasmon-hydron mode.
+Indeed, the result in Eq. (3) involves bare surface response functions, without any renormalization
+due to the presence of the other medium.
+However, the denominator |1 − gegℓ|2 accounts for
+solid-liquid interactions to arbitrary order (at the RPA level) and contains the signature of any
+
+10
+potential strong coupling effects. We find that these effects are indeed important, as removing the
+denominator in Eq. (3) (that is, treating the electron-liquid interactions only to first order) results
+in under-estimation of the liquid-mediated cooling rate by about 30% (Fig. 4a) . In order to gain
+physical insight into the nature of these higher order effects, we may compute the graphene surface
+response function renormalized by the presence of water, which is given by (see SI Sec. 2.3)
+˜ge(q, ω) =
+ge(q, ω)
+1 − ge(q, ω)gℓ(q, ω).
+(7)
+The renormalized surface excitation spectrum Im [˜ge(q, ω)] is plotted in Fig. 4b, for a chemical
+potential µ = 180 meV. We observe that the graphene plasmon now splits into two modes, which
+are both a mixture of the the bare plasmon and water libration. These are in fact analogous to
+the coupled plasmon-phonon modes that have been predicted [7] and measured [8, 9] for graphene
+on a polar substrate. It can be seen in the inset of Fig. 4b that coupling to the water modes also
+increases the spectral density at low frequencies (below the plasmon peak), compared to the bare
+graphene response function. This is in fact the higher-order effect that is mainly responsible for
+the enhancement of the electron cooling rate. As shown in Fig. 4c, taking into account solid-liquid
+interactions to arbitrary order mainly enhances the contribution of low frequencies to the energy
+transfer.
+Conclusions
+We have carried out ultrafast measurements of electron relaxation in graphene, revealing signa-
+tures of direct energy transfer between the graphene electrons and the surrounding liquid. These
+results speak to the importance of electronic degrees of freedom in the dynamics of solid-liquid
+interfaces, particularly interfaces between water and carbon-based materials. Despite conventional
+theories and simulations that describe the interface in terms of atomic-scale Lennard-Jones poten-
+tials [24, 25], or with electronic degrees of freedom in the Born-Oppenheimer approximation [40, 41],
+here we demonstrate experimentally that the dynamics of the water-graphene interface need to be
+considered at the level of collective modes in the terahertz frequency range. In particular, our semi-
+quantitative theoretical analysis attributes the observed cooling dynamics to the strong coupling
+between the graphene plasmon and water libration modes.
+The experimental observation of such a collective mode interaction supports the proposed mech-
+anism for quantum friction at the water-carbon interface, which is precisely based on momentum
+transfer between collective modes [10]. The near-quantitative agreement between the experiment
+
+11
+and theory obtained for energy transfer suggests that a similar agreement should be achieved for
+momentum transfer. We note, however, that quantum friction of water on graphene is typically
+negligible compared to the classical surface roughness contribution, and it is only expected to play
+a role in the presence of a phonon wind [12]. Quantum friction has been predicted to be much
+more important for water on graphite due to the difference in plasmon dispersion between the two
+materials [10]: the investigation of electron-water energy transfer in the case of carbon multilayers
+will be the subject of future work.
+Our results provide yet another example of the water-carbon interface outperforming other solid-
+liquid systems [42]. Indeed, the electronic contribution to the graphene-water thermal boundary
+conductance is as high as λ = 0.25 MW · m−2 · K−1, exceeding the value obtained with the
+other investigated liquids by at least a factor of 2.
+This even exceeds the thermal boundary
+conductance obtained for the graphene-hBN interface, at which particularly fast "super-Planckian"
+energy transfer was observed [33]. Our investigation thus suggests that the density of modes in the
+terahertz frequency range is a key determinant for the thermal conductivity of graphene-containing
+composite materials.
+Acknowledgements
+We acknowledge financial support from the MaxWater initiative of the Max Planck Society. We
+thank Xiaoyu Jia and Hai Wang for carrying out preliminary experiments, and Maksim Grechko
+and Detlev-Walter Scholdei for assisting with the FTIR measurements. X.Y. is grateful for support
+from the China Scholarship Council. K.J.T. acknowledges funding from the European Union’s
+Horizon 2020 research and innovation program under Grant Agreement No. 804349 (ERC StG
+CUHL), RYC fellowship No. RYC-2017-22330, and IAE project PID2019-111673GB-I00. N.K.
+acknowledges support from a Humboldt fellowship.
+The Flatiron Institute is a division of the
+Simons Foundation. We thank Lucy Reading-Ikkanda (Simons Foundation) for help with figure
+preparation.
+[1] Hwang, H. Y. et al. Nonlinear Thz conductivity dynamics in p-type CVD-grown graphene. Journal of
+Physical Chemistry B 117, 15819–15824 (2013).
+[2] Hafez, H. A. et al. Extremely efficient terahertz high-harmonic generation in graphene by hot Dirac
+fermions. Nature 561, 507–511 (2018).
+
+12
+[3] Liu, M. et al. A graphene-based broadband optical modulator. Nature 474, 64–67 (2011).
+[4] Romagnoli, M. et al. Graphene-based integrated photonics for next-generation datacom and telecom.
+Nature Reviews Materials 3, 392–414 (2018).
+[5] Muench, J. E. et al. Waveguide-integrated, plasmonic enhanced graphene photodetectors. Nano Letters
+19, 7632–7644 (2019).
+[6] Pisana, S. et al. Breakdown of the adiabatic Born-Oppenheimer approximation in graphene. Nature
+Materials 6, 198–201 (2007).
+[7] Hwang, E. H., Sensarma, R. & Sarma, S. D. Plasmon-phonon coupling in graphene. Physical Review
+B 82, 195406 (2010).
+[8] Dai, S. et al. Graphene on hexagonal boron nitride as a tunable hyperbolic metamaterial. Nature
+Nanotechnology 10, 682–686 (2015).
+[9] Koch, R. et al. Robust phonon-plasmon coupling in quasifreestanding graphene on silicon carbide.
+Physical Review Letters 116, 106802 (2016).
+[10] Kavokine, N., Bocquet, M.-L. & Bocquet, L. Fluctuation-induced quantum friction in nanoscale water
+flows. Nature 602, 84–90 (2022).
+[11] Bui, A. T., Thiemann, F. L., Michaelides, A. & Cox, S. J. Classical quantum friction at water-carbon
+interfaces. arXiv 2210.14040 .
+[12] Coquinot, B., Bocquet, L. & Kavokine, N. Quantum feedback at the solid-liquid interface: flow-induced
+electronic current and negative friction. arXiv 2205.03250 .
+[13] Marcotte, A. et al.
+Strong electronic winds blowing under liquid flows on carbon surfaces.
+arXiv
+2205.05037 .
+[14] Maali, A., Cohen-Bouhacina, T. & Kellay, H. Measurement of the slip length of water flow on graphite
+surface. Applied Physics Letters 92, 053101 (2008).
+[15] Secchi, E. et al. Massive radius-dependent flow slippage in carbon nanotubes. Nature 537, 210–213
+(2016).
+[16] Xie, Q. et al.
+Fast water transport in graphene nanofluidic channels.
+Nature Nanotechnology 13,
+238–245 (2018).
+[17] George, P. A. et al. Ultrafast optical-pump terahertz-probe spectroscopy of the carrier relaxation and
+recombination dynamics in epitaxial graphene. Nano Letters 8, 4248–4251 (2008).
+[18] Kar, S., Su, Y., Nair, R. R. & Sood, A. K. Probing photoexcited carriers in a few-layer MoS2 laminate
+by time-resolved optical pump terahertz probe spectroscopy. ACS Nano 9, 12004–12010 (2015).
+[19] Mihnev, M. T. et al. Microscopic origins of the terahertz carrier relaxation and cooling dynamics in
+graphene. Nature Communications 7, 11617 (2016).
+[20] Pogna, E. A. et al.
+Hot-carrier cooling in high-quality graphene is intrinsically limited by optical
+phonons. ACS Nano 15, 11285–11295 (2021).
+[21] Zheng, W. et al. Band transport by large Fröhlich polarons in MXenes. Nature Physics 18, 544–550
+(2022).
+
+13
+[22] Tielrooij, K. J. et al. Out-of-plane heat transfer in van der Waals stacks through electron-hyperbolic
+phonon coupling. Nature Nanotechnology 13, 41–46 (2018).
+[23] Phillpot, S. R. & McGaughey, A. J. Introduction to thermal transport. Materials Today 8, 18–20
+(2005).
+[24] Gutierrez-Varela, O., Merabia, S. & Santamaria, R. Size-dependent effects of the thermal transport at
+gold nanoparticle-water interfaces. The Journal of Chemical Physics 157, 084702 (2022).
+[25] Herrero, C., Joly, L. & Merabia, S. Ultra-high liquid-solid thermal resistance using nanostructured gold
+surfaces coated with graphene. Applied Physics Letters 120, 171601 (2022).
+[26] Volokitin, A. I. & Persson, B. N. Near-field radiative heat transfer and noncontact friction. Reviews of
+Modern Physics 79, 1291–1329 (2007).
+[27] Biehs, S.-A. et al. Near-field radiative heat transfer in many-body systems. Reviews of Modern Physics
+93, 025009 (2021).
+[28] Bistritzer, R. & MacDonald, A. H. Electronic cooling in graphene. Physical Review Letters 102, 206410
+(2009).
+[29] Brida, D. et al. Ultrafast collinear scattering and carrier multiplication in graphene. Nature Commu-
+nications 4, 1987 (2013).
+[30] Tomadin, A. et al. The ultrafast dynamics and conductivity of photoexcited graphene at different
+Fermi energies. Science Advances 4, eaar5313 (2018).
+[31] Massicotte, M., Soavi, G., Principi, A. & Tielrooij, K. J. Hot carriers in graphene-fundamentals and
+applications. Nanoscale 13, 8376–8411 (2021).
+[32] Mahan, G. D. Many-Particle Physics, chap. 7 (Springer, 2000).
+[33] Principi, A. et al. Super-Planckian electron cooling in a van der Waals stack. Physical Review Letters
+118, 126804 (2017).
+[34] Rammer, J. & Smith, H. Quantum field-theoretical methods in transport theory of metals. Reviews of
+Modern Physics 58, 323–359 (1986).
+[35] Wise, J. L., Roubinowitz, N., Belzig, W. & Basko, D. M. Signature of resonant modes in radiative heat
+current noise spectrum. Physical Review B 106, 165407 (2022).
+[36] Pendry, J. B. Radiative exchange of heat between nanostructures. Journal of Physics: Condensed
+Matter 11, 6621–6633 (1999).
+[37] Volokitin, A. I. & Persson, B. N. J. Radiative heat transfer between nanostructures. Physical Review
+B 63, 205404 (2001).
+[38] Wunsch, B., Stauber, T., Sols, F. & Guinea, F. Dynamical polarization of graphene at finite doping.
+New Journal of Physics 8, 318–318 (2006).
+[39] Carlson, S., Brunig, F. N., Loche, P., Bonthuis, D. J. & Netz, R. R. Exploring the absorption spectrum
+of simulated water from MHz to infrared. Journal of Physical Chemistry A 124, 5599–5605 (2020).
+[40] Tocci, G., Joly, L. & Michaelides, A. Friction of water on graphene and hexagonal boron nitride from
+ab initio methods: Very different slippage despite very similar interface structures. Nano Letters 14,
+
+14
+6872–6877 (2014).
+[41] Tocci, G., Bilichenko, M., Joly, L. & Iannuzzi, M. Ab initio nanofluidics: disentangling the role of
+the energy landscape and of density correlations on liquid/solid friction. Nanoscale 12, 10994–11000
+(2020).
+[42] Bocquet, L. Nanofluidics coming of age. Nature materials 19, 254–256 (2020).
+
+Supplementary information for:
+Electron cooling in graphene
+enhanced by plasmon-hydron resonance
+X. Yu, A. Principi, K.-J. Tielrooij, M. Bonn and N. Kavokine
+-
+Contents
+1
+Experimental methods
+1
+1.1
+Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+1
+1.2
+OPTP measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+1
+1.3
+FTIR measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+3
+1.4
+Raman measurements
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+5
+2
+Theoretical methods
+5
+2.1
+Interaction Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+6
+2.2
+General theory of electron-boson heat transfer . . . . . . . . . . . . . . . . . . . . .
+7
+2.3
+Application to the graphene-liquid system . . . . . . . . . . . . . . . . . . . . . . .
+8
+arXiv:2301.05095v1  [cond-mat.mes-hall]  12 Jan 2023
+
+Figure S1: Schematic of the OPTP setup.
+1
+Experimental methods
+1.1
+Sample preparation
+CVD-grown graphene samples supported on 1 mm-thick copper substrates were purchased from
+Grolltex Inc. The MilliQ water (18.2 MΩ · cm) was used as obtained from the machine. Cellulose
+acetate butyrate (CAB, average Mn ∼ 12000, Sigma-Aldrich), ammonium persulfate (APS, ACS
+reagent, ≥ 98%, Honeywell FlukaTM) are used as received. CAB was dissolved in ethyl acetate
+(Sigma-Aldrich), producing a 30 mg/mL solution. APS was dissolved in MilliQ water to prepare
+1 M and 0.1 M solutions. The detachable fused silica flow cell was ordered from FireflySci, Inc.
+The flow cell was cleaned by sonication in a hot acetone and ethanol baths for 10 minutes each
+before using.
+We transferred graphene onto the front substrate of the flow cell following a wet transfer
+procedure [1, 2].
+First, we spin-coated graphene samples with CAB at 4000 rpm and baked
+them at 180◦C for 3 minutes. Then, to remove unnecessary graphene on the backside of copper
+substrates, the CAB-coated graphene samples were immersed into a 1 M solution of APS for 10
+minutes and subsequently rinsed with MilliQ water five times. The copper substrates were then
+fully etched by 0.1 M APS solution for 2 hours, followed by a five times rinse with MilliQ water
+to remove the attached ions. Then, the floating CAB-graphene monolayers were "fished" onto the
+flow cell, and the CAB coating was removed by soaking in acetone for 2 hours and in isopropanol
+for one hour.
+1.2
+OPTP measurements
+We probed electron relaxation in graphene using optical pump - terahertz probe (OPTP) spec-
+troscopy. A schematic of the OPTP setup is shown in Fig. S1. The fundamental laser output was
+generated by a regenerative Ti:sapphire amplifier system, which produces 5 W, 50 fs pulses at a
+repetition rate of 1 kHz and a central wavelength of 800 nm. The generated pulses were then split
+into three branches for THz generation, sampling, and optical excitation. A single-cycle THz pulse
+of ∼ 1 ps duration was generated by pumping a 1 mm thick (110) ZnTe crystal with the 800 nm
+fundamental pulses via optical rectification.
+We photoexcited graphene to generate hot carriers by using 800 nm pulses with a diameter of
+1
+
+X
+Polarizer
+Delay stage
+Beam splitter
+P
+Chopper
+ZnTe 入/4
+Wollaston
+sample
+prism
+White foam
+ZnTe
+Differential
+detector0
+2
+4
+6
+8
+0.00
+0.02
+0.04
+0.06
+0.08
+Pump-probe delay (ps)
+N2
+Fluence (
+2)
+ 5.86
+ 4.32
+ 2.85
+ 1.50
+ 0.89
+ 0.29
+c
+b
+a
+0.00
+0.02
+0.04
+0.06
+0.08
+0
+200
+400
+600
+800
+1000
+1200
+Peak value of 
+0
+1
+2
+3
+4
+5
+6
+0.00
+0.02
+0.04
+0.06
+0.08
+ 
+ 
+Peak value of 
+Fluence (μJ/cm2)
+0
+200
+400
+600
+800
+1000
+1200
+T e-T l (K)
+T e-T l (K)
+∆E/E
+∆E/E
+∆E/E
+μJ/cm
+Figure S2: Electron temperature of the pumped graphene layer. a.The OPTP traces
+of graphene in a nitrogen atmosphere with various excitation fluences. b. Peak value of ∆E/E
+as a function of laser fluence and corresponding electron temperature. c. Increase in electron
+temperature (Te) with respect to ambient temperature Tℓ as a function of ∆E/E: a linear relation
+is observed.
+5 mm to ensure a homogeneously photoexcited region. The transmitted THz wave was then recol-
+limated and focused onto a ZnTe detection crystal together with an 800 nm sampling beam, where
+the THz electrical field waveform was detected using the electro-optic sampling method [3]. The
+THz pulse induces birefringence in the ZnTe detection crystal, and the polarization of the sampling
+beam is thus changed. After passing through a quarter-wave plate, the sampling beam changes
+from perfectly circular to slightly elliptical shape. The s and p components of this elliptically
+polarized pulse are separated by a Wollaston prism, and the difference of these two components
+is detected by a balance diode. The signal is collected by a lock-in amplifier that is phase-locked
+to an optical chopper that modulates either the THz generation beam or the pump beam at a
+frequency of 500 Hz. The ultrafast time evolution of the peak intensity of the THz field is tracked
+by varying the time delay between optical pump and THz probe [3, 4]. The setup was purged with
+dry nitrogen during the measurement to avoid the absorption of water vapor.
+The raw data consists in time traces of the pump-induced transmission change at the peak of
+the THz waveform (∆E), normalized by the peak value of the THz transmission without excitation
+(E) (Fig. S2a). Assuming that a fraction γ = 1.6% of the pump pulse energy is absorbed by the
+graphene electrons [5], the maximum electron temperature reached after photoexcitation can be
+related to the pump laser fluence F according to γF = C(Te)Te, where C(Te) is the graphene heat
+capacity at temperature Te. In the limit where the graphene Fermi energy µ is larger than kBTe
+(as relevant for our samples), we may use the approximate expression [6, 7, 8]
+C(Te) = αTe,
+with
+α = 2π
+3
+k2
+Bµ
+(ℏvF)2 ,
+(1)
+where vF is graphene’s constant Fermi velocity. Then,
+Te = T0
+�
+1 + 2γF
+αT 2
+0
+�1/2
+,
+(2)
+where T0 is ambient temperature. The peak value of ∆E/E after photoexcitation increases with
+laser fluence.
+Upon rescaling, we find that the plots of ∆E/E vs.
+F and Te vs.
+F collapse
+upon each other (Fig. S2b), so that we may consider that ∆E/E is proportional to the electron
+temperature within the range of temperatures probed in the experiment, as shown explicitly in
+Fig. S2c.
+The thickness of the liquid layer was set to 50 µm by the geometry of the flow cell. The liquids
+were exchanged using a syringe and the spectroscopic measurement was always carried out at
+2
+
+0
+2
+4
+6
+8
+10
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ no cover
+ with cover
+ 10 μm
+ 20 μm
+ 30 μm
+ 40 μm
+ 50 μm
+ 60 μm
+ 
+1.50
+1.55
+1.60
+1.65
+1.70
+1.75
+1.80
+ Day 1
+ Day 2
+ Day 3
+ 
+ 
+no cover
+with cover
+10 μm
+20 μm
+30 μm
+40 μm
+50 μm
+60 μm
+Cooling time (ps)
+Normalized ∆E/E
+b
+a
+Pump-probe delay (ps)
+Figure S3: Control experiments. a. The OPTP traces of graphene with varying water layer
+thickness. b. Cooling times obtained by exponential fitting of the data in panel a.
+the same spot of the graphene sample. To exclude the effect of beam dispersion in the different
+liquids on the results, we repeated the measurement with different water layer thickness and using
+different Teflon spacers between two fused silica windows (Fig. S3).
+1.3
+FTIR measurements
+We measured the dielectric functions of water, heavy water, ethanol and methanol using Fourier-
+transform infrared (FTIR) spectroscopy.
+We measured the transmitted and reflected infrared
+intensities both for an empty cell (It,cell, Ir,cell) and for a cell filled with liquid (It,liquid, Ir,liquid)
+thanks to an A510/Q-T Reflectance and Transmittance accessory placed in a commercial VERTEX
+70 FTIR spectrometer (Fig. S4a). In order to avoid disassembling the cell when changing liquids,
+we carried out the measurements inside a flow cell, made out of two-silicon wafers separated by a
+10 µm Teflon spacer. We calculated the absorbance A(ω) according to
+A(ω) = − log10
+�
+It,solution(ω)
+It,cell(ω) + Ir,cell(ω) − Ir,solution(ω)
+�
+.
+(3)
+The H2O and D2O show saturated absorption in the range of 3100-3600 and 2200-2700 cm−1,
+respectively. We obtained the data in this frequency range by measuring the spectra without any
+spacer between two CaF2 windows and then rescaled the spectra to overlap with the data with
+spacer (Fig. S4b). The imaginary part k(ω) of the refractive index is related to the absorbance by
+k(ω) = A(ω)ln(10)
+4πωℓ ,
+(4)
+where ℓ is the sample thickness. To accurately determine the thickness of the cell, we calculate
+the absorbance of the empty cell without correction for multiple reflections,
+A2(ω) = − log10
+�
+It,cell(ω)
+It,lamp(ω)
+�
+.
+(5)
+where It,lamp(ω) is the intensity of the lamp of the FTIR source (Fig. S5a). Fourier transformation
+of this spectrum yields a peak at the time ∆t that light takes to travel twice through the cell (Fig.
+S5b), so that ℓ = c∆t/2 = 10.29 µm. We then obtained the real part of the refractive index
+through a numerical Kramers-Krönig transformation:
+n(ω) = n∞ + 2
+π
+� ∞
+0
+dω′ k(ω′)
+ω′ − ω,
+(6)
+3
+
+Figure S4: FTIR data analysis. a. Raw intensity data of empty cell and water-filled cell. b.
+Absorbance of H2O and D2O, measured with spacer and without spacer (the latter is rescaled to
+overlap with the former).
+Figure S5: Determination of the cell thickness. a. Absorbance of empty cell. b. Fourier
+transform of the data in panel a.
+4
+
+a 0.20
+celltransmission
+cellreflection
+0.15
+solutiontransmission
+(a.u.
+solutionreflection
+0.05
+0.00
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+Frequency (cm-1)
+6
+H,0
+b
+:D20
+5
+-H,Owospacer(X5.0)
+bsorbance
+-D,Owospacer(X3.3)
+H,O combined
+3
+D,o combined
+0
+1000
+2000
+3000
+4000
+Frequency (cm-1)a
+1.2
+007
+0.06863ps
+1.0.
+0.06
+0.05
+Absorbance
+itensity
+0.04
+0.6
+0.03
+0.4
+0.02
+0.2.
+0.01.
+0.00
+0.0
+0
+2000
+4000
+6000
+0.04
+0.05
+0.06
+0.07
+0.08
+0.09
+Frequency(cm")
+Time (ps)a
+b
+Figure S6: Raman characterization of graphene sample. a. Spatial map of Raman G
+band frequency for graphene sample in air. b. Distribution of the Raman G band frequency with
+different liquids placed on the graphene surface.
+where n∞ is the refractive index in the high frequency limit, which is obtained by the ATAGO
+Digital Handheld Refractometer: PAL-RI. The measured values for H2O, D2O, methanol, ethanol
+and isopropanol are 1.333, 1.3291, 1.3285, 1.3604, and 1.3706 respectively.
+We then obtain the dielectric function ϵ(ω) = ϵ′(ω) + iϵ′′(ω) according to
+�
+ϵ′(ω) = n(ω)2 − k(ω)2
+ϵ′′(ω) = 2n(ω)k(ω)
+.
+(7)
+1.4
+Raman measurements
+We estimated the Fermi level µ in our liquid-covered graphene samples from the Raman G-band
+frequency, according to the empirical equation [6]
+|µ|(eV) = ωG − 1580 cm−1
+42 cm−1
+.
+(8)
+An example of a spatial map of the Raman G-band frequency is shown in Fig. S6a. The fre-
+quency shows spatial inhomogeneities on the µm scale with an amplitude around 10 cm−1. The
+corresponding distributions are shown in Fig. S6b. The average G-band frequency is essentially
+independent of the nature of the liquid, which excludes a change in charge carrier density as a
+possible mechanism for the liquid effect on the electron cooling rate. To take into account the
+broadness of the distribution, in the theoretical analysis we considered chemical potentials in the
+range µ = 100 − 180 meV. The theoretical prediction is independent of the electron or hole nature
+of the charge carriers.
+2
+Theoretical methods
+In this section, we develop a description of energy transfer between the Dirac fermion charge
+carriers in graphene and a liquid, treated as a bosonic bath, within the non-equilibrium Keldysh
+framework of perturbation theory. For the sake of completeness, and in order to show consistency
+with previous theoretical approaches, we apply the same description to energy transfer between
+the graphene electrons and its optical phonon modes, showing that our formalism recovers the
+results that were previously obtained within a Boltzmann equation approach [9].
+5
+
+We use SI units throughout the text. We adopt the following convention for the n-dimensional
+Fourier transform:
+˜f(q) =
+� +∞
+−∞
+dnr f(r)e−iq·r
+and
+f(r) =
+1
+(2π)n
+� +∞
+−∞
+dnq ˜f(q)eiq·r.
+(9)
+2.1
+Interaction Hamiltonian
+2.1.1
+Electron-hydron interaction
+In this section, r represents a vector in 3D space, and ρ a vector in 2D space.
+The charge
+fluctuations of the liquid in the z > 0 half-space couple to the graphene electrons via the Coulomb
+potential V . In real space, the corresponding Hamiltonian is
+Hew(t) =
+�
+drdr′nw(r, t)V (r − r′)ne(r, t),
+(10)
+where nw and ne are the liquid and graphene instantaneous charge density, respectively.
+Let
+c†
+k,ν, ck,ν be the Dirac fermion creation and annihilation operators in the chiral basis (ν = ±1). A
+2D Fourier transformation then yields
+Hint =
+�
+dq
+(2π)2
+e2
+2ϵ0qns(q, t)
+�
+k,ν,ν′
+⟨k + q, ν|eiqρeqz|k, ν′⟩c†
+k+q,ν(t)ck,ν′(t),
+(11)
+with
+ns(q) =
+�
+dρ
+� +∞
+0
+dz e−iqρe−qznw(ρ, z, t).
+(12)
+As long as we consider wavevectors q such that q−1 is large compared to the extension of the
+carbon pz orbitals perpendicular to the graphene plane, we may approximate
+|⟨k + q, ν|eiqρeqz|k, ν′⟩|2 ≈ |⟨k + q, ν|eiqρ|k, ν′⟩|2 = 1
+2
+�1 + νν′ cos(φk+q − φq)
+� .
+(13)
+2.1.2
+Electron-phonon interaction
+Let d†
+q,α, dq,α be the creation and annihilation operators of phonons in the mode α with frequency
+ωα. The non-interacting electron-phonon system’s Hamiltonian is
+H0 =
+�
+k,ν
+Ek,νc†
+k,νck,ν +
+�
+q,α
+ℏωαd†
+q,αdq,α,
+(14)
+where Ek,ν are the band energies, and �
+k ≡ (1/ABZ)
+�
+BZ dk (ABZ is the area of the 2D Brillouin
+zone). The electron-phonon interaction Hamiltonian has the general form [10]
+Hep =
+�
+α
+�
+BZ
+dq
+(2π)2
+�
+k,ν,ν′
+gνν′
+α,k,k+qc†
+k+qck(d†
+q,α + d−q,α),
+(15)
+Following [9], we consider the Γ point LO and TO phonons that scatter electrons within one valley,
+and the K, K’ point LO phonons that scatter electrons between valleys. The electron-phonon
+matrix elements read
+|gνν′
+Γ,k,k+q|2 = g2
+Γ(1 ± νν′ cos(φk + φk+q − 2φq)),
+(16)
+where the + (−) sign is for LO (TO) phonons; and
+|gνν′
+Γ,k,k+q|2 = g2
+K(1 ∓ νν′ cos(φk − φk+q)),
+(17)
+where the − (+) sign corresponds to scattering from K to K’ (from K’ to K); here, φv is the polar
+angle of the vector v. The values of the coupling constants are gΓ = 0.55 eV·Å and gK = 0.85 eV·Å,
+according to GW calculations [11].
+6
+
+2.1.3
+General form
+We find that for both types of interactions the Hamiltonian has the general form
+Heb =
+�
+dq
+(2π)2 nq(t)ϕq(t),
+(18)
+where nq is an electronic two-particle operator and ϕq is a free bosonic field. In the electron-phonon
+case, we define
+nq =
+�
+k,ν,ν′
+gνν′
+α,k,k+q
+√ℏωα
+c†
+k+q,νck,ν′
+and
+ϕq =
+�
+ℏωα(d†
+q,α + d−q,α);
+(19)
+in the electron-hydron case
+nq =
+�
+Vq
+�
+k,ν,ν′
+⟨k + q, ν|eiqρ|k, ν′⟩c†
+k+q,νck,ν′
+and
+ϕq =
+�
+Vqns(q),
+(20)
+where Vq ≡ e2/(2ϵ0q) is the 2D Fourier-transformed Coulomb potential. With these definitions,
+both nq and φq have dimensionless correlation functions in frequency space.
+2.2
+General theory of electron-boson heat transfer
+2.2.1
+Non-equilibrium perturbation theory
+We consider an initial state of the electron-boson system where the electrons are at a temperature
+Te and the bosons at a temperature Tb. We wish to study the subsequent dynamics. In particular,
+we are interested in the heat flux per unit surface from the electrons to the bosons:
+Q(t) = − 1
+A
+d
+dt⟨Heb(t)⟩.
+(21)
+Since the system is under non-equilibrium conditions, this average value needs to be computed in
+the Keldysh framework. In particular, we may define the Keldysh component of the electron-boson
+correlation function:
+χK
+eb(q, t, t′) = − 1
+A
+i
+ℏ⟨{nq(t), ϕ−q(t′)}⟩.
+(22)
+Then,
+Q(t) = −iℏ
+2
+�
+dq
+(2π)2
+dχK
+eb(q, t, t)
+dt
+.
+(23)
+Form this point on, the computation of the electron-boson correlation function follows the exact
+same steps as in the theory of quantum friction [12], and we reproduce here only the main equations.
+Diagramatically, the correlation function satisfies the following Dyson equation:
+(24)
+where the "bubble" represents the propagator of n (denoted χe), and the dashed line the propagator
+of ϕ (denoted χb). When made explicit in terms of the R, A, K components, the Dyson equation
+becomes
+�
+�
+�
+�
+�
+χK
+eb = χR
+e ⊗ χK
+b + χK
+e ⊗ χA
+b + χR
+e ⊗ χR
+b ⊗ χK
+eb + (χR
+e ⊗ χK
+b + χK
+e ⊗ χA
+b ) ⊗ χA
+eb
+χR,A
+eb
+= χR,A
+e
+⊗ χR,A
+b
++ χR,A
+e
+⊗ χR,A
+b
+⊗ χR,A
+eb
+,
+(25)
+where ⊗ represents time convolution.
+While these equations are extremely general, they are
+impractical to manipulate analytically, unless a number of assumptions are made. In order to
+7
+
+proceed, we will restrict ourselves to cooling dynamics that are slow enough for time-translation
+invariance to hold when it comes to determining the cooling rate. This assumption is expected
+to hold for small enough temperature differences, such that the cooling rate is approximately
+temperature-independent. We will further assume that, in line with experimental observations,
+that electron thermalization is much faster than electron-boson energy transfer, so that the electron
+and boson propagators may be considered as equilibrium propagators, satisfying the fluctuation-
+dissipation theorem: we work within a two-temperature model. We may then carry out Fourier
+transforms in time, so that Eq. (23) becomes
+Q = 1
+2
+� dqdω
+(2π)3 ℏω χK
+eb(q, ω).
+(26)
+The convolutions in Eq. (25) become products in Fourier space. Before proceeding, it is convenient
+to flip the signs of all the correlation functions: we introduce, for all the labels, g ≡ −χ. Then,
+after some algebra, we obtain an explicit expression for Q:
+Q =
+1
+2π3
+�
+dq
+� +∞
+0
+dω ℏω[nB(ω, Te) − nB(ω, Tb)]Im [gR
+e (q, ω)]Im [gR
+b (q, ω)]
+|1 − gR
+e (q, ω)gR
+b (q, ω)|2 ,
+(27)
+where nB(ω, T) ≡ 1/(eℏω/kBT − 1) is the Bose distribution at temperature T. We recover Eq. (3)
+of the main text.
+2.2.2
+Cooling rate
+The cooling dynamics are governed by the equation
+dE(Te)
+dt
+= −Q(Te, Tb),
+(28)
+where E is the total energy per unit surface of the electronic system. We follow ref. [9] in de-
+termining the electronic heat capacity (per unit surface) at constant density C(Te), such that
+dtE = C(Te)dtTe. We may then define the instantaneous cooling rate
+τ(Te, Tb) = C(Te)(Te − Tb)
+Q(Te, Tb)
+.
+(29)
+2.3
+Application to the graphene-liquid system
+2.3.1
+Liquid-mediated cooling
+We first consider electron cooling through the electron-hydron coupling. Using eqs. (12) and (20),
+we find that
+gR
+b (q, t, t′) = − 1
+AVq
+� +∞
+0
+dzdz′ e−q(z+z′)
+�
+− i
+ℏθ(t − t′)⟨[ns(q, z, t), ns(−q, z′, t′)]⟩
+�
+.
+(30)
+This is the microscopic definition of the liquid’s surface response function. In the long wavelength
+limit, it can be expressed in terms of the liquid’s bulk dielectric function ϵ(ω) [12]:
+gR
+b (q, ω) = ϵ(ω) − 1
+ϵ(ω) + 1,
+(31)
+as stated in the main text. The electronic response function gR
+e (q, ω) simply amount to (minus)
+the density-density response function. Taking into account electron-electron interactions at the
+RPA level [13],
+gR
+e (q, ω) = −
+Vqχ0
+e(q, ω)
+1 − Vqχ0e(q, ω).
+(32)
+8
+
+600
+800
+1000
+1200
+Electron temperature (K)
+1
+1.5
+2
+2.5
+Cooling time (ps)
+Theory
+Experiment
+Figure S7: Dependence of water-mediated cooling time on initial electron temperature. The red
+dots are experimental data for graphene in contact with water and the red dots correspond to the
+prediction of Eq. (29) (with µ = 180 meV).
+The non-interacting response function χ0
+e is given by [13]
+χ0
+e(q, ω) = gsgv
+�
+dk
+(2π)2
+�
+ν,ν′
+|⟨k + q, ν|eiqρ|k, ν′⟩|2 nF(Eν
+k, Te) − nF(Eν′
+k+q, Te)
+Eν
+k − Eν′
+k+q + ω + iδ
+,
+(33)
+where gs = gv = 2 are the spin and valley degeneracies of graphene, respectively, Eν
+k = νvF k are
+the band energies in the Dirac fermion approximation, nF(E, T) = 1/(e(E−µ)/kBT +1) is the Fermi
+distribution at chemical potential µ and temperature T, and δ → 0+. The integral is evaluated
+numerically at non-zero temperature.
+With all the above, we may compute theoretical predictions for the liquid-mediated cooling rate
+by numerical integration according to Eq. (27). We considered a graphene chemical potential µ in
+the range 100 − 180 meV (see section 1.4) and an electron temperature Te = 623 K, corresponding
+to the lowest pump laser fluence. Our model is further able to reproduce the dependence of the
+electron cooling time on Te, as shown in Fig. S7.
+We note that Eq. (27) involves bare surface response functions, that contain no effect of the
+presence of the neighboring medium, at least at the RPA level. Nevertheless, the physical response
+function of graphene in the presence of water undergoes RPA renormalization according to
+(34)
+In this diagrammatic equation, when the propagators are interpreted as surface response functions,
+the vertices reduce to unity, so that we obtain the renormalized graphene response function ˜ge as
+˜ge(q, ω) =
+ge(q, ω)
+1 − ge(q, ω)gb(q, ω),
+(35)
+which is Eq. (7) of the main text.
+2.3.2
+Phonon-mediated cooling
+In the phonon case, the boson response function is proportional to the usual phonon propagator:
+gR
+b (q, ω) =
+2ω2
+α
+ω2α − ω2 .
+(36)
+9
+
+The non-interacting electronic response function now involves the electron-phonon matrix elements:
+gR
+e (q, ω) = −gs
+�
+BZ
+dk
+(2π)2
+�
+ν,ν′
+|gνν′
+α,k,k+q|2
+ℏωα
+nF(Eν
+k, Te) − nF(Eν′
+k+q, Te)
+Eν
+k − Eν′
+k+q + ω + iδ
+.
+(37)
+We now show that we recover the results of ref. [9] for the electron-phonon cooling rate obtained in
+a Boltzamann equation framework, if we neglect electron-electron interactions and treat electron-
+phonon interactions to first order. Under these assumptions, Eq. (27) reduces to
+Q =
+1
+2π3
+�
+dq
+� +∞
+0
+dω ℏω[nB(ω, Te) − nB(ω, Tb)]Im [gR
+e (q, ω)]Im [gR
+b (q, ω)].
+(38)
+We notice that
+Im [gR
+b (q, ω)] = πω2
+α[δ(ω − ωα) − δ(ω + ωα)]
+(39)
+and
+Im [gR
+e (q, ω)] = πgs
+�
+BZ
+dk
+(2π)2
+�
+ν,ν′
+|gνν′
+α,k,k+q|2
+ℏωα
+[nF(Eν
+k, Te) − nF(Eν′
+k+q, Te)]δ(Eν
+k − Eν′
+k+q + ω). (40)
+Moreover, upon integration over k and q in Eq. (38), the angle-dependent parts of the electron-
+phonon matrix elements vanish, and the intervalley phonons become formally identical to the
+intravalley phonons: we may introduce the valley degeneracy and carry out integrations over a
+single Dirac cone. Altogether, we obtain
+Q = 2πgsgvωαg2
+α[nB(ωα, Te) − nB(ωα, Tb)] . . .
+· · ·
+�
+ν,ν′
+� dqdk
+(2π)4 [nF(Eν
+k, Te) − nF(Eν′
+q , Te)]δ(Eν
+k − Eν′
+q + ωα).
+(41)
+If we introduce another delta function, according to
+Q = 2πgsgvωαg2
+α[nB(ωα, Te) − nB(ωα, Tb)] . . .
+· · ·
+�
+ν,ν′
+� dqdk
+(2π)4
+� +∞
+−∞
+dϵ[nF(ϵ − ωα, Te) − nF(ϵ, Te)]δ(Eν
+k − ϵ + ωα)δ(ϵ − Eν′
+q ),
+(42)
+we recognize the graphene density of states,
+ν(ϵ) = gsgv
+�
+ν
+�
+dk
+(2π)2 δ(ϵ − Ek,ν) = 2|ϵ|
+πv2
+F
+.
+(43)
+Our result then simplifies according to
+Q = 2πωαg2
+α
+gsgv
+[nB(ωα, Te) − nB(ωα, Tb)]
+� +∞
+−∞
+dϵ[nF(ϵ − ωα, Te) − nF(ϵ, Te)]ν(ϵ)ν(ϵ − ωα),
+(44)
+which is Eq. (18) in the supplementary information of ref. [9].
+References
+[1] Yogeswaran, N. et al. Piezoelectric graphene field effect transistor pressure sensors for tactile
+sensing. Applied Physics Letters 113, 014102 (2018).
+[2] Burwell, G., Smith, N. & Guy, O. Investigation of the utility of cellulose acetate butyrate
+in minimal residue graphene transfer, lithography, and plasma treatments. Microelectronic
+Engineering 146, 81–84 (2015).
+10
+
+[3] Ulbricht, R., Hendry, E., Shan, J., Heinz, T. F. & Bonn, M. Carrier dynamics in semicon-
+ductors studied with time-resolved terahertz spectroscopy. Reviews of Modern Physics 83,
+543–586 (2011).
+[4] Lee, Y.-S. Principles of Terahertz Science and Technology (Springer US, 2009).
+[5] Fu, S. et al.
+Long-lived charge separation following pump-wavelength-dependent ultrafast
+charge transfer in graphene/ws2 heterostructures. Science Advances 7, eabd9061 (2021).
+[6] Shi, S. F. et al.
+Controlling graphene ultrafast hot carrier response from metal-like to
+semiconductor-like by electrostatic gating. Nano Letters 14, 1578–1582 (2014).
+[7] Tielrooij, K. J. et al. Photoexcitation cascade and multiple hot-carrier generation in graphene.
+Nature Physics 9, 248–252 (2013).
+[8] Lui, C. H., Mak, K. F., Shan, J. & Heinz, T. F. Ultrafast photoluminescence from graphene.
+Physical Review Letters 105, 127404 (2010).
+[9] Pogna, E. A. et al. Hot-carrier cooling in high-quality graphene is intrinsically limited by
+optical phonons. ACS Nano 15, 11285–11295 (2021).
+[10] Neto, A. H. C. & Guinea, F. Electron-phonon coupling and raman spectroscopy in graphene.
+Physical Review B 75, 045404 (2007).
+[11] Sohier, T. et al.
+Phonon-limited resistivity of graphene by first-principles calculations:
+Electron-phonon interactions, strain-induced gauge field, and boltzmann equation. Physical
+Review B 90, 125414 (2014).
+[12] Kavokine, N., Bocquet, M.-L. & Bocquet, L.
+Fluctuation-induced quantum friction in
+nanoscale water flows. Nature 602, 84–90 (2022).
+[13] Wunsch, B., Stauber, T., Sols, F. & Guinea, F. Dynamical polarization of graphene at finite
+doping. New Journal of Physics 8, 318–318 (2006).
+11
+

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+ROBUST MACHINE LEARNING PIPELINES FOR TRADING
+MARKET-NEUTRAL STOCK PORTFOLIOS ∗
+THOMAS WONG† AND MAURICIO BARAHONA‡
+Abstract. The application of deep learning algorithms to financial data is difficult due to heavy
+non-stationarities which can lead to over-fitted models that underperform under regime changes.
+Using the Numerai tournament data set as a motivating example, we propose a machine learning
+pipeline for trading market-neutral stock portfolios based on tabular data which is robust under
+changes in market conditions.
+We evaluate various machine-learning models, including Gradient
+Boosting Decision Trees (GBDTs) and Neural Networks with and without simple feature engineer-
+ing, as the building blocks for the pipeline. We find that GBDT models with dropout display high
+performance, robustness and generalisability with relatively low complexity and reduced computa-
+tional cost. We then show that online learning techniques can be used in post-prediction processing
+to enhance the results.
+In particular, dynamic feature neutralisation, an efficient procedure that
+requires no retraining of models and can be applied post-prediction to any machine learning model,
+improves robustness by reducing drawdown in volatile market conditions. Furthermore, we demon-
+strate that the creation of model ensembles through dynamic model selection based on recent model
+performance leads to improved performance over baseline by improving the Sharpe and Calmar ra-
+tios. We also evaluate the robustness of our pipeline across different data splits and random seeds
+with good reproducibility of results.
+Key words.
+Robust Machine Learning, Online Learning, Gradient Boosting Decision Trees,
+Deep Learning, Stock Trading, Tabular Data
+1. Introduction. As investors explore new ways to generate profit, machine
+learning (ML) models are increasingly used as part of trading strategies, e.g., to pre-
+dict the future return of stocks or stock portfolios. In particular, deep learning models
+for time-series data, such as Recurrent Neural Networks (RNNs) and Convolutional
+Neural Networks (CNNs), have been applied to the prediction of future stock re-
+turns [1–3]. However, a major challenge for such methods is the highly stochastic,
+non-stationary and non-ergodic nature of financial data, which violates the assump-
+tions of many algorithms. Furthermore, deep learning models are over-parameterised,
+with numbers of parameters orders of magnitude larger than typical sizes of time series
+data. Therefore, deep models can be easily over-fitted to specific patterns in historical
+market data not present in future market data, and the over-fitting worsens with the
+more complicated neural network architectures, such as Long Short Term Memory
+(LSTM) or Transformer networks. In addition, the continuous influx of data, coupled
+with possible regime changes, requires costly updating and retraining of such models.
+Therefore, such methods can lack reproducibility and robustness for the prediction of
+future market data.
+As pointed out in recent reviews [4,5], replication of ML studies is often difficult
+due to several issues, including data leakage [5], program bugs [6], data and code
+usability [7], and model representation and evaluation [4]. These problems and are
+currently hindering the usage of ML in high-risk decision processes, such as healthcare
+and finance. For trading applications in particular, these issues can have critical effects
+on the validity of results. Data leakage, in the form of look-ahead bias or overlap
+∗Funding: This work is supported by the Wellcome Trust under Grant 108908/B/15/Z and by
+the EPSRC under grant EP/N014529/1.
+†Department of Mathematics,
+Imperial College London,
+London SW7 2AZ, U.K (ming-
+[email protected]).
+‡Department
+of
+Mathematics,
+Imperial
+College
+London,
+London
+SW7
+2AZ,
+U.K
+([email protected],).
+1
+arXiv:2301.00790v1  [q-fin.CP]  30 Dec 2022
+
+2
+THOMAS WONG AND MAURICIO BARAHONA
+of training/test sets [8], can inflate in-sample performance with poor performance
+when deployed live. Furthermore, black-box ML models, such as neural networks,
+can lack robustness as they are highly sensitive to small changes in parameters and
+data, thus resulting in volatile predictions. The non-stationary data and the presence
+of regime changes also mean that ML models need to be re-trained with the latest
+financial data, a task that is not only computationally costly but also introduces
+further uncertainty to the trading models. Yet most studies do not consider model
+performance when trained on different segments of historical market data [1–3,9,10].
+Although reinforcement learning (RL) in online learning settings allows ML models
+to adapt to changing environments, deep reinforcement learning models are complex
+and require large computational resources [11]. Indeed, applying RL to stock trading
+is difficult since the complexity of the action space increases exponentially with the
+number of stocks in the portfolio.
+The above issues suggest the need to further develop robust ML pipelines for
+trading applications possibly based on simpler models that can still operate on non-
+stationary, highly stochastic data under regime changes. Here we consider such a
+pipeline based on tabular data, which allows the use of traditional ML models, such
+as Gradient Boosting Decision Trees (GBDT) and other ensemble methods, to predict
+trading stocks and stock indices [12, 13]. This approach also allows the integration
+of additional sources of data, such as sentiment analysis of news articles to improve
+the prediction accuracy of the direction of stock returns [14]. In particular, we find
+that Gradient Boosting models, which are known to be robust to data perturbations,
+outperform neural network models. Finally, we show that improved robustness of ML
+models and adaptation to regime changes can be attained without the use of deep
+reinforcement learning by employing: (i) dynamic feature neutralisation, a simple
+approach that reduces the linear correlation to a subset of features evolving in time,
+and (ii) dynamic model selection of optimal models from an ensemble based on recent
+performance. These approaches robustly improve trading performances by reducing
+volatility and drawdown during adversarial market regimes.
+To exemplify the above issues, we consider a benchmark financial data platform
+that is continuously updated and curated under the Numerai tournament of stock
+portfolio prediction [15]. Numerai is a hedge fund that organises a data science com-
+petition (as of Oct 2022) and provides free, open-source, high quality standardised
+financial data to all participants. As discussed below in more detail, the data set is
+given in the form of pre-processed temporal tabular data and the task is the predic-
+tion of the relative performances of stocks within an evolving trading universe without
+access to the identity of individual stocks. Unlike other financial research papers that
+use proprietary data sets which can be difficult to validate [9,10], this open financial
+data competition allows researchers to replicate findings transparently and allows us
+to focus on establishing ML end-to-end pipelines to achieve consistent profits trad-
+ing a market-neutral portfolio under changing market regimes. Our pipeline, shown
+in Figure 1, is built upon simple, yet robust methodologies that avoid some of the
+problems of over-fitting and high computational cost inherent to deep methods. The
+robustness of the pipeline is enhanced since each step is implemented independently
+avoiding data leakage, which is common in other methods such as neural networks,
+where the pre-processing and the actual model often share data.
+Key ingredients
+are the post-prediction processing and feature engineering steps, which allow easy
+adaptation of models towards regime changes without expensive retraining.
+The paper is organised as follows. Section 2 introduces the Numerai datasets
+used in this paper. Section
+3 describes and discusses the different computational
+
+ROBUST ML MODELS IN FINANCE
+3
+Fig. 1: Schematic of the Machine Learning pipeline. Starting with the Numerai
+data set, we consider feature engineering methods to augment the dataset and train an
+ML model (several are evaluated, including neural networks, but we settle for gradient
+boosting trees) to obtain the raw predictions. These then go through post-prediction
+processing (e.g., dynamic feature neutralisation) to provide normalised predictions,
+which are then combined through model ensembling and dynamic model selection
+methods to output the predictions that are submitted to the Numerai tournament.
+methods, including online cross-validation, feature engineering and the different ML
+models considered and evaluated for the pipeline. Section 4 presents the results from
+our ML pipeline, including the impact of different design choices on the robustness of
+trading performance. Performances of ML models under different market regimes are
+discussed in Section 5. In Section 6, we introduce adaptations to our ML models
+based on online learning approaches, which can work well under regime changes,
+noting that these adaptations are generic and not limited to specific families of ML
+models. Lastly, we discuss the results of the method, open directions and alternatives
+and provide a study of the robustness of our ML pipeline in Section 9.4.
+2. Numerai dataset and prediction task. Financial data are often available
+in the form of time series. These time series can be treated directly using classic meth-
+ods such as ARIMA models [16] and more recently through deep learning methods
+such as Temporal Fusion Transformers [17]. However, such methods are easily over-
+fitted and lead to expensive retraining for financial data, which are inherently affected
+by regime changes and high stochasticity. Alternatively, one can use various feature
+engineering methods to transform these time series into tabular form through a pro-
+cess sometimes called ‘de-trending’ in the financial industry, where the characteristics
+of a financial asset at a particular time point, including features from its history, are
+represented by a single dimensional data row (i.e., a vector). In this representation,
+the time dimension is not considered explicitly, as the state of the system is captured
+through transformed features at each time point and the continuity of the temporal
+dimension is not used. For example, we can summarise the time series of the return
+of a stock with the mean and standard deviation over different look-back periods.
+Grouping these data rows for different financial assets into a table at a given time
+point we obtain a tabular dataset. If the features are informative, this representation
+can be used for prediction tasks at each time point, and allow us to employ robust
+and widely tested ML algorithms that are applicable to tabular data. The Numerai
+competition is based on a curated tabular data set with high-quality features made
+available to the research community.
+Description of the dataset:. The Numerai dataset is a temporal tabular dataset.
+A temporal tabular dataset is a collection of matrices {Xi}1≤i≤T collected over time
+eras 1 to T. Each matrix Xi represents data available at era i with shape Ni × M,
+
+Machine
+Dataset Creation
+Feature
+Post-Prediction
+Model
+Learning
+(Numerai)
+Engineering
+Processing
+Ensemble
+Model Training4
+THOMAS WONG AND MAURICIO BARAHONA
+where Ni is the number of data samples (number of stocks here) in era i and M
+is the number of features describing the samples. Note that the features are fixed
+throughout the eras, in the sense that the same computational formula is used to
+compute the features in each week. On the other hand, the number of data samples
+(stocks) Ni does not have to be constant across time.
+In the Numerai dataset, the matrices Xi contain M obfuscated global stock mar-
+ket features (computed by Numerai) for Ni stocks, which are updated weekly (i.e., the
+eras are in our case weeks). It is important to remark that the dataset is obfuscated,
+i.e., it is not possible for participants to know the identity of stocks or even which
+stocks are present each week. Each data row is indexed by a hash index, known only
+to Numerai, that maps the data rows to the stocks. As a result, it is not possible
+for participants to concatenate different data rows to create a continuous history of
+a stock. The matrix Xi thus provides a snapshot of the market at week i presented
+as an unknown set of stocks described by a common set of features, such that the
+features are computed consistently across all stocks in the week and also computed
+consistently across different weeks.
+The Numerai dataset starts on 2003-01-03 (Era 1). The tabular set has 1191
+features, which are already normalised into 5 equal-sized integer bins, from 0 to 4.
+There are 28 target labels which are derived from stock returns using 14 proprietary
+normalisation methods (nomi, jerome, janet, ben, alan, paul, george, william, arthur,
+thomas, ralph, tyler, victor, waldo ) over 2 forward-looking periods (20 trading days,
+60 trading days). The main target label to evaluate performance is target-nomi-v4-
+20, i.e., forward 20 trading days return obtained by the nomi normalisation method.
+Other targets are named similarly. The target labels are all scaled between 0 to 1,
+where a smaller value represents a lower forward return, and are also grouped into
+bins. For each normalisation method, the number of bins could be different, 5 to 7 bins
+are created for each target with the bin sizes following a Gaussian-like distribution,
+so that most stocks are within the central bin of 0.5 while only a small amount of
+stocks are in the tail bins of 0 and 1. We transform the features and labels so that
+both become zero-mean. (For features, we subtract 2 from the integer bins so that
+the transformed bins are -2,-1,0,1,2. For the target labels, we subtract 0.5 so that the
+new targets are in the range -0.5 to 0.5).
+Prediction task:. The tournament task is to predict the stock rankings each week,
+ordered from lowest to highest expected return. The scoring is based on Spearman’s
+rank correlation of the predicted rankings with the main target label (target-nomi-v4-
+20). Hence there is a single overall score each week regardless of the number of stocks
+to predict each week. Participants are not scored on the accuracy of the ranking of
+each stock individually. Numerai uses the predicted rankings to construct a market-
+neutral portfolio which is traded every week (As of Sep 2022), i.e., the hedge fund
+buys and short-sells the same dollar amount of stocks. Therefore the relative return
+of stocks is more relevant than the absolute return, hence the prediction task is a
+ranking problem instead of a forecast problem.
+3. Methods.
+3.1. Robustness in Machine Learning pipelines. In this paper, we aim to
+design an ML pipeline focusing on its robustness. Table 1 details issues related to
+robustness and reproducibility, as listed in a recent review [5], and how they are
+addressed in this paper. By preventing look-ahead bias and other data leakage issues,
+our pipeline can be robustly applied to live trading setups.
+In addition to avoiding data leakage, the following design choices are used to im-
+
+ROBUST ML MODELS IN FINANCE
+5
+Issues
+affecting
+robust-
+ness of ML algorithms
+How the issue is addressed here
+‘No test set’
+A robust cross-validation scheme is used.
+‘Pre-processing on train-
+ing and test set’
+Numerai features are already standardised; hence
+minimal pre-processing.
+‘Feature
+selection
+on
+training and test set’
+Feature Engineering is applied to each data row in-
+dependently
+‘Duplicates in datasets’
+A unique id for each data row reduces the chance of
+duplicates in dataset
+‘Model uses features that
+are not legitimate’
+Only data provided by Numerai is used to train ML
+models—no extra features from other resources, and
+no cherry-picking of features.
+‘Temporal leakage’
+We use Grouped Time-Series Cross-Validation with
+no overlap between training/validation/test (Fig. 2).
+Feature Engineering is applied to each data row in-
+dependently, i.e., no data leakage between eras.
+‘Non-independence
+be-
+tween training and test’
+Training and test samples are market data at different
+periods without overlap.
+‘Sampling bias in test dis-
+tribution’
+The stocks trading each week are decided by Numerai
+based on operational and risk considerations.
+Table 1: Data analysis design. Some common issues regarding data leakage in
+machine learning research [4,5] and how these issues are dealt with in this study.
+prove the robustness and reliability of the results. Firstly, the impact of random seeds
+is reduced by reporting results from average predictions over 10 different random seeds
+for each machine learning method. Secondly, the metrics used for model evaluation
+are the same as in the Numerai tournament to avoid researcher bias in discounting
+unfavourable results. Finally, cross-validation is independent of the effects of random
+seeds and other human selection, thus reducing the chance of overfitting models to a
+particular data split.
+For datasets that involve time, standard cross-validation schemes cannot be used
+directly, as a random split of data eras could lead to the training set including data that
+appears later in time than the validation and test sets, hence introducing look-ahead
+bias. To avoid this problem, we use grouped time-series cross-validation, which splits
+data eras according to their chronological order (Figure 2). Note that for financial
+datasets, the target labels often involve future asset returns and are reported with a
+lag. Therefore, we add a gap between the training and validation sets and similarly
+between the validation and test sets.
+3.2. Feature Engineering methods. Feature engineering is a crucial step in
+enhancing the power of tabular methods for the analysis of time series data. Therefore,
+we evaluate different feature engineering methods that can be applied to temporal
+tabular data sets with numerical features, as the Numerai data set only contains
+normalised numerical features.
+New features can be created by applying polynomial transformations such as
+multiplication and addition to the original features. Here we create new features by
+multiplying two features that can be thought of as modelling the joint distribution
+
+6
+THOMAS WONG AND MAURICIO BARAHONA
+Training Data
+V alidation Data
+Test Data
+Time
+gap
+gap
+Fig. 2: Illustration of data split using grouped time-series cross-validation
+of feature pairs. When the number of features is large, we draw a random subset of
+feature pairs to create new features to alleviate the computational cost. Note that
+the computation of these features can be done in parallel for data from each era.
+A simple way of data augmentation is to add randomness to the feature matrix
+with different dropout methods, which are used extensively to reduce over-fitting of
+neural network models [18]. Here we apply dropout by multiplying the original data
+with a Boolean mask so that some numerical features are set to zero. The dropout is
+characterised by its sparsity level (how many features are set to zero) and its sparsity
+structure (how to choose the features set to zero). Since our tabular dataset has no
+local spatial structure, we use a random Boolean matrix with uniform probability.
+This encourages the machine learning methods to learn multiple feature relationships
+and reduces reliance on a small set of important features.
+For our dataset, we first augment the feature matrix by creating additional fea-
+tures obtained by multiplying feature pairs, and then apply dropout with a random
+Boolean mask on the augmented feature matrix. A grid search is used to find optimal
+hyper-parameters for the feature engineering methods, in particular the number of
+feature products and the sparsity level of dropout.
+3.3. Machine Learning algorithms for tabular datasets. Numerous ma-
+chine learning models have been proposed for tabular datasets, and different bench-
+marking studies have shown conflicting views on their performance [18, 19].
+The
+biggest disagreement in the literature is whether gradient-boosting decision trees or
+neural networks are superior in regression and classification tasks of tabular datasets.
+Whereas one paper claims gradient boosting models (XGBoost) outperformed deep
+learning models in 8 out of 11 datasets and none of the deep learning models consis-
+tently outperform others [19], another paper suggests that well-tuned multi-layer per-
+ceptron (MLP) models with regularisation can outperform different gradient boosting
+models such as XGBoost and CatBoost [18]. Both these studies, however, share the
+same view that neural networks with complicated designs, such as attention layers and
+other transformer layers, tend to generalise poorly with a strong drop in performance
+when applied to data sets beyond their original study.
+Importantly, the Numerai
+data set is different from the data sets in the above benchmarking studies in that it
+is growing instead of fixed. Hence the data distribution varies across time periods
+due to market regime effects, and we do not have a homogeneous distribution across
+cross-validation splits. With such a different problem setup, it is thus not possible to
+use the above benchmarking studies to guide our choice of ML method.
+In this study, we benchmark a wide range of machine learning models, including
+different variants of gradient-boosting decision tree models and different neural net-
+work models. The choice of ML models is based on the popularity of usage in data
+
+ROBUST ML MODELS IN FINANCE
+7
+science competitions and code quality, as one of our aims, is the replicability of results.
+We train all machine learning models with a single GPU, the standard setup for most
+participants in data science competitions. Some brief details of the ML models used
+are provided in the following.
+Gradient Boosting Decision Trees. Boosting can be seen as a generalisation of
+generalized additive models (GAM) where the additive components of smooth para-
+metric functions can be replaced by any weak learners such as decision trees [20].
+Historically, various boosting algorithms have been proposed for different loss func-
+tions. For example, AdaBoost [21] was proposed for binary classification problems
+with exponential loss, whereas Gradient Boosting was first proposed by Friedman in
+2001 [22] for any smooth loss functions. Algorithm 9.1 in the SI outlines the iterative
+update equations of gradient boosting.
+Of the various implementations of gradient boosting decision (GBDT) trees in
+Python, we use LightGBM [23] in this paper. CatBoost [24] is not used here as the
+Numerai dataset has no categorical features. XGBoost [25] is not used due to slower
+computation and more memory consumption. Algorithm 9.2 in the SI shows how the
+gradient boosting algorithm is implemented with decision trees being the weak learners
+in LightGBM. LightGBM implements GBDT models with several computational and
+numerical improvements from XGBoost and other implementations. In addition to
+traditional gradient boosting decision trees (LightGBM-gbdt), we consider two other
+implementations of GBDT models:
+• Dropouts meet Multiple Additive Regression Trees (LightGBM-dart) ignores a
+portion of trees when computing the gradient for subsequent trees [26], thus
+avoiding over-specialisation where the later learned trees can only affect a
+few data instances. This reduces the sensitivity of models towards decisions
+made by the first few trees.
+• Gradient-based One-Side Sampling (LightGBM-goss) reduces the number of
+data instances used to build each tree: it keeps data instances with large
+absolute gradients and randomly samples a subset of data with small absolute
+gradients.
+The approximation error of the gradient using LightGBM-goss
+converges to the standard method when the number of data is large, and it
+outperforms other data sampling (e.g., uniform sampling) in most cases.
+For all LightGBM models, we use mean squared error (L2 loss) as the loss function
+for the regression problems. The number of gradients boosting trees and learning rate
+is optimised by hyper-parameter searches. To prevent the over-fitting of trees, the
+maximum depth and number of leaves in each tree and the minimal number of data
+samples in the leaves are tuned for each model. L1 and L2 regularisation are also
+applied. Data and feature sub-sampling are used to reduce similarities between trees:
+before building each tree, a random part of data is selected without re-sampling and
+a random subset of features is chosen to build the tree. For LightGBM-dart models,
+both the probability to apply dropout during the tree-building process and the portion
+of trees to be dropped out are tuned. Early stopping is applied using the validation
+dataset for LightGBM-gbdt models to further prevent the over-fitting of models.
+Neural Networks. The most basic architecture of neural networks, multi-layer
+perceptron (MLP), failed to outperform gradient boosting models in many benchmark
+studies of tabular datasets [19].
+Recently, more complex network architectures have been proposed for tabular
+data sets, as surveyed in
+[27] These new architectures can be classified into two
+major groups:
+• Hybrid models that combine neural networks with other traditional ML meth-
+
+8
+THOMAS WONG AND MAURICIO BARAHONA
+ods, e.g., decision trees. Neural Oblivious Decision Ensembles (NODE) [28] is
+a generalisation of gradient boosting models into differentiable deep decision
+trees allowing end-to-end training with gradient descent optimisers such as
+PyTorch [29]. DeepGBM [30] combines two neural networks, CatNN to han-
+dle sparse categorical features and GBDT2NN to distil tree structures from
+a pre-trained GBDT model to handle numerical features. A major limitation
+of these models is the large memory consumption, which makes them run out
+of memory on the NVIDIA 3080ti GPU. Therefore we do not use them in our
+benchmark analysis.
+• Transformer-based models that use deep attention mechanisms to model com-
+plex feature relationships. TabNet [31] uses sequential attention to perform
+instance-wise feature selection at each decision step, enabling interpretability
+and better learning. AutoInt [32] maps and models feature interactions in a
+low-dimensional space with a multi-head self-attentive neural network with
+residual connections. AutoInt runs out of memory on a single GPU and is
+thus not used in our benchmark. Tabnet also had similar memory issues,
+hence we down-sampled the data by keeping every fifth week of data (i.e.,
+20% of the original data) for the training/validation periods, so that Tab-
+net could be trained on the single GPU used in this study. Our aim is to
+compare performance under modest computational resources attainable by a
+wide class of users.
+In summary, our benchmark analysis includes two NN models: MLP and TabNet
+implemented in PyTorch.
+We use Adam [33] as the gradient optimiser, with the
+learning rate automatically determined by PyTorch. We use mean squared error (L2
+loss) for the regression problems.
+4. Evaluation of Machine Learning methods for the Numerai temporal
+tabular data set. In this section, we study different ML methods applied to the
+Numerai temporal tabular data set for the prediction of stock rankings aimed at
+market-neutral stock portfolios.
+Data Split. We use the latest version (v4) of the Numerai dataset. The training
+period is fixed between 2003-01-03 (Era 1) to 2012-07-27 (Era 500), and the validation
+period is fixed between 2012-12-21 (Era 521) and 2014-11-14 (Era 620). The test
+period starts on 2015-05-15 (Era 646) and ends on 2022-09-23 (Era 1030). We apply a
+1-year gap between training and validation periods to reduce the effect of recency bias
+so that the performance of the validation period will better reflect future performance.
+The gap between the validation period and test period is set to 26 weeks to allow for
+sufficient time to deploy trained machine learning models.
+Evaluation of performance. For each configuration of each ML method, we aver-
+age over the predictions of different targets before scoring. The predictions are scored
+in each era by calculating the correlation (Corr) between the rank-normalised pre-
+dictions and the actual (binned) stock ranking. The mean and standard deviation
+(volatility) of Corr are reported for both the validation and test periods. To measure
+the downside risk of the model, we also compute the Maximum Drawdown, defined
+as the largest drop suffered by an investor starting at any time during the valida-
+tion/test period. As summary measures, we compute two standard ratios: (i) the
+Sharpe ratio, defined as the ratio of the mean and standard deviation of Corr; and
+(ii) the Calmar ratio, defined as the ratio of mean Corr over Maximum Drawdown.
+Good performance is characterised by large values of both of these ratios,
+
+ROBUST ML MODELS IN FINANCE
+9
+Model Training. We use Optuna [34] to perform the hyper-parameter search (see
+section 9.3 in Supplementary Information) and select the hyper-parameters with the
+highest Sharpe ratio for the main target (target-nomi-v4-20) in the validation period.
+The optimised hyper-parameters for each ML method are so fixed, and we then train
+10 models, starting the algorithms from 10 different random seeds. We report the
+average prediction from these 10 models for evaluation.
+Baseline Model. As a baseline, we consider a factor momentum model which is
+created by linear combinations of signed features, where the sign of each feature is
+determined by the sign of the 52-week moving average of correlations of that feature
+with the target. This simple baseline linear model is then compared with the ML
+models, which can capture non-linearity in the data.
+Comparative results of the ML algorithms and Feature Engineering. Table 2 shows
+the performance on the validation and test sets for the different algorithms.
+We
+concentrate on methods that achieve the highest mean Corr, and Sharpe and Calmar
+ratios.
+(a) Performance over the validation period (2012-12-21 to 2014-11-14)
+Method
+Mean
+Volatility
+Max Draw
+Sharpe
+Calmar
+Factor Momentum (baseline)
+0.0229
+0.0170
+0.0691
+1.3495
+0.3314
+MLP with FE
+0.0423
+0.0208
+0.0241
+2.0338
+1.7552
+MLP without FE
+0.0443
+0.0201
+0.0065
+2.2058
+6.8154
+TabNet without FE
+0.0362
+0.0189
+0.0199
+1.9125
+1.8191
+LightGBM-gbdt with FE
+0.0483
+0.0229
+0.0307
+2.1144
+1.5733
+LightGBM-gbdt without FE
+0.0500
+0.0224
+0.0235
+2.2335
+2.1277
+LightGBM-dart with FE
+0.0496
+0.0223
+0.0215
+2.2274
+2.3070
+LightGBM-dart without FE
+0.0475
+0.0199
+0.0079
+2.3883
+6.0127
+LightGBM-goss with FE
+0.0288
+0.0219
+0.0687
+1.3136
+0.4192
+LightGBM-goss without FE
+0.0302
+0.0234
+0.0877
+1.2877
+0.3444
+(b) Performance over the test period (2015-05-15 to 2022-09-23)
+Method
+Mean
+Volatility
+Max Draw
+Sharpe
+Calmar
+Factor Momentum (baseline)
+0.0080
+0.0275
+0.7877
+0.2923
+0.0102
+MLP with FE
+0.0237
+0.0330
+0.2912
+0.7189
+0.0814
+MLP without FE
+0.0258
+0.0289
+0.1668
+0.8931
+0.1547
+TabNet without FE
+0.0161
+0.0296
+0.5811
+0.5431
+0.0277
+LightGBM-gbdt with FE
+0.0253
+0.0327
+0.3064
+0.7731
+0.0826
+LightGBM-gbdt without FE
+0.0262
+0.0321
+0.2378
+0.8140
+0.1102
+LightGBM-dart with FE
+0.0265
+0.0319
+0.2151
+0.8313
+0.1232
+LightGBM-dart without FE
+0.0278
+0.0284
+0.1622
+0.9791
+0.1714
+LightGBM-goss with FE
+0.0169
+0.0297
+0.5539
+0.5695
+0.0305
+LightGBM-goss without FE
+0.0156
+0.0318
+0.7528
+0.4896
+0.0207
+Table 2: Performance of different machine learning methods with and without feature
+engineering on the Numerai dataset for (a) validation period and (b) test period.
+The three top methods according to Sharpe ratio and Maximum Drawdown over
+the validation period are shown in italics in (a). The top method according to the
+Sharpe ratio and Maximum Drawdown over the test period is shown in boldface in
+(b). For TabNet, the pipeline with feature engineering cannot be run due to memory
+constraints.
+
+10
+THOMAS WONG AND MAURICIO BARAHONA
+Firstly, we see that almost all ML models performed substantially better than the
+factor momentum model (baseline), in both validation and test periods. Whereas the
+factor momentum model relies on linear relationships, the capability of ML models
+to learn non-linear relationships, in addition to linear ones, adds to their robustness
+and improved performance under different, often volatile, market regimes.
+Secondly, we observe that Feature Engineering does not improve the performance
+of ML models. Although, in principle, Feature Engineering allows GBDT-based meth-
+ods to model feature interactions more easily, our results suggest that these interac-
+tions are over-fitted during the training process. For neural network-based models,
+feature engineering is not strictly necessary, as dropout is already embedded in net-
+work architectures.
+Thirdly, we note that all ML models scored better in the validation period than
+the test period. This is expected, as it is well known that the performance of trading
+models deteriorates over time due to overcrowding and regime changes (a phenomenon
+known as alpha decay). Models that are over-fitted to recent training data will ex-
+perience greater alpha decay than properly regularised models.
+To select the ML
+method, we consider the top models according to the Sharpe and Calmar ratios over
+the validation period: a high Sharpe ratio ensures the model has good overall perfor-
+mance, whereas a high Calmar ratio ensures good performance against the worst-case
+scenario, thus capturing the tail risks of the trading model.
+Indeed, we find that
+LightGBM-dart without feature engineering generalises well to the test period further
+into the future.
+Finally, we note that LightGBM-gbdt has better generalisation to the test period
+than neural network-based models (TabNet), suggesting over-fitting in these complex
+deep NN models. This indicates that although over-parameterised models can learn
+non-linear relationships in temporal tabular data sets, these relationships may be
+difficult to generalise under non-stationary data environments. On the other hand, our
+results suggest that, despite their relative simplicity, gradient Boosting models capture
+non-linearity in a more robust and controlled manner, with early trees capturing linear
+relationships and non-linear relationships captured by the later trees, thus reducing
+the risk of catastrophic forgetting [35].
+In summary, we find that the best performing model in our set is LightGBM-
+dart without feature engineering. In the rest of the paper, we will use this model to
+illustrate how the pipeline can be further modified with online learning to account
+for regime effects. To demonstrate the robustness of our pipeline, and how it can
+be applied to improve the performance of any ML model, we will also report the
+performance of two other models: a similar GBDT model (LightGBM-gbdt without
+feature engineering) and a neural network model (MLP without feature engineering).
+5. Dealing with regime effects in the ML pipeline. Financial data are
+heavily influenced by regime changes.
+Growth (‘Bull’) markets are characterised
+by low volatility and positive expected return, whereas high volatility and negative
+expected returns are characteristic of adverse (‘bear’) markets.
+Switches between
+regimes can be triggered by externalities, such as pandemics, economic recessions,
+etc. From the perspective of the Numerai data set, such regime effects affect model
+performance. Volatility is detrimental to long-term performance due to the negative
+compounding of investment losses, a phenomenon known as ‘volatility tax’. Given
+that hedge funds are leveraged, we consider consistent models with reasonably good
+performance under different market regimes, rather than models that have excellent
+performance in one market regime but fail in others.
+
+ROBUST ML MODELS IN FINANCE
+11
+In this section, we focus on how to deal with regime effects when using ML
+models for financial tabular temporal data sets.
+Specifically, we consider feature
+neutralisation, and reducing the dependence on the initial trees in gradient boosting
+models.
+Classification into high and low volatility regimes. To classify the financial market
+into regimes, we consider an intrinsic measure derived directly from the Numerai
+dataset. In particular, we first compute the Numerai Market Index (NMI), i.e., the
+weekly performance of the baseline (linear) factor momentum portfolio, and we then
+calculate the Numerai Realised Volatility Index (NRVIX), defined as the standard
+deviation of NMI rolling over 52 weeks (Fig. 3). The eras are then classified into high
+and low volatility, based on a threshold of NRVIX=0.025, the mean over the first 7
+years of data (2003-01-03 to 2010-02-26). According to this intrinsic characterisation,
+low volatility regimes have stable linear relationships of features to stock returns, often
+associated with a good performance by ML models. On the other hand, high Volatility
+regimes correspond to unstable linear relationships of features to stock returns leading
+to poor model performance. Figure 3 shows that high/low NRVIX regimes are well
+aligned with macroeconomic events: high volatility regimes include the financial crisis
+(2007-2009), the Euro crisis (2011-2012), and the Covid pandemic (2020), whereas low
+volatility regimes correspond to benign market conditions with no significant macro
+event risks, during which the factor momentum baseline portfolio had good returns.
+2005-01-28
+2008-11-28
+2012-09-28
+2016-07-29
+2020-05-29
+0.050
+0.025
+0.000
+0.025
+0.050
+0.075
+0.100
+(a) NMI
+2005-01-28
+2008-11-28
+2012-09-28
+2016-07-29
+2020-05-29
+0.015
+0.020
+0.025
+0.030
+0.035
+0.040
+(b) NRVIX
+Fig. 3: High and low volatility regimes in the Numerai data. (a) Numerai
+Market Index (NMI) for the period between 2005-01-28 (Era 109) and 2022-09-23
+(Era 1016); (b) the computed Numerai Realised Volatility Index (NRVIX) used to
+identify the high and low volatility regimes. The high volatility regime refers to weeks
+where NRVIX is higher than 0.25 and the low volatility regime refers to weeks where
+NRVIX is lower than 0.25.
+5.1. Feature Neutralisation. Feature neutralisation is the general term to
+denote the elimination of the effect of particular features in the model, thus reducing
+the risk of over-relying on certain individual features. Because the predictive ability
+of individual features is highly dependent on market regimes, this can lead to long
+periods of drawdown when there is a regime change. It is therefore undesirable to
+have ML models that could have heavy (linear)-dependence on certain features.
+We start by evaluating here the feature neutralisation suggested by the Numerai
+tournament. Numerai recommends that participants reduce model exposure to 420
+
+12
+THOMAS WONG AND MAURICIO BARAHONA
+‘risky features’ (out of the 1191 features).
+This list of risky features can be used
+for feature neutralisation by subtracting the linear correlation using the formula for
+Feature Neutral Correlation (FNC). Specifically, given a week of data with n stocks,
+let X ∈ Rn×420 be the matrix of risky features and y ∈ Rn the predicted rankings ob-
+tained from a model. For a given neutralisation strength β, 0 ≤ β ≤ 1, the neutralised
+predicted ranking ˆy is calculated as ˆy = y − β XX†y, where X† is the pseudo-inverse
+of X. FNC is then calculated as the correlation of the neutralised predicted rankings.
+Using this procedure, we reduce the linear dependencies of predictions on features.
+(a) LightGBM-dart without FE
+Regime
+Feature Neutral
+Mean
+Volatility
+Max Draw
+Sharpe
+Calmar
+All
+Yes
+0.0215
+0.0182
+0.1153
+1.1806
+0.1865
+No
+0.0278
+0.0284
+0.1622
+0.9791
+0.1714
+High Vol
+Yes
+0.0227
+0.0163
+0.0223
+1.3888
+1.0179
+No
+0.0314
+0.0251
+0.0657
+1.2510
+0.4779
+Low Vol
+Yes
+0.0206
+0.0195
+0.1153
+1.0576
+0.1787
+No
+0.0252
+0.0305
+0.1622
+0.8257
+0.1554
+(b) LightGBM-gbdt without FE
+Regime
+Feature Neutral
+Mean
+Volatility
+Max Draw
+Sharpe
+Calmar
+All
+Yes
+0.0204
+0.0211
+0.1998
+0.9665
+0.1021
+No
+0.0262
+0.0321
+0.2378
+0.8140
+0.1102
+High Vol
+Yes
+0.0217
+0.0198
+0.0364
+1.0953
+0.5962
+No
+0.0308
+0.0293
+0.1123
+1.0497
+0.2743
+Low Vol
+Yes
+0.0194
+0.0220
+0.1998
+0.8820
+0.0971
+No
+0.0227
+0.0338
+0.2378
+0.6727
+0.0955
+(c) MLP without FE
+Regime
+Feature Neutral
+Mean
+Volatility
+Max Draw
+Sharpe
+Calmar
+All
+Yes
+0.0179
+0.0203
+0.2606
+0.8798
+0.0687
+No
+0.0258
+0.0289
+0.1668
+0.8931
+0.1547
+High Vol
+Yes
+0.0196
+0.0193
+0.0326
+1.0191
+0.6012
+No
+0.0298
+0.0276
+0.1247
+1.0802
+0.2390
+Low Vol
+Yes
+0.0165
+0.0210
+0.2606
+0.7875
+0.0633
+No
+0.0228
+0.0296
+0.1668
+0.7721
+0.1367
+Table 3: The effect of feature neutralisation. Performance of different ML meth-
+ods on the Numerai v4 dataset over the test period (2014-06-27 to 2022-09-23) with
+and without feature neutralisation under different market regimes: the whole test
+period (all), high volatility regime (high-vol), and low volatility regime (low-vol).
+In Table 3, we compare the performance of the LightGBM-dart, LightGBM-gbdt
+and MLP with and without feature neutralisation under different market regimes (all,
+high volatility, low volatility). The neutralisation strength β is set to 1 throughout.
+We find that the variance of models is consistently reduced by feature neutralisa-
+tion, suggesting an overall reduction of risk. Further, feature neutralisation improves
+the Sharpe and Calmar ratios of LightGBM-dart and LightGBM-gbdt under different
+market regimes, but does not improve the performance of MLP models.
+Importantly, this default feature neutralisation procedure suggested by Numerai
+
+ROBUST ML MODELS IN FINANCE
+13
+is not optimal, and we will show in Section 6 how online learning approaches can be
+used to improve the procedure.
+5.2. Pruning initial trees in Gradient Boosting models. For gradient-
+boosting tree models, we also consider a specific procedure consisting of pruning
+initial trees during prediction to reduce feature dependencies. Specifically, we perform
+a grid search over the number of initial trees to be pruned off in the trained LightGBM
+models, and we cap the number of trees to be pruned to not more than half of the
+trees to ensure our models do not degenerate.
+Table 4 compares the performance of LightGBM-dart and LightGBM-gbdt models
+pruning different numbers of initial trees before feature neutralisation. Pruning ini-
+tial trees during prediction improves the Sharpe and Calmar ratios of both LightGBM
+models, but LightGBM-gbdt models see a bigger improvement than LightGBM-dart
+models. This is expected as LightGBM-dart models already employ a similar fun-
+damental idea during training, i.e., the trained trees in LightGBM-dart models are
+already optimised. Our numerics also suggest that there is a limit of trees to be pruned
+such that there is little improvement in model performance once over a threshold of
+around 100-250 trees.
+(a) LightGBM-dart without FE
+Prune Trees
+Mean
+Volatility
+Max Draw
+Sharpe
+Calmar
+0
+0.0278
+0.0284
+0.1622
+0.9791
+0.1714
+100
+0.0272
+0.0264
+0.1384
+1.0293
+0.1965
+250
+0.0264
+0.0255
+0.1299
+1.0336
+0.2032
+500
+0.0249
+0.0238
+0.1166
+1.0459
+0.2136
+(b) LightGBM-gbdt without FE
+Prune Trees
+Mean
+Volatility
+Max Draw
+Sharpe
+Calmar
+0
+0.0262
+0.0321
+0.2378
+0.8140
+0.1102
+100
+0.0265
+0.0291
+0.1835
+0.9106
+0.1444
+250
+0.0253
+0.0259
+0.1490
+0.9769
+0.1698
+500
+0.0253
+0.0259
+0.1490
+0.9765
+0.1698
+Table 4: The effect of tree pruning. Strategy Performance of different LightGBM
+models in the test period (2014-06-27 to 2022-09-23) when pruning different numbers
+of initial trees.
+5.3. Joint effect of feature neutralisation and tree pruning. We then
+considered the joint effect of feature neutralisation and pruning initial trees. Table
+5 compares the performance (FNC) of LightGBM-dart and LightGBM-gbdt models
+pruning a different number of initial trees after feature neutralisation. The effect of
+pruning on model performance for both LightGBM models after feature neutralisation
+is at best modest. As FNC is a measure of the effect of non-linear relationships, this
+suggests that in gradient boosting models, early weak learners (trees) mostly capture
+linear relationships whereas most of the non-linear relationships are captured in the
+later weak learners (trees). Therefore, pruning initial trees can be thought of as a
+model-dependent feature neutralisation method.
+
+14
+THOMAS WONG AND MAURICIO BARAHONA
+(a) LightGBM-dart without FE with Feature Neutralisation
+Prune Trees
+Mean
+Volatility
+Max Draw
+Sharpe
+Calmar
+0
+0.0215
+0.0182
+0.1153
+1.1806
+0.1865
+100
+0.0208
+0.0174
+0.1079
+1.1998
+0.1928
+250
+0.0200
+0.0168
+0.1103
+1.1918
+0.1813
+500
+0.0183
+0.0156
+0.1044
+1.1748
+0.1753
+(b) LightGBM-gbdt without FE with Feature Neutralisation
+Prune Trees
+Mean
+Volatility
+Max Draw
+Sharpe
+Calmar
+0
+0.0204
+0.0211
+0.1998
+0.9665
+0.1021
+100
+0.0206
+0.0200
+0.1912
+1.0293
+0.1077
+250
+0.0194
+0.0188
+0.2058
+1.0307
+0.0943
+500
+0.0193
+0.0188
+0.2063
+1.0301
+0.0936
+Table 5: The joint effect of feature neutralisation and tree pruning. Perfor-
+mance of different LightGBM models after neutralisation in the test period (2014-06-
+27 to 2022-09-23) when pruning different numbers of initial trees.
+6. Online Learning to improve post-prediction processing and model
+ensembles. As a further improvement to the ML pipeline, we apply online learning
+approaches to both feature neutralisation and model ensembles to produce improved
+versions called dynamic feature neutralisation and dynamic model selection. Dynamic
+feature neutralisation acts by applying statistical rules to determine subsets of fea-
+tures to neutralise predictions in each era. Dynamic model selection acts by updating
+regularly the choice of model(s) from a model ensemble based on recent model per-
+formance.
+The aim of online learning is to derive an optimal procedure to select ML mod-
+els and parameters as data arrives continuously. In a continuous time setting, the
+Hamilton-Jacobi-Bellman (HJB) equation is solved to find the optimal determinis-
+tic control for the decision problem [36]. The discrete-time equivalent, the Bellman
+equation, is used in reinforcement learning to derive optimal policies of agents [37].
+For the Numerai tournament, we consider online learning in the discrete-time
+setting, since data and predictions are required once per week.
+For each week t
+(1 ≤ t ≤ T), we have a state (data) process Xt, which contains all the infor-
+mation we know about the environment (Numerai datasets and trained ML model
+parameters) up to week t. Our task is then to derive a deterministic decision pro-
+cess Dt(βt) described by parameters βt := βt(Xt), subject to the objective function
+VT = maxDt
+�T
+t=1 q(Xt, Dt), where q(Xt, Dt) represents the utility at time instant t
+given the data and decision process.
+(Deep) Reinforcement learning algorithms are commonly used to solve online
+learning problems. However, they are not used here due to the following reasons:
+1. Limited data: Available data is not enough to train reinforcement learning
+models, such as Deep Q Networks (DQN) [38], Proximal Policy Optimisation
+(PPO) [39] and Soft Actor-Critic (SAC) [40]). Generating a large number of
+samples is difficult here since we must avoid look-ahead bias.
+2. Expanding action space: Most implementations of reinforcement learning al-
+
+ROBUST ML MODELS IN FINANCE
+15
+gorithms, as found in Ray-RLlib [11], cannot adapt naturally to an expanding
+action space. For the dynamic model selection problem, the number of po-
+tential models is unbounded, as newer models can be trained with the latest
+data available and added to the candidate list. Rule-based models, on the
+other hand, can handle the issue of expanding action space easily.
+3. Actions have negligible impact on environment: Highly successful reinforce-
+ment learning algorithms are usually targetted at robotics and Atari games
+[41], where agent actions can modify the environment. However, for the trad-
+ing models considered here, the trading activities are assumed to have zero or
+negligible market impact, and reinforcement learning algorithms thus reduce
+to an online learning prediction problem.
+4. Large, correlated feature sets for neutralisation: To improve feature neutral-
+isation, we use a different subset of features to neutralise predictions in each
+era. Yet the size of the set of risky features (420 features) makes it computa-
+tionally infeasible to learn feature subsets through supervised ML methods or
+reinforcement learning, as it is difficult to construct a robust reward function
+for correlated features. Heuristic methods thus provide suitable alternatives
+to learn interpretable and robust feature neutralisation schemes.
+5. Model ensembling can be simplified in the Numerai problem: The model en-
+semble step of the pipeline assigns portfolio weightings to different ML mod-
+els. Although similar to a multi-armed bandit problem, in our problem ex-
+ploration is not needed for the agent to learn the distribution of rewards from
+different choices since the performance of all ML models up to the decision
+time are known to the Numerai tournament participant. Hence there is less
+need to employ trial-and-error as in multi-armed bandit algorithms.
+As a consequence, instead of reinforcement learning algorithms, we use heuristics
+which are shown to be effective in improving the robustness of the ML pipeline. These
+heuristics can be interpreted as strong priors in Bayesian learning that greatly simplify
+our problem.
+6.1. Dynamic Feature neutralisation. In Section 5, the subset of ‘risky fea-
+tures’ that are used to neutralise ML models is fixed throughout the whole validation
+and test periods. As market conditions are variable, we suggest choosing a different
+set of features to neutralise in each era to adapt our ML models without the need for
+expensive re-training of models. Specifically, each week we update the set of features
+to neutralise based on rolling statistical properties of features, as follows. For each
+feature in the dataset, we calculate the correlation of the feature with the target (fea-
+ture Corr) and then compute lagged moving average statistics, with a lag of 6 weeks
+to account for the lagged reporting of future performance. The look-back window to
+compute statistical properties of feature Corr is 52 weeks. We consider 5 different
+criteria to select the subset of features to be neutralised:
+1. ‘Fixed’: 420 features provided by the portfolio optimiser in Numerai, as in
+Section 5 above
+2. ‘Low Mean’: 420 features that are least correlated to the target recently
+3. ‘High Mean’: 420 features that are most correlated to the target recently
+4. ‘Low Volatility’: 420 features that have correlations least volatile recently
+5. ‘High Volatility’: 420 features that have correlations most volatile recently
+Table 6 compares the performance obtained by the different dynamic feature
+neutralisation schemes on LightGBM-dart, LightGBM-gbdt and MLP models. All
+Dynamic Feature Neutralisation methods perform better than using a fixed set of
+
+16
+THOMAS WONG AND MAURICIO BARAHONA
+features but the ‘Low Mean’ neutralisation method has the best Sharpe and Calmar
+ratios for all ML models, followed by neutralisation of ‘High Volatility’ features. The
+worse performance of ‘High Mean’ and ’Low Volatility’ neutralisations suggests that a
+large part of the model risks can be attributed to recently underperforming and high
+volatility features.
+(a) LightGBM-dart without FE
+Dynamic Feature Neutral.
+Mean
+Volatility
+Max Draw
+Sharpe
+Calmar
+Fixed
+0.0215
+0.0182
+0.1153
+1.1806
+0.1865
+Low Mean
+0.0240
+0.0164
+0.0350
+1.4595
+0.6857
+High Mean
+0.0218
+0.0185
+0.0986
+1.1783
+0.2211
+Low Vol
+0.0244
+0.0200
+0.0538
+1.2220
+0.4535
+High Vol
+0.0226
+0.0169
+0.0341
+1.3411
+0.6628
+(b) LightGBM-gbdt without FE
+Dynamic Feature Neutral.
+Mean
+Volatility
+Max Draw
+Sharpe
+Calmar
+Fixed
+0.0204
+0.0211
+0.1998
+0.9665
+0.1021
+Low Mean
+0.0234
+0.0184
+0.0495
+1.2737
+0.4727
+High Mean
+0.0199
+0.0212
+0.1469
+0.9381
+0.1355
+Low Vol
+0.0224
+0.0228
+0.1852
+0.9797
+0.1210
+High Vol
+0.0182
+0.1633
+0.0487
+1.1986
+0.4476
+(c) MLP without FE
+Dynamic Feature Neutral.
+Mean
+Volatility
+Max Draw
+Sharpe
+Calmar
+Fixed
+0.0179
+0.0203
+0.2606
+0.8798
+0.0687
+Low Mean
+0.0211
+0.0185
+0.0806
+1.1387
+0.2618
+High Mean
+0.0186
+0.0201
+0.1283
+0.9256
+0.1450
+Low Vol
+0.0206
+0.0215
+0.0878
+0.9598
+0.2346
+High Vol
+0.0191
+0.0172
+0.0730
+1.1150
+0.2616
+Table 6: The effect of Dynamic Feature Neutralisation. Performance of differ-
+ent ML models in the test period (2014-06-27 to 2022-09-23) with different dynamic
+feature neutralisation methods
+Next we compared the performance obtained by different dynamic feature neu-
+tralisations under different market regimes, as defined in Section
+5.
+The results
+can be found in Tables 9 and 8 in the Supplementary Information. Neutralisation
+by ‘Low Mean’ performs better than Neutralisation by ‘High Mean’ in low volatility
+regimes, but not in high volatility regimes. Under high volatility regimes, neutralisa-
+tion by ‘Low Volatility’ features in the models performs better than neutralisation by
+‘Low Mean’. Under a low volatility regime, neutralisation by ‘Low Mean’ performs
+significantly better than others.
+Based on the above, we make the following observations: In a low volatility
+regime, factors that are performing well recently continue to do so in the near future
+as the feature correlation structure is more stable in low volatility regime.
+This
+works until there is a regime change. In a high volatility regime, the ML models
+after neutralisation of ‘Low Volatility’ features have a much higher Mean Corr than
+models obtained by other neutralisation methods. ‘Low Volatility’ represents features
+
+ROBUST ML MODELS IN FINANCE
+17
+that have a low variance, and stable performance in the last 52 weeks. During volatile
+regimes, these features will underperform. Models that neutralise these features can
+then outperform when there is market stress.
+6.2. Dynamic model selection. In practice, it is not possible to know the best
+dynamic feature engineering methods in advance. Therefore, we propose an online
+learning procedure to select the dynamic feature engineering method during the test
+period consisting of two steps. The first step is to have a warm-up period to collect
+data on model performances, during which all 5 feature neutralisation methods (fixed,
+low mean, high mean, low vol, high vol) have equal weighting. The second step is to
+allocate weights to the optimal model based on recent performance according to the
+following criteria:
+• ‘Average’: Using all five feature neutralisation methods with equal weighting
+• ‘Momentum’: Using the feature neutralisation method with the highest Mean
+Corr in the last 52 weeks
+• ‘Sharpe’: Using the feature neutralisation method with the highest Sharpe
+Ratio in the last 52 weeks
+• ‘Calmar’: Using the feature neutralisation method with the highest Calmar
+ratio in the last 52 weeks
+In Table 7, we use these criteria to select the optimal dynamic feature engineering
+method based on recent performance. As above, a lag of 6 weeks is applied to account
+for data delays.
+The online learning procedure can thus select the optimal dynamic feature engi-
+neering method to outperform the ‘Average’ selection in most cases. For all three ML
+models (LightGBM-dart/LightGBM-gbdt/MLP), the ‘Momentum’ selection method
+has higher mean Corr and Calmar ratio than the‘Average’ (baseline) and ‘Sharpe’
+methods. This shows that the ‘Momentum’ method, a very simple model selection
+method that chooses the recent best-performing model, can adapt a trained ML model
+towards different market regimes efficiently. For LightGBM-dart and LightGBM-gbdt
+models, the ‘Calmar’ selection method gives a higher Calmar ratio than the ‘Momen-
+tum’ method but with a lower mean Corr. For MLP models, the ‘Calmar’ selection
+method significantly under-performs other model selection methods, with a much
+higher Max Drawdown. This suggests that selection based on historical drawdown is
+not robust, especially under situations with regime changes.
+In summary, the proposed online learning procedure to select optimal dynamic
+feature engineering methods can significantly reduce trading risks and improve the
+robustness of trading models, outperforming the baseline selection method that takes
+a simple average of all available models.
+7. Discussion. Motivated by the Numerai tournament, we have designed here
+an ML pipeline that can be applied to tabular temporal data of stock prices to under-
+pin strategies for trading of market-neutral stock portfolios. The various steps in the
+ML pipeline are carefully designed for robustness against regime changes and to avoid
+information leakage through time. We thus aim to obtain models with relatively low
+complexity, so as to reduce the danger of over-fitting, and with high robustness to
+changes in hyper-parameters and other choices in the algorithms. Another aim is to
+Regarding the choice of ML models, we find that gradient-boosting decision tree
+models are both more robust and interpretable than neural network-based models,
+and they allow more consistent performance under different market regimes.
+We also find that post-prediction processing, which is model-agnostic, is an effec-
+tive means of adapting trained ML models towards new situations without the need
+
+18
+THOMAS WONG AND MAURICIO BARAHONA
+(a) LightGBM-dart without FE
+Model Selection
+Mean
+Volatility
+Max Draw
+Sharpe
+Calmar
+Average
+0.0229
+0.0160
+0.0619
+1.4323
+0.3700
+Momentum
+0.0246
+0.0180
+0.0533
+1.3654
+0.4615
+Sharpe
+0.0234
+0.0165
+0.0533
+1.4148
+0.4390
+Calmar
+0.0225
+0.0171
+0.0350
+1.3122
+0.6429
+(b) LightGBM-gbdt without FE
+Model Selection
+Mean
+Volatility
+Max Draw
+Sharpe
+Calmar
+Average
+0.0216
+0.0177
+0.0710
+1.2165
+0.3042
+Momentum
+0.0228
+0.0201
+0.0729
+1.1342
+0.3128
+Sharpe
+0.0224
+0.0187
+0.0729
+1.1966
+0.3073
+Calmar
+0.0216
+0.0195
+0.0508
+1.1102
+0.4252
+(c) MLP without FE
+Model Selection
+Mean
+Volatility
+Max Draw
+Sharpe
+Calmar
+Average
+0.0195
+0.0175
+0.0918
+1.1149
+0.2124
+Momentum
+0.0212
+0.0191
+0.0878
+1.1124
+0.2415
+Sharpe
+0.0207
+0.0186
+0.0878
+1.1110
+0.2358
+Calmar
+0.0187
+0.0201
+0.1973
+0.9309
+0.0948
+Table 7: The effect of dynamic model selection. Performance of different ML
+models in the test period (2014-06-27 to 2022-09-23) with different online learning
+procedures selecting the optimal dynamic feature neutralisation method.
+to re-train ML models and introduce additional model uncertainty. Using dynamic
+feature neutralisation produces models with different flavours in an interpretable way,
+which also have better risk-adjusted performance than models with fixed feature neu-
+tralisation.
+Staking is commonly used in ML competitions to improve the robustness of mod-
+els. The method suggested in this study, dynamic model selection can be applied to
+online ML problems in guiding the selection of an optimal model(s) from a growing
+model ensemble. We find that a simple design, such as equal-weighted models, has
+a robust performance under different market regimes, but selecting the best model
+based on recent performance provides an improvement compared to the baseline as
+it switches to a lower-risk model during more volatile market regimes. It remains an
+open research area into how reinforcement learning or other online learning methods
+can be used to learn optimal staking weights between different ML models, given their
+historical performance and correlations.
+We also studied the robustness of our ML pipeline under different random seeds
+and changes in data splits for cross-validation. The results are presented in Section
+9.4 in the Supplementary Information, where we show that LightGBM dart mod-
+els are robust against these changes. The statistical rules used in dynamic feature
+neutralisation are also shown to perform better than features chosen at random.
+In the following, we discuss some ideas for further work to improve the ML pipeline
+
+ROBUST ML MODELS IN FINANCE
+19
+we designed. The diversity of models within a model ensemble is a key ingredient
+for dynamic model selection and other model ensemble/staking methods.
+A new
+metric could be designed to study the impact of a new ML model on an existing
+model ensemble. This metric could then be used to train new ML models that are
+uncorrelated to existing ones.
+The simple feature engineering methods used in our present study could not
+improve the performance of ML models.
+Identifying robust relationships between
+features over different market regimes is difficult but generative models, such as Vari-
+ational Autoencoders [42], could be used to create new features that summarise non-
+linear relationships in existing features.
+The Gradient Boosting models used in our pipeline are suitable for distributed
+learning, where large datasets are split into smaller batches to train on different ma-
+chines, often with various computational resource constraints. Data science compe-
+titions like the Numerai tournament rely on community efforts of individual data
+scientists to create a meta-model. This approach to crowd sourcing depends on the
+assumption that a complicated ML model that needs to be trained with advanced
+hardware can be approximated by combining a number of ML models (each trained
+with fewer data or features). Studying the convergence of model performance would
+be important for organising the data science competition as it decides how many
+participants are needed to maintain a well-diverse pool of models to create the meta-
+model.
+Overall, our results suggest using simple, well-established ML models such as
+gradient-boosting decision trees instead of specialised neural network models for this
+tasks.
+Rather than using a single neural network to perform feature engineering,
+model training/inference and post-prediction transformations, the modularised de-
+sign of the ML pipeline in this study offers increased robustness and transparency.
+Researchers can add, modify or delete a component without affecting the rest of the
+pipeline. Creating model ensembles improves model performances by reducing id-
+iosyncratic variance from individual ML models. The simple model selection rules
+based on recent performances provide a baseline that works well under different mar-
+ket regimes, whereas various portfolio metrics such as Sharpe and Calmar ratios are
+improved by using the recently best-performing models.
+8. Data and Code Availability . The data and code used in this paper is
+available at https://github.com/ThomasWong2022/numerai-benchmark.
+REFERENCES
+[1] J. Arosemena, N. Perez, D. Benitez, D. Riofrio, and R. Flores-Moyano, “Stock price analy-
+sis with deep-learning models,” in 2021 IEEE Colombian Conference on Applications of
+Computational Intelligence (ColCACI), 2021, pp. 1–6.
+[2] S. Selvin, R. Vinayakumar, E. A. Gopalakrishnan, V. K. Menon, and K. P. Soman, “Stock price
+prediction using lstm, rnn and cnn-sliding window model,” in 2017 International Confer-
+ence on Advances in Computing, Communications and Informatics (ICACCI), 2017, pp.
+1643–1647.
+[3] K. A. Althelaya, E.-S. M. El-Alfy, and S. Mohammed, “Evaluation of bidirectional lstm for
+short-and long-term stock market prediction,” in 2018 9th International Conference on
+Information and Communication Systems (ICICS), 2018, pp. 151–156.
+[4] J. Hullman, S. Kapoor, P. Nanayakkara, A. Gelman, and A. Narayanan, “The worst of both
+worlds: A comparative analysis of errors in learning from data in psychology and machine
+learning,” 2022. [Online]. Available: https://arxiv.org/abs/2203.06498
+[5] S. Kapoor and A. Narayanan, “Leakage and the reproducibility crisis in ml-based science,”
+2022. [Online]. Available: https://arxiv.org/abs/2207.07048
+
+20
+THOMAS WONG AND MAURICIO BARAHONA
+[6] E. Rivera-Landos, F. Khomh, and A. Nikanjam, “The challenge of reproducible ml: an empirical
+study on the impact of bugs,” 2021. [Online]. Available: https://arxiv.org/abs/2109.03991
+[7] [Online]. Available: https://arxiv.org/abs/2006.12117
+[8] J. Bollen, H. Mao, and X. Zeng, “Twitter mood predicts the stock market,” Journal
+of
+Computational
+Science,
+vol.
+2,
+no.
+1,
+pp.
+1–8,
+2011.
+[Online].
+Available:
+https://www.sciencedirect.com/science/article/pii/S187775031100007X
+[9] J. P. N and B. Vasudevan, “Effective implementation of neural network model with tune pa-
+rameter for stock market predictions,” in 2021 2nd International Conference on Smart
+Electronics and Communication (ICOSEC), 2021, pp. 1038–1042.
+[10] S. Singh and S. Sharma, “Forecasting stock price using partial least squares regression,” in
+2018 8th International Conference on Cloud Computing, Data Science & Engineering
+(Confluence), 2018, pp. 587–591.
+[11] P. Moritz, R. Nishihara, S. Wang, A. Tumanov, R. Liaw, E. Liang, M. Elibol, Z. Yang,
+W. Paul, M. I. Jordan, and I. Stoica, “Ray: A distributed framework for emerging ai
+applications,” 2017. [Online]. Available: https://arxiv.org/abs/1712.05889
+[12] F.
+Zhou,
+Q.
+Zhang,
+D.
+Sornette,
+and
+L.
+Jiang,
+“Cascading
+logistic
+regression
+onto
+gradient
+boosted
+decision
+trees
+for
+forecasting
+and
+trading
+stock
+indices,”
+Applied
+Soft
+Computing,
+vol.
+84,
+p.
+105747,
+2019.
+[Online].
+Available:
+https:
+//www.sciencedirect.com/science/article/pii/S1568494619305289
+[13] C. Bockel-Rickermann,
+“Predicting day-ahead stock returns using search engine query
+volumes:
+An application of gradient boosted decision trees to the s&p 100,” 2022.
+[Online]. Available: https://arxiv.org/abs/2205.15853
+[14] R. Akita, A. Yoshihara, T. Matsubara, and K. Uehara, “Deep learning for stock prediction using
+numerical and textual information,” in 2016 IEEE/ACIS 15th International Conference
+on Computer and Information Science (ICIS), 2016, pp. 1–6.
+[15] Numerai. Numerai Hedge Fund. (2022, Apr 12). [Online]. Available: https://numerai.fund/
+[16] D. B. Percival and A. T. Walden, Spectral Analysis for Univariate Time Series, ser. Cambridge
+Series in Statistical and Probabilistic Mathematics.
+Cambridge University Press, 2020.
+[17] B.
+Lim,
+S.
+O.
+Arik,
+N.
+Loeff,
+and
+T.
+Pfister,
+“Temporal
+fusion
+transformers
+for
+interpretable
+multi-horizon
+time
+series
+forecasting,”
+2019.
+[Online].
+Available:
+https://arxiv.org/abs/1912.09363
+[18] A. Kadra, M. Lindauer, F. Hutter, and J. Grabocka, “Well-tuned simple nets excel on tabular
+datasets,” 2021. [Online]. Available: https://arxiv.org/abs/2106.11189
+[19] R. Shwartz-Ziv and A. Armon, “Tabular data:
+Deep learning is not all you need,” 2021.
+[Online]. Available: https://arxiv.org/abs/2106.03253
+[20] S. B. Kotsiantis, “Decision trees: a recent overview,” Artificial Intelligence Review, vol. 39, pp.
+261–283, 2011.
+[21] Y. Freund and R. E. Schapire, “A decision-theoretic generalization of on-line learning and
+an application to boosting,” Journal of Computer and System Sciences, vol. 55, no. 1,
+pp. 119–139, 1997. [Online]. Available:
+https://www.sciencedirect.com/science/article/
+pii/S002200009791504X
+[22] J. H. Friedman, “Greedy function approximation: A gradient boosting machine,” The Annals
+of statistics, vol. 29, no. 5, pp. 1189–1232, 2001.
+[23] G.
+Ke,
+Q.
+Meng,
+T.
+Finley,
+T.
+Wang,
+W.
+Chen,
+W.
+Ma,
+Q.
+Ye,
+and
+T.-Y.
+Liu,
+“Lightgbm:
+A highly efficient gradient boosting decision tree,”
+in Advances
+in Neural Information Processing Systems,
+I. Guyon,
+U. V. Luxburg,
+S. Bengio,
+H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, Eds., vol. 30.
+Curran
+Associates, Inc., 2017. [Online]. Available:
+https://proceedings.neurips.cc/paper/2017/
+file/6449f44a102fde848669bdd9eb6b76fa-Paper.pdf
+[24] L. Prokhorenkova, G. Gusev, A. Vorobev, A. V. Dorogush, and A. Gulin, “Catboost: unbiased
+boosting with categorical features,”
+in Advances in Neural Information Processing
+Systems, S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and
+R. Garnett, Eds., vol. 31.
+Curran Associates, Inc., 2018. [Online]. Available:
+https:
+//proceedings.neurips.cc/paper/2018/file/14491b756b3a51daac41c24863285549-Paper.pdf
+[25] T. Chen and C. Guestrin, “Xgboost:
+A scalable tree boosting system,” in Proceedings of
+the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data
+Mining, ser. KDD ’16.
+New York, NY, USA: Association for Computing Machinery,
+2016, p. 785-794. [Online]. Available: https://doi.org/10.1145/2939672.2939785
+[26] R. Korlakai Vinayak and R. Gilad-Bachrach, “DART: Dropouts meet Multiple Additive
+Regression Trees,” in Proceedings of the Eighteenth International Conference on Artificial
+Intelligence and Statistics, ser. Proceedings of Machine Learning Research, G. Lebanon
+and S. V. N. Vishwanathan, Eds., vol. 38.
+San Diego, California, USA: PMLR,
+
+ROBUST ML MODELS IN FINANCE
+21
+09–12 May 2015, pp. 489–497. [Online]. Available:
+https://proceedings.mlr.press/v38/
+korlakaivinayak15.html
+[27] V.
+Borisov,
+T.
+Leemann,
+K.
+Se¨uler,
+J.
+Haug,
+M.
+Pawelczyk,
+and
+G.
+Kasneci,
+“Deep
+neural
+networks
+and
+tabular
+data:
+A
+survey,”
+2021.
+[Online].
+Available:
+https://arxiv.org/abs/2110.01889
+[28] S. Popov, S. Morozov, and A. Babenko, “Neural oblivious decision ensembles for deep learning
+on tabular data,” 2019. [Online]. Available: https://arxiv.org/abs/1909.06312
+[29] [Online]. Available: https://arxiv.org/abs/1912.01703
+[30] G. Ke, Z. Xu, J. Zhang, J. Bian, and T.-Y. Liu, “Deepgbm:
+A deep learning framework
+distilled by gbdt for online prediction tasks,” in Proceedings of the 25th ACM SIGKDD
+International Conference on Knowledge Discovery & Data Mining, ser. KDD ’19.
+New
+York, NY, USA: Association for Computing Machinery, 2019, p. 384-394. [Online].
+Available: https://doi.org/10.1145/3292500.3330858
+[31] S. ˜A. Arik and T. Pfister, “Tabnet: Attentive interpretable tabular learning,” Proceedings of
+the AAAI Conference on Artificial Intelligence, vol. 35, no. 8, pp. 6679–6687, May 2021.
+[Online]. Available: https://ojs.aaai.org/index.php/AAAI/article/view/16826
+[32] W. Song, C. Shi, Z. Xiao, Z. Duan, Y. Xu, M. Zhang, and J. Tang, “AutoInt,” in Proceedings
+of the 28th ACM International Conference on Information and Knowledge Management.
+ACM, nov 2019. [Online]. Available: https://doi.org/10.11452F3357384.3357925
+[33] D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” 2014. [Online].
+Available: https://arxiv.org/abs/1412.6980
+[34] T.
+Akiba,
+S.
+Sano,
+T.
+Yanase,
+T.
+Ohta,
+and
+M.
+Koyama,
+“Optuna:
+A
+next-
+generation hyperparameter optimization framework,” 2019. [Online]. Available:
+https:
+//arxiv.org/abs/1907.10902
+[35] J. Kirkpatrick, R. Pascanu, N. Rabinowitz, J. Veness, G. Desjardins, A. A. Rusu, K. Milan,
+J. Quan, T. Ramalho, A. Grabska-Barwinska, D. Hassabis, C. Clopath, D. Kumaran, and
+R. Hadsell, “Overcoming catastrophic forgetting in neural networks,” Proceedings of the
+National Academy of Sciences, vol. 114, no. 13, pp. 3521–3526, 2017. [Online]. Available:
+https://www.pnas.org/doi/abs/10.1073/pnas.1611835114
+[36] D. E. Kirk, Optimal control theory : an introduction.
+Mineola, N.Y: Dover Publications, 2004
+- 1970.
+[37] R. S. Sutton, Reinforcement learning : an introduction, second edition. ed., ser. Adaptive
+computation and machine learning series.
+Cambridge, Massachusetts: The MIT Press,
+2018.
+[38] V. Mnih, K. Kavukcuoglu, D. Silver, A. A. Rusu, J. Veness, M. G. Bellemare, A. Graves,
+M. Riedmiller, A. K. Fidjeland, G. Ostrovski, S. Petersen, C. Beattie, A. Sadik,
+I. Antonoglou,
+H. King,
+D. Kumaran,
+D. Wierstra,
+S. Legg,
+and D. Hassabis,
+“Human-level control through deep reinforcement learning,” Nature, vol. 518, no. 7540,
+pp. 529–533, Feb 2015. [Online]. Available: https://doi.org/10.1038/nature14236
+[39] J. Schulman,
+F. Wolski,
+P. Dhariwal,
+A. Radford,
+and O. Klimov,
+“Proximal policy
+optimization algorithms,” 2017. [Online]. Available: https://arxiv.org/abs/1707.06347
+[40] T. Haarnoja, A. Zhou, P. Abbeel, and S. Levine, “Soft actor-critic:
+Off-policy maximum
+entropy deep reinforcement learning with a stochastic actor,” in Proceedings of the 35th
+International Conference on Machine Learning, ser. Proceedings of Machine Learning
+Research, J. Dy and A. Krause, Eds., vol. 80.
+PMLR, 10–15 Jul 2018, pp. 1861–1870.
+[Online]. Available: https://proceedings.mlr.press/v80/haarnoja18b.html
+[41] J.
+Schulman,
+P.
+Moritz,
+S.
+Levine,
+M.
+Jordan,
+and
+P.
+Abbeel,
+“High-dimensional
+continuous control using generalized advantage estimation,” 2015. [Online]. Available:
+https://arxiv.org/abs/1506.02438
+[42] D. P. Kingma and M. Welling, “Auto-encoding variational bayes,” in 2nd International
+Conference on Learning Representations, ICLR 2014, Banff, AB, Canada, April 14-16,
+2014, Conference Track Proceedings, Y. Bengio and Y. LeCun, Eds., 2014. [Online].
+Available: http://arxiv.org/abs/1312.6114
+[43] P. Buhlmann and T. Hothorn,
+“Boosting algorithms:
+Regularization,
+prediction and
+model fitting,”
+Statistical Science,
+vol. 22,
+no. 4,
+nov 2007. [Online]. Available:
+https://doi.org/10.1214%2F07-sts242
+
+22
+THOMAS WONG AND MAURICIO BARAHONA
+9. Supplementary Information.
+9.1. Additional results for dynamic feature neutralisation. Here we show
+the performance of dynamic feature neutralisation for low and high volatility regimes.
+(a) LightGBM-dart without FE
+Feature Neutralisation
+Mean
+Volatility
+Max Draw
+Sharpe
+Calmar
+Fixed
+0.0206
+0.0195
+0.1153
+1.0576
+0.1787
+Low Mean
+0.0255
+0.0175
+0.0350
+1.4578
+0.7286
+High Mean
+0.0207
+0.0206
+0.0986
+1.0033
+0.2099
+Low Vol
+0.0238
+0.0221
+0.0538
+1.0793
+0.4424
+High Vol
+0.0235
+0.0180
+0.0341
+1.3069
+0.6891
+(b) LightGBM-gbdt without FE
+Feature Neutralisation
+Mean
+Volatility
+Max Draw
+Sharpe
+Calmar
+Fixed
+0.0194
+0.0220
+0.1998
+0.8820
+0.0971
+Low Mean
+0.0251
+0.0188
+0.0495
+1.3328
+0.5071
+High Mean
+0.0184
+0.0228
+0.1469
+0.8053
+0.1253
+Low Vol
+0.0214
+0.0247
+0.1852
+0.8657
+0.115
+High Vol
+0.0225
+0.0188
+0.0487
+1.1939
+0.4620
+(c) MLP without FE
+Feature Neutralisation
+Mean
+Volatility
+Max Draw
+Sharpe
+Calmar
+Fixed
+0.0165
+0.0210
+0.2606
+0.7875
+0.0633
+Low Mean
+0.0215
+0.0187
+0.0496
+1.1496
+0.4335
+High Mean
+0.0170
+0.0210
+0.1283
+0.8118
+0.1325
+Low Vol
+0.0194
+0.0229
+0.0878
+0.8487
+0.2210
+High Vol
+0.0194
+0.0177
+0.0730
+1.0990
+0.2658
+Table 8: Performance of ML models in the test period (2014-06-27 to 2022-09-23)
+with different dynamic feature neutralisation methods in low volatility regime
+
+ROBUST ML MODELS IN FINANCE
+23
+(a) LightGBM-dart without FE
+Feature Neutralisation
+Mean
+Volatility
+Max Draw
+Sharpe
+Calmar
+Fixed
+0.0227
+0.0163
+0.0223
+1.3888
+1.0179
+Low Mean
+0.0220
+0.0148
+0.0199
+1.4907
+1.1055
+High Mean
+0.0233
+0.0151
+0.0206
+1.5372
+1.1311
+Low Vol
+0.0252
+0.0168
+0.0330
+1.4980
+0.7636
+High Vol
+0.0215
+0.0152
+0.0143
+1.4077
+1.5035
+(b) LightGBM-gbdt without FE
+Feature Neutralisation
+Mean
+Volatility
+Max Draw
+Sharpe
+Calmar
+Fixed
+0.0217
+0.0198
+0.0364
+1.0953
+0.5962
+Low Mean
+0.0212
+0.0176
+0.0380
+1.2039
+0.5579
+High Mean
+0.0218
+0.0186
+0.0334
+1.1728
+0.6527
+Low Vol
+0.0237
+0.0201
+0.0306
+1.1792
+0.7745
+High Vol
+0.0209
+0.0173
+0.0308
+1.2068
+0.6786
+(c) MLP without FE
+Feature Neutralisation
+Mean
+Volatility
+Max Draw
+Sharpe
+Calmar
+Fixed
+0.0196
+0.0193
+0.0326
+1.0191
+0.6012
+Low Mean
+0.0205
+0.0183
+0.0806
+1.1212
+0.2543
+High Mean
+0.0170
+0.0210
+0.1283
+0.8118
+0.1325
+Low Vol
+0.0222
+0.0194
+0.0397
+1.1442
+0.5592
+High Vol
+0.0187
+0.0165
+0.0336
+1.1368
+0.5565
+Table 9: Performance of ML models in the test period (2014-06-27 to 2022-09-23)
+with different dynamic feature neutralisation methods in high volatility regime
+9.2. Pseudocode for algorithms in the text. For completeness, we present
+here brief pseudocode for some of the main methods in the paper with the appropriate
+references.
+Algorithm 9.1 Gradient boosting algorithm [22,43]
+Given N data samples (xi, yi), 1 ≤ i ≤ N with the aim to find an increasing better
+estimate ˆf(x) of the minimising function f(x) which minimise the loss L(f) between
+targets and predicted values. L(f) = �
+i l(yi, f(xi)) where l is a given loss function
+such as mean square losses for regression problems. Function f is restricted to the
+class of additive models f(x) = �K
+k=1 wkh(x, αk) where h(·, α) is a weak learner
+with parameters α and wk are the weights.
+Initialise f0(x) = arg minα0
+�N
+i=1 l(yi, h(xi, α0))
+For k = 1 : K Compute the gradient residual using gik = −
+�
+∂l(yi,fk−1(xi))
+∂fk−1(xi)
+�
+Use the weak learner to compute αk which minimises �N
+i=1(gik − h(xi, αk))2
+Update with learning rate λ fk(x) = fk−1(x) + λh(x, αk)
+Return f(x) = fK(x)
+
+24
+THOMAS WONG AND MAURICIO BARAHONA
+Algorithm 9.2 Gradient boosting tree algorithm implemented in LightGBM [22,23,
+43]
+Initialise f0(x) = arg minα0
+�N
+i=1 l(yi, x, α0)
+For k = 1 :
+K For i
+=
+1, 2, . . . N,
+compute the gradient residual using
+gik = −
+�
+∂l(yi,fk−1(xi))
+∂fk−1(xi)
+�
+Fit a decision tree to the targets gik giving terminal leaves Rkj, j = 1, 2, . . . Jk, where
+Jk is the number of terminal leaves.
+For j = 1, 2, . . . Jk, compute αjk = arg minα
+�
+xi∈Rkj l(yi, fk−1(xi) + α)
+Update boosting trees with learning rate λ fk(x) = fk−1(x) + λ �Jk
+j=1 αkjI(x ∈ Rkj)
+Return fK(x)
+
+ROBUST ML MODELS IN FINANCE
+25
+9.3. Hyper-parameter search space for different ML models. We ran all
+experiments on a GPU cluster, each node of which contains a NVIDIA GeForce RTX
+2080 Ti GPU, running with 4352 CUDA cores and 11GB memory. Hyper-parameter
+search is performed using Optuna [34]. For each Feature Engineering/ML pipeline,
+hyper-parameter search is ran for at most 8 hours or at most 100 configurations,
+whichever came first. The default TPE sampler in Optuna is used to perform the
+hyper-parameter search. In Figure 4 and 5, we list the Hyper-parameter search pa-
+rameters defined in Optuna [34] for different ML models used in the main text to
+train the models.
+• Feature Engineering
+– Numerai Basic Feature Engineering
+∗ dropout pct: low:0.05, high:0.25, step:0.05,
+∗ no product features: low:50, high:1000, step:50,
+• ML Models
+– LightGBM-gbdt
+∗ n estimators: low:50, high:1000, step:50
+∗ learning rate: low:0.005, high:0.1, log:True
+∗ min data in leaf: low:2500, high:40000, step:2500
+∗ lambda l1: low:0.01, high: 1, log:True
+∗ lambda l2: low:0.01, high: 1, log:True
+∗ feature fraction: low:0.1, high:1, step:0.05
+∗ bagging fraction: low:0.5, high:1, step:0.05
+∗ bagging freq: low:10, high:50, step:10
+– LightGBM-dart
+∗ n estimators: low:50, high:1000, step:50
+∗ learning rate: low:0.005, high:0.1, log:True
+∗ min data in leaf: low:2500, high:40000, step:2500
+∗ lambda l1: low:0.01, high: 1, log:True
+∗ lambda l2: low:0.01, high: 1, log:True
+∗ feature fraction: low:0.1, high:1, step:0.05
+∗ bagging fraction: low:0.5, high:1, step:0.05
+∗ bagging freq: low:10, high:50, step:10
+∗ drop rate: low:0.1, high:0.5, step:0.1
+∗ skip drop: low:0.1, high:0.8, step:0.1
+– LightGBM-goss
+∗ n estimators: low:50, high:1000, step:50
+∗ learning rate: low:0.005, high:0.1, log:True
+∗ min data in leaf: low:2500, high:40000, step:2500
+∗ lambda l1: low:0.01, high: 1, log:True
+∗ lambda l2: low:0.01, high: 1, log:True
+∗ feature fraction: low:0.1, high:1, step:0.05
+∗ bagging fraction: low:0.5, high:1, step:0.05
+∗ bagging freq: low:10, high:50, step:10
+∗ top rate: low:0.1, high:0.4, step:0.05
+∗ other rate: low:0.05, high:0.2, step:0.05
+Fig. 4: Hyper-parameter Space for ML models
+
+26
+THOMAS WONG AND MAURICIO BARAHONA
+• Machine Learning
+– MLP
+∗ max epochs: low:10, high:100, step:5
+∗ patience: low:5, high:20, step:5
+∗ num layers: low:2, high:7, step:1
+∗ neurons: low:64, high:1024, step:64
+∗ neuron scale: low:0.3, high:1, log:True
+∗ dropout: low:0.1, high:0.9, log:True
+∗ batch size: low:10240, high:40960, step:10240
+– TabNet
+∗ max epochs: low:10, high:100, step:5
+∗ patience: low:5, high:20, step:5
+∗ batch size: low:1024, high:4096, step:1024
+∗ num d: low:4, high:16, step:4
+∗ num a: low:4, high:16, step:4
+∗ num steps: low:1, high:3, step:1
+∗ num shared: low:1, high:3, step:1
+∗ num independent: low:1, high:3, step:1
+∗ gamma : low:1, high:2, step:0.1
+∗ momentum: low:0.01, high:0.4, step:0.01
+∗ lambda sparse: low:0.0001, high:0.01, log:True
+Fig. 5: Hyper-parameter Space for ML models
+
+ROBUST ML MODELS IN FINANCE
+27
+9.4. Robustness of ML pipeline. One of the aims in this work was to provide
+a robust pipeline for tabular temporal data under regime changes. Here we present
+additional results of the robustness of the method under different scenarios and sources
+of variability.
+Robustness under changes of random seeds in the learning algorithms. In Ta-
+ble 10, we report the variability of the performance of the LightGBM-dart, LightGBM-
+gbdt and MLP models trained starting from 10 different initial random seeds. The
+performance is generally robust to the change in random seeds, with small variances
+in the prediction of the mean Corr and volatility and moderate for the Maximum
+Drawdown.
+Model
+Mean
+Volatility
+Max Draw
+Sharpe
+Calmar
+LightGBM-dart without FE
+mean
+0.0254
+0.0266
+0.1567
+0.9593
+0.1639
+sd
+0.0006
+0.0007
+0.0158
+0.0365
+0.0175
+LightGBM-gbdt without FE
+mean
+0.0253
+0.0312
+0.2338
+0.8104
+0.1100
+sd
+0.0006
+0.0006
+0.0296
+0.0278
+0.0153
+MLP without FE
+mean
+0.0233
+0.0271
+0.1643
+0.8600
+0.1446
+sd
+0.0009
+0.0011
+0.0248
+0.0365
+0.0219
+Table 10: Variability of the performance of ML models in the test period (2014-06-
+27 to 2022-09-23). The mean and standard deviation of each portfolio metrics are
+calculated over models with 10 different random seeds for each method
+A general strategy to reduce the variance is to combine different ML models.
+There are two ways to do so: (i) averaging over models, by calculating the average
+performance of different models, and (ii) averaging over predictions, by calculating the
+average predictions from each model and then scoring the average predictions against
+the target. Table 11 shows that averaging over predictions gives higher mean Corr
+and Sharpe/Calmar ratios than averaging over models.
+Therefore, this averaging
+method is used to compute model performances in Table 2 in the main text.
+Model
+Average
+Mean
+Volatility
+Max Draw
+Sharpe
+Calmar
+LightGBM-dart without FE
+Over models
+0.0254
+0.0266
+0.1567
+0.9593
+0.1639
+Over predictions
+0.0278
+0.0284
+0.1622
+0.9791
+0.1714
+LightGBM-gbdt without FE
+Over models
+0.0253
+0.0312
+0.2338
+0.8104
+0.1100
+Over predictions
+0.0262
+0.0321
+0.2378
+0.8140
+0.1102
+MLP without FE
+Over models
+0.0233
+0.0271
+0.1643
+0.8600
+0.1446
+Over predictions
+0.0258
+0.0289
+0.1668
+0.8931
+0.1547
+Table 11: Performance of different ML methods on Numerai v4 dataset in the test
+period (2014-06-27 to 2022-09-23) with different averaging methods
+Robustness under different cross-validation data splits. As financial data are regime
+dependent, an important measure of model robustness is to measure the performance
+of ML models that have been trained using different cross-validation splits of the data
+and compute how much the model performance changes over different test periods.
+To ascertain the robustness of data splits, we have carried out 3 cross-validation
+splits (CV 1, CV 2, CV 3) as shown in Table 12. The hyper-parameters are optimised
+under CV 1, which is the cross-validation used to generate the model performances
+in the main text. These hyper-parameters are fixed for the models trained under
+the CV 2 and CV 3 splits. For ML methods that require early stopping, the data
+
+28
+THOMAS WONG AND MAURICIO BARAHONA
+in the validation period (different for each split) are used to regularise the models.
+Therefore, by reusing the optimised hyper-parameters across all splits, we evaluate
+the robustness of the model performance to the optimisation of hyper-parameters. We
+then compute the performance when applying the models to shifted cross-validation
+datasets in the walk-forward CV 2 and CV 3 data splits.
+Our results show good
+consistency in performance across CV 2 and CV 3, with only a small deterioration of
+the results as compared to CV 1 (over which the hyperparameters were optimised).
+We also find that LightGBM-dart with FE, the ML method that has the highest
+mean Corr in CV 1, has the greatest return and best Sharpe and Calmar ratios also
+in other cross-validations, as seen in Table 13.
+Train Start
+Train End
+Validation Start
+Validation End
+Enter Ensemble
+CV 1
+2003-01-03
+2012-07-27
+2012-12-21
+2014-11-14
+2015-05-15
+CV 2
+2003-01-03
+2014-06-27
+2014-11-21
+2016-10-14
+2017-04-14
+CV 3
+2003-01-03
+2016-05-27
+2016-10-21
+2018-09-14
+2019-03-15
+Table 12: Various cross-validation schemes to train ML models on different parts of
+the data. CV 1 is the cross-validation used for hyper-parameter optimisation and
+training ML models in the main text.
+(a) CV 1 (2015-05-15 to 2022-09-23)
+Method
+Mean
+Volatility
+Max Draw
+Sharpe
+Calmar
+LightGBM-dart without FE
+0.0278
+0.0284
+0.1622
+0.9791
+0.1714
+LightGBM-gbdt without FE
+0.0262
+0.0321
+0.2378
+0.8140
+0.1102
+MLP without FE
+0.0258
+0.0289
+0.1668
+0.8931
+0.1547
+(b) CV 2 (2017-04-14 to 2022-09-23)
+Method
+Mean
+Volatility
+Max Draw
+Sharpe
+Calmar
+LightGBM-dart without FE
+0.0250
+0.0278
+0.1817
+0.8990
+0.1376
+LightGBM-gbdt without FE
+0.0231
+0.0324
+0.3227
+0.7104
+0.0716
+MLP without FE
+0.0215
+0.0289
+0.2307
+0.7446
+0.0932
+(c) CV 3 (2019-03-15 to 2022-09-23)
+Method
+Mean
+Volatility
+Max Draw
+Sharpe
+Calmar
+LightGBM-dart without FE
+0.0264
+0.0297
+0.1380
+0.8140
+0.1913
+LightGBM-gbdt without FE
+0.0261
+0.0336
+0.1584
+0.7772
+0.1648
+MLP without FE
+0.0224
+0.0240
+0.1171
+0.9339
+0.1913
+Table 13: Performance of selected machine learning methods on the Numerai dataset
+in the test period for various walk-forward cross-validation schemes, (a) CV 1, (b)
+CV 2 and (c) CV 3
+Robustness under feature selection for dynamic feature neutralisation. A fixed
+set of 420 features to be neutralised was given by the Numerai organisers based on
+internal evaluations of parameters. In Section 6, we introduce several statistical rules
+that allow us to select a varying subset of features to be neutralised in each era based
+on empirical heuristic criteria motivated by financial modelling.
+
+ROBUST ML MODELS IN FINANCE
+29
+To evaluate the robustness of the proposed statistical rules, we draw 100 subsets
+of 420 features selected at random. and use each set to neutralise the raw predictions
+from ML models. We then evaluate the performance of ML models based on each of
+the random subsets. Using the procedure described in section 6.2 we then select the
+optimal dynamic feature neutralisation method and compute the performance of the
+top 10 models of the highest mean Corr, Sharpe and Calmar ratio over the test period.
+The results are reported in Table 14 and should be compared to the performance of
+the same models in Table 7, which were obtained with dynamic feature neutralisation
+using the statistical rules defined in section 6.2.
+The mean Corr of models obtained with random feature neutralisation for each
+rule (Momentum/Sharpe/Calmar) are lower than those obtained using the statistical
+rules in Table 7. On the other hand, the Sharpe ratio of models for models with
+random feature neutralisation is slightly higher, as expected due to the variance re-
+duction effect by averaging over 10 different models. For models selected based on the
+Calmar rule, the models obtained with statistical rules have a much higher Calmar
+ratio than random feature neutralisation. It suggests the statistical rules defined can
+effectively reduce model risks by reducing linear exposure to undesirable features.
+(a) LightGBM-dart without FE
+Feature Neutralisation
+Mean
+Volatility
+Max Draw
+Sharpe
+Calmar
+Average
+0.0214
+0.0147
+0.0482
+1.4547
+0.4440
+Momentum
+0.0216
+0.0149
+0.0472
+1.4522
+0.4576
+Sharpe
+0.0213
+0.0147
+0.0459
+1.4474
+0.4641
+Calmar
+0.0214
+0.0148
+0.0453
+1.4504
+0.4724
+(b) LightGBM-gbdt without FE
+Feature Neutralisation
+Mean
+Volatility
+Max Draw
+Sharpe
+Calmar
+Average
+0.0203
+0.0167
+0.0664
+1.2140
+0.3057
+Momentum
+0.0208
+0.0167
+0.0641
+1.2457
+0.3245
+Sharpe
+0.0206
+0.0168
+0.0618
+1.2267
+0.3333
+Calmar
+0.0216
+0.0195
+0.0508
+1.1102
+0.2743
+(c) MLP without FE
+Feature Neutralisation
+Mean
+Volatility
+Max Draw
+Sharpe
+Calmar
+Average
+0.0176
+0.0165
+0.0831
+1.0658
+0.2118
+Momentum
+0.0179
+0.0165
+0.0790
+1.0842
+0.2266
+Sharpe
+0.0177
+0.0164
+0.0762
+1.0751
+0.2323
+Calmar
+0.0175
+0.0167
+0.0825
+1.0511
+0.2121
+Table 14: Performance of different ML models in the test period (2015-05-15 to 2022-
+09-23) obtained with random feature neutralisation. These are averages obtained by
+selecting the top 10 models under the different online learning procedures over the
+test period.
+

F9FKT4oBgHgl3EQfbS5P/content/tmp_files/2301.11811v1.pdf.txt

@@ -0,0 +1,944 @@
+Indonesian Journal of Electrical Engineering and Computer Science 
+Vol. 28, No. 1, October 2022, pp. 328~338 
+ISSN: 2502-4752, DOI: 10.11591/ijeecs.v28.i1.pp328-338 
+     328 
+ 
+ 
+Journal homepage: http://ijeecs.iaescore.com 
+A systematic review of structural equation modeling in 
+augmented reality applications 
+ 
+ 
+Vinh The Nguyen1, Chuyen Thi Hong Nguyen2 
+1Faculty of Information Technology, TNU-University of Information and Communication Technology, Thai Nguyen, Vietnam 
+2Faculty of Primary Education, Thai Nguyen University of Education, Thai Nguyen, Vietnam 
+ 
+ 
+Article Info 
+ 
+ABSTRACT  
+Article history: 
+Received Mar 26, 2022 
+Revised Jun 21, 2022 
+Accepted Jul 14, 2022 
+ 
+ 
+The purpose of this study is to present a comprehensive review of the use of 
+structural equation modeling (SEM) in augmented reality (AR) studies in the 
+context of the COVID-19 pandemic. IEEE Xplore Scopus, Wiley Online 
+Library, Emerald Insight, and ScienceDirect are the main five data sources 
+for data collection from Jan 2020 to May 2021. The preferred reporting 
+items for systematic reviews and meta-analyses (PRISMA) approach was 
+used to conduct the analysis. At the final stage, 53 relevant publications were 
+included for analysis. Variables such as the number of participants in the 
+study, 
+original 
+or 
+derived 
+hypothesized 
+model, 
+latent 
+variables, 
+direct/indirect contact with users, country, limitation/suggestion, and 
+keywords were extracted. The results showed that a variety of external 
+factors were used to construct the SEM models rather than using the 
+parsimonious ones. The reports showed a fair balance between the direct and 
+indirect methods to contact participants. Despite the COVID-19 pandemic, 
+few publications addressed the issue of data collection and evaluation 
+methods, whereas video demonstrations of the augmented reality (AR) apps 
+were utilized. The current work influences new AR researchers who are 
+searching for a theory-based research model in their studies. 
+Keywords: 
+Augmented reality 
+COVID-19 
+External factors 
+Structural equation modeling 
+Theory-based research 
+This is an open access article under the CC BY-SA license. 
+ 
+Corresponding Author: 
+Vinh The Nguyen 
+Faculty of Information Technology, TNU-University of Information and Communication Technology 
+Z115 Street, Quyet Thang Commune, Thai Nguyen, Vietnam 
+Email: [email protected] 
+ 
+ 
+1. 
+INTRODUCTION  
+Augmented reality (AR) is a technology that has attracted a lot of attention in various domains [1]-
+[3]. Unlike virtual reality (VR) which allows users to be totally immersed in a virtual environment, AR 
+enriches the real world with virtual artifacts [4]. The primary value of AR is that it allows digital objects to 
+be blended more seamlessly into a person’s perception of the real world than simply displaying data on a 
+screen. Market research [5] anticipates that AR’s market will reach USD 88.4 billion, growing 31.5% from 
+2021 to 2026. In addition, in response to the COVID-19 pandemic, more companies and organizations have 
+adopted remote work and are utilizing augmented reality technology [6]. What that means is that a huge 
+number of AR applications are being developed, especially in electrical engineering and computer science 
+[1]-[3], [7], [8]. 
+Assessment is one of the key factors in ensuring the success of an AR application, especially when it 
+is involved with end-users. However, literature work reported that only a few studies afforded time for this 
+type of evaluation (only 8% of published papers) [9]. One plausible explanation was that AR 
+researchers/developers had to devote their time to solving technical issues [10]. Moreover, the lack of 
+methods or theory-driven research on evaluating AR apps, considering end users’ involvement, contributed 
+
+cC
+BY
+SAIndonesian J Elec Eng & Comp Sci  
+ISSN: 2502-4752 
+ 
+ 
+A systematic review of structural equation modeling in augmented reality applications (Vinh The Nguyen) 
+329 
+to the scarcity of AR evaluation [11]. In addition, after the COVID-19 outbreak, many conferences (e.g., 
+ISMAR) encouraged researchers to find alternative means of evaluating AR apps rather than canceling the 
+submissions due to social distancing. There has been no study addressing this issue so far, thus it remains a 
+gap in the literature. To close this gap, this paper-based on prior AR studies–provided an overview of theory-
+based methods that can effectively be used for AR assessment. Among many other end-user evaluation 
+methods, the scope of the current study focused on structural equation modeling (SEM), a model commonly 
+used in behavioral science. SEM is a comprehensive statistical method that examines relationships between 
+observed and latent factors [12]. It has been widely used in confirmatory factory analysis in many topics and 
+fields [13]-[15].  
+A number of review studies on SEM applications have been conducted in various research domains, 
+including ecology [16], social science [17], psychological research [18], and strategic management [19]. It 
+indicated that a review study would be valuable for new researchers to quickly acquire knowledge in the field 
+effectively. Yet, it also implies that it would be important to look at SEM from AR’s perspective since AR is 
+one of the emerging trends in the digital transformation era. However, there is no study of SEM for AR 
+applications other than previously mentioned review studies. Thus, the current research is unique on its own 
+by the AR’s topic and the outcomes of this study can be used as a referencguidene for researchers in similar 
+studies, particularly in electrical engineering and computer science. More specifically, the present study tries 
+to answer to following research questions: i) What are the preferred theory-driven models being used in prior 
+AR studies amid the COVID-19 pandemic? ii) What are the dimensions or variables being investigated by 
+AR researchers so far? iii) How do researchers of prior AR studies communicate with end-users for 
+evaluation? vi) How many participants are typically involved in a study? Would this number still be 
+considered appropriate from the literature? v) What are the main drawbacks of tR studies? Do they suffer 
+from the COVID-19 pandemic? 
+ 
+ 
+2. 
+METHOD 
+This study involves a review of SEM in AR applications; thus, the preferred reporting items for 
+systematic reviews and meta-analyses (PRISMA) statement was applied [20]. The PRISMA statement aims 
+to assist scholars in improving the reporting of scientific reviews and meta-analyses. It is an evidence-based 
+minimum set of elements for systematic review reports that are intended to assist systematic reviewers in 
+clearly explaining why the review was conducted and what the authors performed. It has previously been 
+used to target comparable research objectives [21], [22]. 
+ 
+2.1.  Source selection 
+IEEE Xplore, Scopus, Wiley Online Library, Emerald Insight, and ScienceDirect databases were 
+used to build the corpus, encompassing titles, abstracts, and keywords. These five databases are regarded as 
+essential and dependable sources of high-quality articles in the fields of computer science and engineering 
+[21], [23]. Although, some other indexing databases are available (i.e., Scholar) but they are out of scope in 
+the current study. 
+ 
+2.2.  Search criteria 
+To add articles to our corpus, both of the following related criteria need to be fulfilled, i) Structural 
+equation modeling search term: at least one SEM-related term must appear in an article’s title, abstract, or 
+author keywords (i.e., structural equation modeling, SEM, planned behavior, theory of planned behaviour 
+(TPB), motivational model, Michaelis–Menten (MM), reasoned action, theory of reasoned action (TRA), 
+social cognitive, SCT, diffusion of innovation, IDT); and ii) Augmented Reality search term: terms include 
+augmented reality, AR. Using the aforementioned criteria, 16 articles were discovered in IEEE Xplore, 107 
+articles in Scopus, 197 papers in Wiley Online Library, 68 papers in Emerald Insight, and 695 papers in 
+ScienceDirect. The corpus was collected between June 3, 2021, and June 12, 2021. 
+ 
+2.3.  Eligibility assessment for the final analysis corpus 
+To determine the acceptability of the obtained papers, the first researcher personally reviewed the 
+entry criteria mentioned below by reviewing the titles and abstracts of the obtained publications. When a 
+clear judgment could not be reached, other aspects of the publication, particularly the method and data 
+acquisition descriptions, were discussed in conjunction with the second author. Only items that meet the 
+following criteria are retained in the corpus: i) Peer-reviewed: The paper was peer-reviewed in the two 
+indexing databases. This is due to the trustworthiness of peer-reviewed journals and the rigorous peer-review 
+processes, only articles in these databases are considered for this review; ii) Topic relevant: The topic of an 
+article is pertinent to the applications of SEM in AR; iii) Language: Publication was reported in English; and 
+vi) Duration: Paper was published between Jan 2020 and May 2021. 
+
+      
+  
+ 
+       ISSN: 2502-4752 
+Indonesian J Elec Eng & Comp Sci, Vol. 28, No. 1, October 2022: 328-338 
+330 
+If the article meets any of the following criteria, it will be excluded from the corpus: i) Books and 
+cover page, abstract only, poster; ii) The paper was not written in English; iii) Application of SEM is not for 
+AR; and vi) Paper was published before Jan 2020 and after May 2021.  
+Figure 1 depicts the flow of information through the different phases of the systematic review 
+utilizing PRISMA approach. 1,083 records were found in all data sources. Duplications were removed based 
+on the titles. Each paper was screened individually to remove items that are out of scope. Then 230 records 
+were excluded. As such, 309 candidates left for full-text retrieval. Of these remaining items, 9 records cannot 
+be retrieved due to access restrictions. The authors examined each report for eligibility and removed 247 
+studies. In the end, 53 items were included in this research. The remaining papers were examined 
+individually to extract interesting variables such as the number of participants, original or derived 
+hypothesized model, latent variables, direct/indirect contact with the user, country of origin, 
+limitation/suggestion (if any), and keywords. 
+ 
+ 
+ 
+ 
+Figure 1. The flow diagram represents the movement of information through the various stages of a 
+systematic review 
+ 
+ 
+2.4.  Data coding and analysis 
+To extract the data, all articles were loaded into NVivo software, and a coding scheme was created. 
+NVivo is a program that facilitates qualitative analytical method research. This tool enables researchers to 
+organize, analyze and explore unstructured or qualitative data, including interviews, reviews, articles, social 
+media, and web content. Codes included authors, journal name, year of publication, countries of authorship, 
+title, abstract, author keywords, method, objectives, findings and limitations on how SEM was used. 
+ 
+ 
+3. 
+RESULTS AND DISCUSSION 
+3.1.  What are the preferred theory-based driven models being used in prior AR studies amid the 
+COVID-19 pandemic? 
+Figure 2 depicts the distribution of papers over hypothesized models. Most publications fall into the 
+SEM category (accounted for 58.49%), followed by eTAM and TAM with 20.76% and 11.32% respectively. 
+Although the UTAUT model was developed recently, the result shows less popularity of adopting this model 
+(only 3.77%), which is the same as the SOR model. 
+Technology acceptance model (TAM): originally developed by Davis [24], TAM is known as a 
+theory of information systems that describes how consumers come to accept and use technology. Real system 
+usage is the point at which people interact with technology. People utilize technology because of their 
+behavioral intentions. In this survey, 6 articles (11.32%) used original TAM for their research. 
+Extended technology acceptance model (eTAM): In this category, 11 publications (20.75%) 
+extended TAM with external variables such as perceived task-technology fit [25]-[28]–which asserted that 
+
+Identification ofnew studies viadatabases
+Recordsidentifiedfromdatabases:
+Recordsremovedbeforescreening:
+N=1.083
+N=544
+Records screened:
+Records excluded:
+N=539
+N=230
+Reports soughtfor retrieval
+Reports not retrieved:
+N=309
+N=9
+Reports excluded:247
+Reports assessed for eligibility:
+NotinvolveSEM(N=27)
+N=300
+NotinvolveAR(N=127)
+ReviewOnly(N=93)
+Reports of included studies:
+N=53Indonesian J Elec Eng & Comp Sci  
+ISSN: 2502-4752 
+ 
+ 
+A systematic review of structural equation modeling in augmented reality applications (Vinh The Nguyen) 
+331 
+the technology must be utilized and a good fit with the tasks it supports to have positive impacts on 
+individual performance, perceived visual design/appeal [25]-[31] which assumed that beauty is important, 
+and it impacts decisions that should not be influenced by aesthetics, perceived enjoyment [32]-[35]-which 
+refers to the hedonic value of new technology and expresses how pleasurable a person finds its use.  
+ 
+ 
+ 
+ 
+Figure 2. Models distribution across prior studies 
+ 
+ 
+The unified theory of acceptance and use of technology (UTAUT): Venkatesh et al. [36] developed 
+the UTAUT after reviewing and consolidating the components of eight previous models used to describe 
+information system user behavior. In this review, several external variables were incorporated into the 
+existing UTAUT model (eUTAUT) such as innovativeness, reward, trust, enjoyment, hedonic motivation, 
+habit, and gamification [37], [38]. 
+Stimulus-organism-response (SOR): Mehrabian-stimulus Russell's model [39] depicts the 
+occurrence of a person's response to environmental stimuli. Qin et al. [40] decomposed stimulus into two 
+external factors (i.e., Interactivity, Virtuality), Organism into 4 variables (i.e., Hedonic, Utilitarian, 
+Informativeness, and Ease of Use), and Response into 2 factors including Attitude and Behavioral Intention. 
+Similarly in the scope of this review,  Qin et al. [40] also included (critical mass, social interaction, 
+information timelines, content richness) into stimulus, (attachment, conformity) into Organism, and (visiting 
+intention, continue intention) into Response.  
+Structural equation modeling (SEM): This category contains the largest portion of the papers 
+included in our investigation (58.49%). Authors in this group mainly adapted constructs, measures in the 
+literature to form hypothesis. As such, PLS-SEM was utilized as an analytical method to conduct 
+confirmatory factor analysis and path analysis. Confirmatory factor analysis, which originates in 
+psychometrics, aims to quantify underlying psychological characteristics such as attitude and satisfaction. 
+Path analysis, on the other hand, has its origins in biometrics and is intended to discover the causal link 
+between variables by drawing a path diagram [41]. 
+ 
+ 
+3.2.  What are dimensions or variables being investigated by AR researchers so far? 
+Figure 3 depicts 77 unique constructs/latent variables from hypothesized models. There are 184 
+unique constructs found in this study. Behavioral intention, usefulness, ease of use, attitude, user behavior, 
+and enjoyment are the most frequent items used in the hypothesized models. 
+ 
+ 
+ 
+ 
+Figure 3. Wordcloud depicts 77 unique constructs from all hypothesized models 
+
+Count of SEM
+35
+31
+30
+25
+20
+15
+11
+10
+6
+5
+2
+0
+eTAM
+eUTAUT
+SEM
+SOR
+TAMHedonic
+EaseOf Use
+Novelty ConceptualUnderstanding
+Performance Anxiety Quality
+Technology
+SocialInteraction
+Enjoyment
+Responses
+ntention
+Interactivity
+UseBehavior
+Knowledge Gain
+Immersion
+Voluntariness TaskSatisfaction Aesthetics
+Reievance
+Sublective Norms
+Control
+Behavioral Intentionsli-Efiacy
+TaskTechnologyFit
+Embedding
+Environmental Motivation
+Trust Fit
+Learning
+Playfulness
+Behaviora
+Game
+Involvement
+Value
+BenefitJob
+Attitude
+ExperienceVisual
+Presence
+Effort Richness
+Perceived
+Informativeness Soclal ActualUsage
+Engagement
+Simulated
+Purchase Intention
+Behavior
+System
+Augmentation
+information
+Education
+Usefulness
+Expectancy
+Service
+Image Entertainment      
+  
+ 
+       ISSN: 2502-4752 
+Indonesian J Elec Eng & Comp Sci, Vol. 28, No. 1, October 2022: 328-338 
+332 
+Figure 4 captures the top 14 dominant keywords in the collection of papers in this study. Aside from 
+“augmented reality”, TAM is the most popular term that the authors used for indexing their papers. In total, 
+this study extracted 319 keywords with 230 unique terms, indicating that there is a high variation of 
+topics/techniques used. However, in terms of their broad contents, the major theme of these collected papers 
+can be categorized as the “social marketing” theme as they were mainly focused on “Intention to Purchase” 
+or “Intention to Visit”. 
+ 
+ 
+ 
+ 
+Figure 4. Frequency of keywords extracted from publications 
+ 
+ 
+3.3.  How do researchers of prior AR studies communicate with end-users for evaluation? 
+Table 1 reports the communication channels used to gather data from respondents. Results showed 
+that there is a fair balance between the direct (45.28%) and indirect (50.94%) methods. Here, the indirect 
+method means that the research teams did not contact participants directly (e.g., lab setting, or field study). 
+Instead, they contact users via online channels (e.g., social network, email, discussion group). On the other 
+hand, the direct method requires subjects to be at the site of the study for the experiment.  
+ 
+ 
+Table 1. Communication channels to collect data from respondents 
+Communication channel 
+Count 
+Percentage 
+Indirect 
+27 
+50.94 
+Direct  
+24 
+45.28 
+Direct and Indirect 
+2 
+3.77 
+Total 
+53 
+100 
+ 
+ 
+Figure 5 depicts the spatial locations of authors researching AR utilizing the SEM method across the 
+globe. It can be observed that most publications were conducted in the United States although this country 
+was suffered heavily from the COVID-19 pandemic. However, 8 out of 10 papers utilized the indirect 
+research method to recruit and gather data, meaning that the study was conducted remotely, and opinions 
+were collected through online tools. 
+ 
+ 
+
+'SocialMarketing':AugmentedRealityappearsmostoften.
+AugmentedReality
+TechnologyAcceptance Model
+BehavioralIntentions
+Pokemon Go
+VirtualReality
+Social Marketing
+GeneralizedStructuredComponentAnalysis
+Presence
+TAM
+TechnologyAdoption
+Interactivity
+User Experience
+MobileAugmented Reality
+A-Frame
+MobileAugmented RealityApplications
+0
+5
+10
+15
+20
+25
+30
+35
+40
+Social MarketingIndonesian J Elec Eng & Comp Sci  
+ISSN: 2502-4752 
+ 
+ 
+A systematic review of structural equation modeling in augmented reality applications (Vinh The Nguyen) 
+333 
+ 
+ 
+Figure 5. Spatial locations of authors researching on AR utilizing SEM in 2020-2021 
+ 
+ 
+3.4.  How many participants are typically involved in a study? Would this number still be considered 
+appropriate from the literature? 
+Figure 6 shows the distribution of sample size across peer-reviewed papers. The whisker plot 
+indicates that on average the sample size (the number of participants) who took part in the studies was 
+approximately 300 subjects considering 4 extreme values (or outliers). The minimum sample size is 9 and the 
+maximum is 1,566. The median indicates that most papers recruited around 200 users for their studies. When 
+the four extreme values were not considered, the average sample size for direct communication with 
+participant was 142 (median=113, range=340, min=24, max=364), and indirect method was 286 
+(median=302, range=710, min=9, max=719). 
+ 
+ 
+ 
+ 
+Figure 6. Distribution of sample size in the peer-reviewed papers 
+ 
+ 
+Sample size is a debating subject in the literature. As such, the determination of sample size varies 
+from study to study. Some researchers advocate a minimum sample size of 100–200 per a study, an 
+acceptable sample size can range between 300 and 500, or with criteria such as acceptable of five cases per 
+free parameter, moderate of ten cases per free parameter [12], and ideal of 20 instances per free parameter in 
+the model. Kock and Hadaya [42] proposed a technique for determining an adequate sample size based on 
+“inverse square root” and “gamma-exponential” approaches which were adapted by Nikhashemi et al. [43] 
+included in this study. To some extent, Figure 6 reflects the balance of sample size recommendation in the 
+literature. Interestingly, the median sample size calculated in this study (Median=200) was aligned with the 
+findings based on reviews of studies in different research areas, including operations management, education 
+and psychology. 
+ 
+3.5.  What are the main drawbacks of the AR studies? Do they suffer from the COVID-19 pandemic 
+Table 2 reports the frequency of limitations addressed by authors in the collected publications. The 
+most common flaw that needs to be examined further in future studies is the failure to incorporate additional 
+external components (39.62%) in the postulated model, followed by convenience sampling (35.85%), multi-
+level analysis (32.08%) and limited to one region (30.19%). In terms of convenience sampling drawback, 
+
+United States
+10
+Germany
+n
+Taiwan
+4
+Greece
+4
+UnitedKingdom
+China.
+SouthKorea
+Indonesia
+2
+Thailand
+2
+Romania
+2
+Vietnam
+2
+Australia
+2
+France
+1
+Italy
+1
+Spain
+1
+HongKong
+1
+Ireland
+1
+Turkey
+Netherlands
+1
+India
+1
+Oman
+1
+Portugal
+1
+Malaysia
+1
+Powered by Bing
+Tom,WikipediaDistribution of sample size across publications
+1800
+1600
+·1566
+1400
+1200
+:1183
+:1192
+1000
+800
+719
+600
+400
+412
+X298.8113208
+200
+200
+68      
+  
+ 
+       ISSN: 2502-4752 
+Indonesian J Elec Eng & Comp Sci, Vol. 28, No. 1, October 2022: 328-338 
+334 
+many authors acknowledged that they used the non-probability method to acquire sample data through their 
+networks of interest. As such, their reports/findings cannot be generalized to the population.  
+ 
+ 
+Table 2. Frequency of limitations addressed by the authors in the collected publications 
+Limitations 
+References 
+Not consider other factors (21) 
+[25], [26], [30], [31], [32], [38], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], 
+[55], [56], [57] 
+Convenience sampling (19) 
+[32], [35], [37], [40], [43], [45], [46], [50], [52], [53], [55], [57], [58], [59], [60], [61], [62], [63], 
+[64] 
+Multi levels analysis (17) 
+[25], [37], [44], [45], [46], [48], [49], [51], [55], [56], [59], [62], [65], [66], [67], [68], [69] 
+Limited to one region (16) 
+[30], [32], [33], [37], [38], [46], [47], [48], [49], [54], [56], [58], [59], [63], [68], [70]  
+Tailored to a specific AR product 
+(14) 
+[45], [46], [47], [52], [53], [54], [57], [61], [62], [64], [65], [67], [68], [70]  
+Small Sample Size (10) 
+[30], [32], [33], [38], [40], [47], [50], [54], [60], [71] 
+Short term effect (10) 
+[29], [31], [38], [43], [45], [58], [63], [65], [69], [72] 
+Not specified (9) 
+[34], [41], [50], [73], [74], [75], [76], [77], [78] 
+Only Intention Model (6) 
+[31], [51], [52], [56], [58], [79] 
+Lack of AR features (6) 
+[25], [29], [32], [48], [63], [71]  
+Lack of functions (4) 
+[25], [26], [29], [32] 
+Self-Administered Survey (3) 
+[58], [66], [79] 
+Use Videos for demonstrations (3) 
+[25], [26], [65] 
+Technical challenges (2) 
+[27], [28] 
+Standardized tools (2) 
+[29], [52] 
+Single Analysis technique (2) 
+[33], [48] 
+Lab setting (2) 
+[55], [64] 
+Not consider privacy concerns (2) 
+[25], [60] 
+Others (8) 
+[25], [26], [29], [32], [56], [59], [58], [70] 
+ 
+ 
+Along with convenience sampling, limited study to one region is another shortcoming that is often 
+mentioned with non-probability method limitation. Unlike convenience sampling drawback that subjects may 
+come from different parts of the world, the regional issue was arising where the study was intentionally 
+designed for a specific region through a case study or in the lab setting [55], [64]. A large portion of the 
+published work was carried out with the help of pre-existing AR products. This evaluation includes examples 
+such as IKEA Place, YouCam Makeup, and Pokémon Go. Participants were asked if they had any experience 
+with these AR apps, and if so, they were encouraged to take part in the survey. Furthermore, the authors' 
+capacity to extend the study to additional products/services was limited because they did not have control or 
+flexibility over the AR apps. 
+The results show that though the sample size was a sufficiently addressed problem by the 
+researchers, the proportion of this limitation was just 18.87%. Without considering publications that did not 
+report limitations in their work (i.e., not specified (9)), 77.27% (34/44 papers) of the research group justified 
+their sample size using an analytical tool/method, a sample size recommendation in the literature, and the use 
+of PLS-SEM, which can work with small sample sizes. As a result, sample estimation was deemed sufficient. 
+Another issue worth mentioning is the short-term effect addressed by 10 author groups (18.87%). The short-
+term impact was explained by the fact that the experiments were only conducted for a limited period. As a 
+result, the theorized models can only explain variables impacting user behavior at that point in time. The 
+authors emphasized that because technology has evolved drastically over the years, the question of whether 
+their proposed models stand up remained unresolved. In addition, people's perspectives shift throughout time 
+as they gain experience [36], as a consequence, long-term research was suggested to validate the models.  
+In terms of the indirect method to conduct an experiment with users, four studies administered their 
+AR applications through video demonstrations [25], [26], [31], [65]. In this regard, instead of asking 
+participants to download or use the AR apps directly, the authors created videos demonstrating the features of 
+their studied AR apps. Based on the evidence of previous studies using video depictions of AR prototypes 
+[80], [81], these authors argued that the technology itself was not available for participants to interact with at 
+the time, and the purpose of the hypothesized models was to examine the influential factors that affect 
+behavioral intention before releasing the actual AR product to the market. As such in this category, studies in 
+[26], [29], [52] recommended that there is a need to have a tool or new evaluation method to overcome the 
+current issue. 
+In summary, compared with previous studies [16]-[19], this study has some similarities and 
+differences as: First, it is the selection of model, our report also shows similar results, that is, many different 
+types of models and variables are applied to the research. There has not yet been a general consensus set to 
+guide new researchers to follow. The difference is that the variables in this study revolve around technology 
+
+Indonesian J Elec Eng & Comp Sci  
+ISSN: 2502-4752 
+ 
+ 
+A systematic review of structural equation modeling in augmented reality applications (Vinh The Nguyen) 
+335 
+rather than ecology, social science, psychology, and management. Second is the issue of limitations. While 
+similar studies only listed restrictions that exist in articles, our study quantified these limitations by specific 
+numbers and arranges them in descending order. As such, interested researchers can rely on it to cover the 
+information more broadly. The third consideration is the study’s time span. This investigation was carried out 
+in the context of digital transformation and the influence of COVID-19. Many new factors emerge and exert 
+effect that have received little consideration in prior research (see Figure 3). Summarizing these factors will 
+help researchers have more options instead of reading different articles. And finally, by synthesizing how the 
+experiments were carried out during the pandemic, not only new researchers can adapt prior evaluation 
+approach in the current situations but also improve them in the subsequent studies. 
+ 
+ 
+4. 
+CONCLUSION 
+This paper presented a systematic review of the use of SEM in AR studies during the COVID-19 
+pandemic. The PRISMA model was adapted as a guideline for doing the research. Five data sources were 
+used for data retrieval. After a series of preprocessing steps, 53 publications were included in the study. The 
+results showed that authors used a variety of external factors to form the generative hypothesized models 
+(SEM), followed by the extension of TAM. The diversity of external factors indicated that there is no 
+consensus among AR scholars for using common factors influencing AR adoption, thus opening a huge 
+potential research gap for the AR community. Interestingly, United States was the most active country in 
+conducting AR studies during the Covid-19 pandemic, however 80% of its studies were conducted through 
+indirect communication channels. Hence, they were not affected by the pandemic. A large portion of AR 
+studies focused on understanding factors influencing user behavioral toward using third-party AR apps. As 
+such, participants were required to download and use the apps then answer the survey questionnaires. Sample 
+size, in this regard, cannot be excused due to social distancing. Only few studies examined user behavioral 
+through developed AR apps and the corresponding authors suggested that there is a need to have an 
+alternative approach to conduct user study rather than the traditional face-to-face fashion. Watching two 
+separate videos (one with AR and one without AR) was currently be used as an alternative method to 
+alleviate the issue but not a plausible approach in the long run. Therefore, this research gap remains open and 
+needs to be addressed in further studies. Thus, the outcomes of this study can be used as a reference guideline 
+for researchers in similar studies where there is a lack of theoretical framework for assessment, particular in 
+electrical engineering and computer science. 
+ 
+ 
+ACKNOWLEDGEMENTS 
+This research is supported by project T2022-07-09 undertaken at the TNU–University of 
+Information and Communication Technology, Thai Nguyen, Vietnam. 
+ 
+ 
+REFERENCES 
+[1] 
+K. Awang, S. N. W. Shamsuddin, I. Ismail, N. A. Rawi, and M. M. Amin, “The usability analysis of using augmented reality for 
+linus students,” Indonesian Journal of Electrical Engineering and Computer Science, vol. 13, no. 1, pp. 58–64, 2019, doi: 
+10.11591/ijeecs.v13.i1.pp58-64. 
+[2] 
+A. Ihsan, N. Fadillah, and C. R. Gunawan, “Acehnese traditional clothing recognition based on augmented reality using hybrid 
+tracking method,” Indonesian Journal of Electrical Engineering and Computer Science, vol. 20, no. 2, pp. 1030–1036, 2020, doi: 
+10.11591/ijeecs.v20.i2.pp1030-1036. 
+[3] 
+A. N. Rosman, N. A. Samsudin, A. Ismail, M. S. Aripin, and S. K. A. Khalid, “Augmented reality application for location finder 
+guidance,” Indonesian Journal of Electrical Engineering and Computer Science, vol. 13, no. 3, pp. 1237–1242, 2019, doi: 
+10.11591/ijeecs.v13.i3.pp1237-1242. 
+[4] 
+R. T. Azuma, “A survey of augmented reality,” Foundations and Trends in Human-Computer Interaction, vol. 8, no. 2–3, pp. 73–
+272, 2014, doi: 10.1561/1100000049. 
+[5] 
+reportthinker.com, “Augmented Reality Market with COVID-19 Impact Analysis, by Device Type, Offering, Application, 
+Technology 
+And 
+Geography 
+- 
+Global 
+Forecast 
+to 
+2026,” 
+2021. 
+[Online]. 
+Available: 
+https://www.reportlinker.com/p05026084/Augmented-Reality-Market-by-Offering-and-Software-Device-Type-Application-and-
+Geography-Global-forecast-to.html. 
+[6] 
+K. Nesenbergs, V. Abolins, J. Ormanis, and A. Mednis, “Use of augmented and virtual reality in remote higher education: a 
+systematic umbrella review,” Education Sciences, vol. 11, no. 1, p. 8, Dec. 2020, doi: 10.3390/educsci11010008. 
+[7] 
+F. Redzuan, A.-N. A. Khairuddin, and N. A. Daud, “Emotional augmented reality-based mobile learning design elements: a 
+kansei engineering approach,” Indonesian Journal of Electrical Engineering and Computer Science, vol. 14, no. 1, p. 413, Apr. 
+2019, doi: 10.11591/ijeecs.v14.i1.pp413-420. 
+[8] 
+S. Nasir, M. N. Zahid, T. A. Khan, K. Kadir, and S. Khan, “Augmented reality an economical solution for engineers and 
+designers,” Indonesian Journal of Electrical Engineering and Computer Science, vol. 17, no. 2, pp. 833–844, 2019, doi: 
+10.11591/ijeecs.v17.i2.pp834-844. 
+[9] 
+A. Dünser, R. Grasset, and M. Billinghurst, “Survey of evaluation techniques used in augmented studies,” in ACM SIGGRAPH 
+ASIA 2008 Courses, SIGGRAPH Asia’08, 2008, no. 64 3, pp. 766–792, doi: 10.1145/1508044.1508049. 
+ 
+
+      
+  
+ 
+       ISSN: 2502-4752 
+Indonesian J Elec Eng & Comp Sci, Vol. 28, No. 1, October 2022: 328-338 
+336 
+[10] A. Dünser and M. Billinghurst, “Evaluating Augmented reality systems,” in Handbook of Augmented Reality, New York, NY: 
+Springer New York, 2011, pp. 289–307. 
+[11] D. Zhang, M. Wang, and J. G. Wu, “Design and Implementation of augmented reality for english language education,” in 
+Springer Series on Cultural Computing, 2020, pp. 217–234. 
+[12] R. B. Kline, “Principles and practice of structural equation modeling,” in Methodology in the Social Sciences, Guilford 
+publications, 2005, pp. 1–554. 
+[13] A. Aramja, O. Kamach, and R. Elmeziane, “Companies’ perception toward manufacturing execution systems,” International 
+Journal of Electrical and Computer Engineering (IJECE), vol. 11, no. 4, p. 3347, Aug. 2021, doi: 10.11591/ijece.v11i4.pp3347-
+3355. 
+[14] V. T. Nguyen, “The perceptions of social media users of digital detox apps considering personality traits,” Education and 
+Information Technologies, pp. 1–24, Mar. 2022, doi: 10.1007/s10639-022-11022-7. 
+[15] M. S. Arman, R. Akter, I. Mahmud, and T. Ramayah, “Modelling turn away intention of information technology professionals in 
+Bangladesh: A partial least squares approach,” International Journal of Electrical and Computer Engineering, vol. 10, no. 5, pp. 
+4973–4981, 2020, doi: 10.11591/IJECE.V10I5.PP4973-4981. 
+[16] Y. Fan et al., “Applications of structural equation modeling (SEM) in ecological studies: an updated review,” Ecological 
+Processes, vol. 5, no. 1, p. 19, Dec. 2016, doi: 10.1186/s13717-016-0063-3. 
+[17]  de J. Carvalho and O. F. Chima, “Applications of structural equation modeling in social sciences research,” American 
+International Journal of Contemporary Research, vol. Vol. 4, no. 1, pp. 6–11, 2014. 
+[18] R. C. MacCallum and J. T. Austin, “Applications of structural equation modeling in psychological research,” Annual Review of 
+Psychology, vol. 51, pp. 201–226, 2000, doi: 10.1146/annurev.psych.51.1.201. 
+[19] L. J. Williams, M. B. Gavin, and N. S. Hartman, “Structural equation modeling methods in strategy research: Applications and 
+issues, in Research methodology in strategy and management,” in Research Methodology in Strategy and Management, vol. 1, 
+2004, pp. 303–346. 
+[20] D. Moher, A. Liberati, J. Tetzlaff, and D. G. Altman, “Preferred reporting items for systematic reviews and meta-analyses: the 
+PRISMA statement,” Journal of clinical epidemiology, vol. 62, no. 10, pp. 1006–1012, 2009, doi: 10.1016/j.jclinepi.2009.06.005. 
+[21] H. Yousuf, M. Lahzi, S. A. Salloum, and K. Shaalan, “A systematic review on sequence-to-sequence learning with neural network 
+and its models,” International Journal of Electrical and Computer Engineering, vol. 11, no. 3, pp. 2315–2326, 2021, doi: 
+10.11591/ijece.v11i3.pp2315-2326. 
+[22] V. T. Nguyen, K. Jung, and V. Gupta, “Examining data visualization pitfalls in scientific publications,” Visual Computing for 
+Industry, Biomedicine, and Art, vol. 4, no. 1, p. 27, Dec. 2021, doi: 10.1186/s42492-021-00092-y. 
+[23] C. W. Okonkwo and A. Ade-Ibijola, “Chatbots applications in education: A systematic review,” Computers and Education: 
+Artificial Intelligence, vol. 2, p. 100033, 2021, doi: 10.1016/j.caeai.2021.100033. 
+[24] F. D. Davis, “A technology acceptance model for empirically testing new end-user information systems: Theory and results,” 
+Massachusetts Institute of Technology, 1985. 
+[25] V. T. Nguyen, K. Jung, and T. Dang, “BlocklyAR: A visual programming interface for creating augmented reality experiences,” 
+Electronics, vol. 9, no. 8, p. 1205, Jul. 2020, doi: 10.3390/electronics9081205. 
+[26] K. Jung, V. T. Nguyen, and J. Lee, “BlocklyXR: An interactive extended reality toolkit for digital storytelling,” Applied Sciences, 
+vol. 11, no. 3, p. 1073, Jan. 2021, doi: 10.3390/app11031073. 
+[27] K. Jung, V. T. Nguyen, D. Piscarac, and S.-C. Yoo, “Meet the virtual jeju dol harubang—the mixed vr/ar application for cultural 
+immersion in korea’s main heritage,” ISPRS International Journal of Geo-Information, vol. 9, no. 6, p. 367, Jun. 2020, doi: 
+10.3390/ijgi9060367. 
+[28] K. Jung, V. T. Nguyen, S.-C. Yoo, S. Kim, S. Park, and M. Currie, “PalmitoAR: the last battle of the U.S. civil war reenacted 
+using augmented reality,” ISPRS International Journal of Geo-Information, vol. 9, no. 2, p. 75, Jan. 2020, doi: 
+10.3390/ijgi9020075. 
+[29] T. A. Mikropoulos et al., “Acceptance and user experience of an augmented reality system for the simulation of sensory overload 
+in children with autism,” in 2020 6th International Conference of the Immersive Learning Research Network (iLRN), Jun. 2020, 
+pp. 86–92, doi: 10.23919/iLRN47897.2020.9155113. 
+[30] M. Yavuz, E. Çorbacıoğlu, A. N. Başoğlu, T. U. Daim, and A. Shaygan, “Augmented reality technology adoption: Case of a 
+mobile application in Turkey,” Technology in Society, vol. 66, 2021, doi: 10.1016/j.techsoc.2021.101598. 
+[31] C. Goebert and G. P. Greenhalgh, “A new reality: Fan perceptions of augmented reality readiness in sport marketing,” Computers 
+in Human Behavior, vol. 106, p. 106231, May 2020, doi: 10.1016/j.chb.2019.106231. 
+[32] R. G. Boboc, R. L. Chiriac, and C. Antonya, “How augmented reality could improve the student’s attraction to learn 
+mechanisms,” Electronics (Switzerland), vol. 10, no. 2, pp. 1–24, Jan. 2021, doi: 10.3390/electronics10020175. 
+[33] A. Elshafey, C. C. Saar, E. B. Aminudin, M. Gheisari, and A. Usmani, “Technology acceptance model for Augmented Reality and 
+Building Information Modeling integration in the construction industry,” Journal of Information Technology in Construction, vol. 
+25, pp. 161–172, Mar. 2020, doi: 10.36680/j.itcon.2020.010. 
+[34] E. Holdack, K. Lurie-Stoyanov, and H. F. Fromme, “The role of perceived enjoyment and perceived informativeness in assessing 
+the acceptance of AR wearables,” Journal of Retailing and Consumer Services, vol. 65, p. 102259, Mar. 2022, doi: 
+10.1016/j.jretconser.2020.102259. 
+[35] C. Papakostas, C. Troussas, A. Krouska, and C. Sgouropoulou, “User acceptance of augmented reality welding simulator in 
+engineering training,” Education and Information Technologies, vol. 27, no. 1, pp. 791–817, Jan. 2022, doi: 10.1007/s10639-020-
+10418-7. 
+[36] V. Venkatesh, M. G. Morris, G. B. Davis, and F. D. Davis, “User acceptance of information technology: Toward a unified view,” 
+MIS Quarterly: Management Information Systems, vol. 27, no. 3, pp. 425–478, 2003, doi: 10.2307/30036540. 
+[37] V. Saprikis, G. Avlogiaris, and A. Katarachia, “Determinants of the intention to adopt mobile augmented reality apps in shopping 
+malls among university students,” Journal of Theoretical and Applied Electronic Commerce Research, vol. 16, no. 3, pp. 491–
+512, Nov. 2021, doi: 10.3390/jtaer16030030. 
+[38] H. Alqahtani, M. Kavakli-Thorne, and M. Alrowaily, “The impact of gamification factor in the acceptance of cybersecurity 
+awareness augmented reality game (cybar),” Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial 
+Intelligence and Lecture Notes in Bioinformatics), vol. 12210 LNCS, pp. 16–31, 2020, doi: 10.1007/978-3-030-50309-3_2. 
+[39] A. Mehrabian and J. A. Russell, An Approach to Environmental Psychology. MIT Press, 1974. 
+[40] H. Qin, D. A. Peak, and V. Prybutok, “A virtual market in your pocket: How does mobile augmented reality (MAR) influence 
+consumer decision making?,” Journal of Retailing and Consumer Services, vol. 58, p. 102337, Jan. 2021, doi: 
+10.1016/j.jretconser.2020.102337. 
+
+Indonesian J Elec Eng & Comp Sci  
+ISSN: 2502-4752 
+ 
+ 
+A systematic review of structural equation modeling in augmented reality applications (Vinh The Nguyen) 
+337 
+[41] C. Pasalidou and N. Fachantidis, “Teachers’ perceptions towards the use of mobile augmented reality: the case of greek 
+educators,” in Advances in Intelligent Systems and Computing, vol. 1192 AISC, 2021, pp. 1039–1050. 
+[42] K. N and H. P, “Minimum sample size estimation in PLS-SEM: the inverse square root and gamma-exponential methods,” 
+Information Systems Journal, vol. 28, no. 1, p. 227, 2018. 
+[43] S. R. Nikhashemi, H. H. Knight, K. Nusair, and C. B. Liat, “Augmented reality in smart retailing: A (n) (A) Symmetric Approach 
+to continuous intention to use retail brands’ mobile AR apps,” Journal of Retailing and Consumer Services, vol. 60, p. 102464, 
+May 2021, doi: 10.1016/j.jretconser.2021.102464. 
+[44] X. Fan, Z. Chai, N. Deng, and X. Dong, “Adoption of augmented reality in online retailing and consumers’ product attitude: A 
+cognitive perspective,” Journal of Retailing and Consumer Services, vol. 53, p. 101986, Mar. 2020, doi: 
+10.1016/j.jretconser.2019.101986. 
+[45] P. Kowalczuk, C. Siepmann (née Scheiben), and J. Adler, “Cognitive, affective, and behavioral consumer responses to augmented 
+reality in e-commerce: A comparative study,” Journal of Business Research, vol. 124, pp. 357–373, 2021, doi: 
+10.1016/j.jbusres.2020.10.050. 
+[46] M. 
+Thongmak, 
+“Determinants 
+of 
+intention 
+to 
+play 
+Pokémon 
+Go,” 
+Heliyon, 
+vol. 
+6, 
+no. 
+12, 
+2020, 
+doi: 
+10.1016/j.heliyon.2020.e03895. 
+[47] Mailizar and R. Johar, “Examining students’ intention to use augmented reality in a project-based geometry learning 
+environment,” International Journal of Instruction, vol. 14, no. 2, pp. 773–790, 2021, doi: 10.29333/iji.2021.14243a. 
+[48] C. L. Chiu, H. C. Ho, T. Yu, Y. Liu, and Y. Mo, “Exploring information technology success of Augmented Reality Retail 
+Applications in retail food chain,” Journal of Retailing and Consumer Services, vol. 61, p. 102561, Jul. 2021, doi: 
+10.1016/j.jretconser.2021.102561. 
+[49] R. Vongurai, “Factors influencing experiential value toward using cosmetic AR try-on feature in Thailand,” Journal of 
+Distribution Science, vol. 19, no. 1, pp. 75–87, 2021, doi: 10.15722/jds.19.1.202101.75. 
+[50] F. Schuster, B. Engelmann, U. Sponholz, and J. Schmitt, “Human acceptance evaluation of AR-assisted assembly scenarios,” 
+Journal of Manufacturing Systems, vol. 61, pp. 660–672, Oct. 2021, doi: 10.1016/j.jmsy.2020.12.012. 
+[51] S. Han, J. H. Yoon, and J. Kwon, “Impact of experiential value of augmented reality: The context of heritage tourism,” 
+Sustainability (Switzerland), vol. 13, no. 8, p. 4147, Apr. 2021, doi: 10.3390/su13084147. 
+[52] M. Daassi and S. Debbabi, “Intention to reuse AR-based apps: The combined role of the sense of immersion, product presence 
+and perceived realism,” Information and Management, vol. 58, no. 4, 2021, doi: 10.1016/j.im.2021.103453. 
+[53] T. T. Haile and M. Kang, “Mobile Augmented reality in electronic commerce: investigating user perception and purchase intent 
+amongst educated young adults,” Sustainability, vol. 12, no. 21, p. 9185, Nov. 2020, doi: 10.3390/su12219185. 
+[54] T. Jung, M. Claudia Tom Dieck, H. Lee, and N. Chung, “Relationships among beliefs, attitudes, time resources, subjective norms, 
+and intentions to use wearable augmented reality in art galleries,” Sustainability (Switzerland), vol. 12, no. 20, pp. 1–17, Oct. 
+2020, doi: 10.3390/su12208628. 
+[55] A. R. Smink, E. A. van Reijmersdal, G. van Noort, and P. C. Neijens, “Shopping in augmented reality: The effects of spatial 
+presence, personalization and intrusiveness on app and brand responses,” Journal of Business Research, vol. 118, pp. 474–485, 
+Sep. 2020, doi: 10.1016/j.jbusres.2020.07.018. 
+[56] T. H. Jung, S. Bae, N. Moorhouse, and O. Kwon, “The impact of user perceptions of AR on purchase intention of location-based 
+AR navigation systems,” Journal of Retailing and Consumer Services, vol. 61, p. 102575, Jul. 2021, doi: 
+10.1016/j.jretconser.2021.102575. 
+[57] S. Bueno, M. D. Gallego, and J. Noyes, “Uses and gratifications on augmented reality games: an examination of Pokémon Go,” 
+Applied Sciences, vol. 10, no. 5, p. 1644, Mar. 2020, doi: 10.3390/app10051644. 
+[58] P. Saleme, T. Dietrich, B. Pang, and J. Parkinson, “A gamified approach to promoting empathy in children,” Journal of Social 
+Marketing, vol. 10, no. 3, pp. 321–337, Jun. 2020, doi: 10.1108/JSOCM-11-2019-0204. 
+[59] Y. Qin, “Attractiveness of game elements, presence, and enjoyment of mobile augmented reality games: The case of Pokémon 
+Go,” Telematics and Informatics, vol. 62, p. 101620, Sep. 2021, doi: 10.1016/j.tele.2021.101620. 
+[60] H. Lee, Y. Xu, and A. Porterfield, “Consumers’ adoption of AR-based virtual fitting rooms: from the perspective of theory of 
+interactive media effects,” Journal of Fashion Marketing and Management: An International Journal, vol. 25, no. 1, pp. 45–62, 
+Feb. 2021, doi: 10.1108/JFMM-05-2019-0092. 
+[61] M. Park and J. Yoo, “Effects of perceived interactivity of augmented reality on consumer responses: A mental imagery 
+perspective,” Journal of Retailing and Consumer Services, vol. 52, p. 101912, Jan. 2020, doi: 10.1016/j.jretconser.2019.101912. 
+[62] E. (Christine) Sung, “The effects of augmented reality mobile app advertising: Viral marketing via shared social experience,” 
+Journal of Business Research, vol. 122, pp. 75–87, Jan. 2021, doi: 10.1016/j.jbusres.2020.08.034. 
+[63] M. Dehghani, S. H. (Mark) Lee, and A. Mashatan, “Touching holograms with windows mixed reality: Renovating the consumer 
+retailing services,” Technology in Society, vol. 63, p. 101394, Nov. 2020, doi: 10.1016/j.techsoc.2020.101394. 
+[64] A. Leopardi et al., “X-reality technologies for museums: a comparative evaluation based on presence and visitors experience 
+through user studies,” Journal of Cultural Heritage, vol. 47, pp. 188–198, Jan. 2021, doi: 10.1016/j.culher.2020.10.005. 
+[65] J. Brannon Barhorst, G. McLean, E. Shah, and R. Mack, “Blending the real world and the virtual world: Exploring the role of 
+flow in augmented reality experiences,” Journal of Business Research, vol. 122, pp. 423–436, Jan. 2021, doi: 
+10.1016/j.jbusres.2020.08.041. 
+[66] X.-F. Lin, D. Tang, W. Shen, Z.-M. Liang, Y. Tang, and C.-C. Tsai, “Exploring the relationship between perceived technology-
+assisted teacher support and technology-embedded scientific inquiry: the mediation effect of hardiness,” International Journal of 
+Science Education, vol. 42, no. 8, pp. 1225–1252, May 2020, doi: 10.1080/09500693.2020.1755475. 
+[67] T. L. Huang, “Restorative experiences and online tourists’ willingness to pay a price premium in an augmented reality 
+environment,” Journal of Retailing and Consumer Services, vol. 58, 2021, doi: 10.1016/j.jretconser.2020.102256. 
+[68] C.-H. Hsiao and K.-Y. Tang, “Who captures whom – Pokémon or tourists? A perspective of the Stimulus-Organism-Response 
+model,” International Journal of Information Management, vol. 61, p. 102312, Dec. 2021, doi: 10.1016/j.ijinfomgt.2021.102312. 
+[69] S. Park and B. Stangl, “Augmented reality experiences and sensation seeking,” Tourism Management, vol. 77, p. 104023, Apr. 
+2020, doi: 10.1016/j.tourman.2019.104023. 
+[70] C. Hinsch, R. Felix, and P. A. Rauschnabel, “Nostalgia beats the wow-effect: Inspiration, awe and meaningful associations in 
+augmented reality marketing,” Journal of Retailing and Consumer Services, vol. 53, 2020, doi: 10.1016/j.jretconser.2019.101987. 
+[71] H. C. K. Lin, Y. H. Lin, T. H. Wang, L. K. Su, and Y. M. Huang, “Effects of incorporating augmented reality into a board game 
+for high school students’ learning motivation and acceptance in health education,” Sustainability (Switzerland), vol. 13, no. 6, p. 
+3333, Mar. 2021, doi: 10.3390/su13063333. 
+ 
+ 
+
+      
+  
+ 
+       ISSN: 2502-4752 
+Indonesian J Elec Eng & Comp Sci, Vol. 28, No. 1, October 2022: 328-338 
+338 
+[72] D. Urbano, M. de Fátima Chouzal, and M. T. Restivo, “Evaluating an online augmented reality puzzle for DC circuits: Students’ 
+feedback and conceptual knowledge gain,” Computer Applications in Engineering Education, vol. 28, no. 5, pp. 1355–1368, 
+2020, doi: 10.1002/cae.22306. 
+[73] E. Lacka, “Assessing the impact of full-fledged location-based augmented reality games on tourism destination visits," Current 
+Issues in Tourism ", vol. 28, no. 5, pp. 1355–1368, 2020. 
+[74] S. H. Hadi et al., “Developing augmented reality-based learning media and users’ intention to use it for teaching accounting 
+ethics,” Education and Information Technologies, vol. 27, no. 1, pp. 643–670, 2022, doi: 10.1007/s10639-021-10531-1. 
+[75] G. D. Voinea, C. C. Postelnicu, M. Duguleana, G. L. Mogan, and R. Socianu, “Driving performance and technology acceptance 
+evaluation in real traffic of a smartphone-based driver assistance system,” International Journal of Environmental Research and 
+Public Health, vol. 17, no. 19, pp. 1–20, 2020, doi: 10.3390/ijerph17197098. 
+[76] H.-N. Do, W. Shih, and Q.-A. Ha, “Effects of mobile augmented reality apps on impulse buying behavior: An investigation in the 
+tourism field,” Heliyon, vol. 6, no. 8, p. e04667, Aug. 2020, doi: 10.1016/j.heliyon.2020.e04667. 
+[77] A. Kumar and A. Mantri, “Evaluating the attitude towards the intention to use ARITE system for improving laboratory skills by 
+engineering educators,” Education and Information Technologies, vol. 27, no. 1, pp. 671–700, Jan. 2022, doi: 10.1007/s10639-
+020-10420-z. 
+[78] H. Y. Chang, J. C. Liang, and C. C. Tsai, “Students’ context-specific epistemic justifications, prior knowledge, engagement, and 
+socioscientific reasoning in a mobile augmented reality learning environment,” Journal of Science Education and Technology, 
+vol. 29, no. 3, pp. 399–408, 2020, doi: 10.1007/s10956-020-09825-9. 
+[79] D. Harborth and S. Pape, “Investigating privacy concerns related to mobile augmented reality Apps – A vignette based online 
+experiment,” Computers in Human Behavior, vol. 122, p. 106833, Sep. 2021, doi: 10.1016/j.chb.2021.106833. 
+[80] A. C. Haugstvedt and J. Krogstie, “Mobile augmented reality for cultural heritage: A technology acceptance study,” ISMAR 2012 
+- 11th IEEE International Symposium on Mixed and Augmented Reality 2012, Science and Technology Papers, pp. 247–255, 
+2012, doi: 10.1109/ISMAR.2012.6402563. 
+[81] J. Cheon, S. Lee, S. M. Crooks, and J. Song, “An investigation of mobile learning readiness in higher education based on the 
+theory of planned behavior,” Computers & Education, vol. 59, no. 3, pp. 1054–1064, Nov. 2012, doi: 
+10.1016/j.compedu.2012.04.015. 
+ 
+ 
+BIOGRAPHIES OF AUTHORS  
+ 
+ 
+Dr. Vinh The Nguyen 
+ 
+ 
+ 
+ is currently a lecturer at the Faculty of Information 
+Technology, University of Information and Communication Technology. He is also a senior 
+visiting lecturer at FPT University Greenwich, Hanoi branch. He graduated with a master's 
+degree in information systems management from Oklahoma State University, USA (under 
+scholarship 322). He completed his PhD program under Project 911 in 2020 at Texas Tech 
+University, USA. His main research interests are Computer Vision, Computer Visualization, 
+and Computer in Human Behavior. He has authored or coauthored more than 35 publications 
+with 10 H-index and more than 250 citations. He can be contacted at email: 
+[email protected]. 
+ 
+ 
+ 
+Chuyen Thi Hong Nguyen 
+ 
+ 
+ 
+ is currently a lecturer at the Faculty of Primary 
+Education, Thai Nguyen University of Education, Vietnam. She graduated with a master's 
+degree in Theory and History of Education from Hanoi University of Education, Vietnam 
+(2008). She completed her PhD program in 2016 at The Vietnam Institute of educational 
+Sciences, Vietnam. Her main research interests are method teaching, assessment in primary 
+education, computational thinking, learning style, and augmented reality in education. She can 
+be contacted at email: [email protected]. 
+ 
+ 
+ 
+ 
+
+pp

FtE3T4oBgHgl3EQftQtg/content/tmp_files/2301.04674v1.pdf.txt

@@ -0,0 +1,1488 @@
+Prepared for submission to JHEP
+IPARCOS-23-002
+Late vacuum choice and slow roll approximation in
+gravitational particle production during reheating
+Jose A. R. Cembranos,a Luis J. Garay,a Álvaro Parra-Lópeza and Jose M. Sánchez
+Velázquezb
+aDepartamento de Física Teórica and IPARCOS, Facultad de Ciencias Físicas,
+Universidad Complutense de Madrid, Ciudad Universitaria, 28040 Madrid, Spain
+bInstituto de Física Teórica UAM/CSIC, c/ Nicolás Cabrera 13-15,
+Cantoblanco, 28049, Madrid, Spain
+E-mail: [email protected], [email protected], [email protected],
+[email protected]
+Abstract: In the transition between inflation and reheating, the curvature scalar typically
+undergoes oscillations which have significant impact on the density of gravitationally
+produced particles. The commonly used adiabatic vacuum prescription for the extraction
+of produced particle spectra becomes a non-reliable definition of vacuum in the regimes for
+which this oscillatory behavior is important. In this work, we study particle production for
+a scalar field non-minimally coupled to gravity, taking into account the complete dynamics
+of spacetime during inflation and reheating. We derive an approximation for the solution
+to the mode equation during the slow-roll of the inflaton and analyze the importance of
+Ricci scalar oscillations in the resulting spectra. Additionally, we propose a prescription for
+the vacuum that allows to safely extrapolate the result to the present, given that the test
+field interacts only gravitationally. Lastly, we calculate the abundance of dark matter this
+mechanism yields and compare it to observations.
+Keywords: Cosmology of Theories beyond the SM, Effective Field Theories, Classical
+Theories of Gravity
+arXiv:2301.04674v1  [gr-qc]  11 Jan 2023
+
+Contents
+1
+Introduction
+1
+2
+Dynamics of a scalar field in flat FLRW cosmologies
+3
+3
+Background dynamics
+5
+3.1
+Inflationary era - Slow-roll approximation
+6
+3.2
+Late reheating
+7
+4
+Particle production
+8
+4.1
+Solution to the mode equation
+8
+4.2
+Choice of reference vacua
+9
+4.3
+Slow-roll approximation for the solution to the mode equation
+11
+4.4
+Adiabaticity and oscillations
+16
+5
+Spectra of particles and total density
+17
+6
+Conclusions
+21
+A Parameters
+23
+1
+Introduction
+The theory of quantum fields in curved spacetimes accommodates a plethora of unexpected
+phenomena such as Hawking radiation [1], the Unruh effect [2], or entanglement across
+horizons [3–6], that have changed our perspective on the interplay between quantum fields
+and gravity. Gravitational particle production due to the spacetime dynamics [7, 8] is one of
+these phenomena and can be particularly important during the early stages of the universe,
+since it may be able to explain the dark matter abundance, as it has been extensively
+discussed in the literature. The rapidly evolving spacetime during inflation [9–11] and
+the consequent transient to reheating [12–16] produce a significant abundance of particles
+for any field that is coupled to the geometry. Since this is the only requirement for this
+process to occur, it is of particular interest to analyze it from the perspective of dark
+matter production mechanisms. Because of the absence of interactions with other fields, the
+abundance of dark matter produced in the early universe due to the expansion of spacetime
+is not diluted as a consequence of thermalization with other fields. It remains then as a relic
+abundance, so that this mechanism alone can in fact explain current observations. This
+has been mostly explored for scalar fields that are non-minimally coupled to gravity in a
+myriad of works, such as [17–20] for supermassive dark matter candidates (WIMPZillas),
+or, more recently, in references [21–24], where the importance of the oscillatory behavior of
+– 1 –
+
+the background geometry was incorporated. On the other hand, gravitational production
+of more general fields, such as fermion and vector fields, has also been analyzed in [25, 26].
+Usually, the dark matter candidate is regarded as a spectator field [27, 28] which does not
+source gravity, and with no direct coupling to the inflationary fields. However, it is generally
+non-minimally coupled to the geometry via the curvature scalar, and interactions with
+other fields are disregarded. In all these works, it is customary to make use of the adiabatic
+prescription to define the vacuum state of the dark matter field in order to calculate the
+gravitational production. This definition seems to hold after a few oscillations of the
+inflaton in the reheating stage, but only in the case of very large masses of the dark matter
+candidate. In the regime of low masses, however, this vacuum provides a correct prediction
+only when considering very late times, after many oscillations have occured. Importantly,
+this oscillating behavior influences gravitational production [24]. It is worth mentioning
+that the type of dark matter produced in this way is adiabatic [23, 29], and therefore the
+observational constraints on isocurvature perturbations [30] do not have to be considered.
+In this work, we study the gravitational production of a massive scalar field ϕ described
+by a Klein-Gordon action that includes a non-minimal coupling to the Ricci curvature scalar
+R through a term of the form ξRϕ2. The strength of this coupling is determined by the
+parameter ξ. In an attempt to accommodate the arguments put forward in refs. [24, 31–33]
+concerning vacuum instability, overproduction, and quantum cosmology analyses, we restrict
+ourselves to the range 1/6 ≤ ξ ≤ 1 for the coupling constant ξ. As inflationary model, we
+consider a single inflaton field φ that slowly rolls down a quadratic potential and starts
+oscillating around its minimum, leading then to a reheating phase. The dynamics for the
+inflaton is analytically solved at the onset of inflation, while the transition to the reheating
+epoch is modeled numerically. Our scalar field is assumed to be in the Bunch-Davies
+vacuum state when inflation starts. In order to extract the gravitational production, the
+Klein-Gordon equation of the field ϕ is solved from that point in time until well inside
+the reheating era. This moment effectively corresponds to the time where the dynamics
+enters the adiabatic regime and particle production becomes negligible. Moreover, one also
+needs to provide a definition of vacuum for this instant, for which the adiabatic prescription
+is usually adopted. We discuss its validity and introduce as well an averaged vacuum
+that produces the same density of particles but allows to obtain the correct result much
+earlier than the time at which adiabaticity is reached. This is particularly helpful when
+considering masses way below the inflaton mass for our scalar field, which are of great
+interest concerning dark matter candidates. Also, we stress the importance of taking into
+account the first few hundreds of oscillations of the inflaton in the final prediction and
+present the results in the form of spectra and total density of produced particles for different
+values of the scalar field mass m and its coupling ξ to the Ricci scalar.
+The remainder of this paper is organized as follows. In section 2, we introduce the field
+that is coupled to the expanding geometry, and work out the formalities of Bogoliubov-
+like particle production in this context. In order to determine the complete form of the
+mode equation, we need to provide the background dynamics coming from the particular
+inflationary model in consideration, which we do in section 3. With all these ingredients, we
+explore the gravitational production for the scalar field in section 4, analyzing the solution
+– 2 –
+
+to the mode equation in the different regimes and studying the influence of the oscillations
+of the curvature scalar in the final result. Moreover, we discuss the importance of the
+vacuum choice when obtaining the numer density of produced particles. Lastly, we present
+our results in the form of spectra and total density of particles in section 5 and elaborate
+our conclusions in section 6.
+Notation. We set MP =
+√
+G, ℏ = c = kB = 1, and use the metric signature (−, +, +, +).
+Furthermore, greek indices µ, ν run from 0 to 3, while latin indices i, j run from 1 to 3.
+2
+Dynamics of a scalar field in flat FLRW cosmologies
+We will consider a massive scalar field ϕ non-minimally coupled to gravity in a Friedmann-
+Lemaître-Robertson-Walker (FLRW) spacetime with vanishing spatial curvature [34–40].
+We will not consider here any coupling of the derivatives of the scalar field (see [41]).
+The dynamics of our scalar field is encoded in the action
+S = −1
+2
+�
+d4x√−g
+�
+∂µϕ∂µϕ +
+�
+m2 + ξR
+�
+ϕ2�
+,
+(2.1)
+where g is the determinant of the metric, m is the bare mass of the field, and ξ is the
+coupling to the Ricci curvature scalar R. The geometry is determined by the spatially flat
+FLRW line element
+ds2 = a2(η)
+�
+−dη2 + dx2 + dy2 + dz2�
+,
+(2.2)
+where we have considered Cartesian coordinates for the flat spatial sections, and η is the
+conformal time, related to cosmological time by a(η)dη = dt.
+It is convenient to work with the auxiliary field
+χ(η, x) = a(η)ϕ(η, x),
+(2.3)
+whose equation of motion can be obtained from the action (2.1),
+χ′′(η, x) −
+�
+∆ + a′′(η)
+a(η) − a2(η)
+�
+m2 + ξR
+��
+χ(η, x) = 0,
+(2.4)
+where ∆ is the Laplace operator, the prime denotes derivative with respect to conformal
+time, and R = 6a′′/a3.
+We can use the eigenfunctions of the Laplace operator, which in our case are Fourier
+modes, as a basis of functions to expand the scalar field χ,
+χ(η, x) =
+�
+d3k
+(2π)2/3
+�akvk(η) + a∗
+−kv∗
+k(η)
+� eikx,
+(2.5)
+where the coefficients ak, a∗
+k become creation and annihilation operators upon quantization
+of the field, with the standard commutation relations [42–45]. The time-dependent mode
+functions vk(η) and v∗
+k(η) satisfy a harmonic oscillator equation
+v′′
+k(η) + ω2
+k(η)vk(η) = 0,
+(2.6)
+– 3 –
+
+with k =
+√
+k2 and a time-dependent frequency
+ω2
+k(η) = k2 + a2(η)
+�
+m2 + (ξ − 1/6)R(η)
+�
+.
+(2.7)
+The solutions to (2.6) have to fulfill the normalization condition
+vkv′ ∗
+k − v′
+kv∗
+k = i,
+(2.8)
+so that they are compatible with the standard commutation relations of creation and
+annihilation operators.
+For a given evolution of the background geometry, encoded in the scale factor a(η) and
+the Ricci scalar R(η), both (2.8) and (2.6) are sufficient to determine vk(η), v∗
+k(η), which is
+a basis of the space of solutions of the mode equations. Since any other solution can be
+expressed as a linear combination of vk(η) and v∗
+k(η), any two sets of solutions vk(η) and
+uk(η) must be related by uk = αkvk + βkv∗
+k, where normalization (2.8) on the temporal
+modes implies the relation |αk|2 − |βk|2 = 1 for the complex coefficients αk and βk, which
+are known as Bogoliubov coefficients [42]. Note that the expansion (2.5) can be carried out
+using either basis of solutions.
+Upon quantization of the field, both sets of coefficients ak and bk (associated with the
+basis vk and uk, respectively) and their complex conjugates become operators that give rise
+to two different definitions on quanta and vacua [43],
+ˆak |0a⟩ = 0
+and
+ˆbk |0b⟩ = 0,
+∀ k.
+(2.9)
+These two quantizations are related by the Bogoliubov transformation ˆbk = α∗
+kˆak − β∗
+kˆa†
+k.
+The mean number density of b-particles in the a-vacuum, which will be, in general, a
+non-vacuum state according to the ˆbk operators, is given by
+⟨0a| ˆnb
+k |0a⟩ = |βk|2.
+(2.10)
+Integrating over all modes, we find the total mean density
+� d3k |βk|2, which will remain
+finite as long as |βk|2 → 0 faster than k−3 for increasing k.
+Let us now associate each basis of solutions to two observers living at different times
+ta < tb. If spacetime is static, the frequency (2.7) is constant, so that the solution to (2.6)
+takes the same form at all times. As a consequence, observers at different times have
+the same notion of particle, and therefore βk = 0. However, if geometry undergoes an
+expansion, two observers living at different times (before and after the expansion) have
+different notions of vacuum. Thus, βk ̸= 0 and therefore nb ̸= 0, which can be understood
+as the number density of particles produced out of the original vacuum state due to the
+expansion of spacetime.
+For the problem at hand, the goal is to extract the number of produced particles after
+the evolution of the universe during inflation and reheating, once these stages have finished.
+Then, as long as the test particle is not (strongly) interacting, this will be related to the
+abundance one observer would measure today only by the expansion dilution. Hence, we
+will take the Bunch-Davies vacuum as initial state, as defined by the solution of the mode
+– 4 –
+
+equation at very early times. In our case, we will take the geometry to approach de Sitter
+spacetime at the beginning of inflation. On the other hand, the notion of vacuum for an
+inertial observer after reheating will be different. If the evolution of spacetime is sufficiently
+adiabatic after this phase, we can assume this is the same vacuum we observe nowadays.
+Therefore, the corresponding operators will measure the number of particles created in the
+evolution.
+The specific form of the scale factor and the Ricci scalar will be determined by the
+specific inflationary model under consideration, which we describe in the next section.
+3
+Background dynamics
+We will describe the early epoch of the universe with a chaotic inflationary model consisting
+of a single scalar field φ with a quadratic potential of the form V (φ) = 1
+2m2
+φφ2, where
+mφ denotes the inflaton mass. The equation of motion for the inflaton is, if we assume
+homogeneity and isotropy,
+0 = ¨φ + 3H(t) ˙φ + ∂φV (φ),
+(3.1)
+where H(t) ≡ ˙a(t)/a(t) is the Hubble parameter. Note that in this context it is customary
+to work with cosmological time t. We will assume that the inflaton contribution to the
+total energy-momentum tensor is dominant when deriving the corresponding Friedmann
+equation,
+H2 =
+4π
+3M2
+P
+� ˙φ2 + 2V (φ)
+�
+.
+(3.2)
+We will also need the Ricci curvature scalar in order to properly describe the frequency of
+the mode equation (2.6), which in terms of the inflaton field reads
+R = 8π
+M2
+P
+�
+4V (φ) − ˙φ2�
+.
+(3.3)
+Equation (3.1), together with (3.2), has no analytic solution in general. However,
+one can find approximations for certain regimes. When this is not possible, we must rely
+on numerical computation. We analyze two different regions which, in conformal time,
+correspond to
+η =
+�
+�
+�
+ηi ≤ η < η∗,
+Slow-roll approximation,
+η∗ ≤ η ≤ ηf,
+Numerical solution.
+(3.4)
+For the inflationary period, we can use the well-known slow-roll approximation to obtain a
+solution to the inflaton equation of motion, as we describe in subsection (3.1). However,
+during the transition between inflation and reheating, the dynamics of the inflaton has to
+be obtained numerically. Both the inflaton field φ and the Ricci scalar R start to oscillate
+with decreasing amplitude, as can be observed in figure 1, where φ(η) and R(η) are depicted
+for an interval of time during the transition phase. This is the epoch in which most of the
+particles are produced and the inflaton dynamics is solved until a numerically accessible time
+ηf is reached, when production becomes negligible. For late times, deep in the reheating
+era, we can also use an analytic approximation for the solution of the inflaton equation of
+– 5 –
+
+-3
+-2
+-1
+0
+1
+2
+3
+0
+2
+4
+6
+-2
+-1
+0
+1
+2
+3
+4
+0
+5
+10
+15
+20
+1
+2
+3
+-0.25
+0
+0.25
+0.5
+Figure 1. Inflaton field φ(η) (left panel) and curvature scalar R(η) (right panel) as functions of
+conformal time. The range of time corresponds to the end of inflation and the beginning of reheating.
+The parameters used for all figures in this article are given in Appendix A.
+motion, given in subsection 3.2, which —although not used in our calculations— will be
+used to make some remarks in section 4.
+3.1
+Inflationary era - Slow-roll approximation
+We will choose the inflationary period to start at the negative, initial time ti. Inflation
+requires that the inflaton field changes slowly in comparison to the potential. Within the
+slow-roll approximation [46, 47], we can neglect the derivative of the field in favor of the
+potential, namely ˙φ2 ≪ |V (φ)|. When this condition is satisfied, the field slowly rolls over
+until it falls to a minimum and starts oscillating. At this point, inflation ends. With this
+assumption, we can approximately write (3.2) during the slow roll as
+H ≃
+�
+8π
+3M2
+P
+V (φ).
+(3.5)
+A slowly-varying inflaton implies that H ∼ constant for this regime. Hence, the expansion
+of spacetime is said to be quasi-exponential, as it resembles the pure de Sitter solution.
+Usually, one also assumes a small rate of change for the (already slow) velocity of φ, such
+that |¨φ| ≪ 3H| ˙φ|. This allows the slow-roll condition to be maintained long enough to solve
+the flatness and horizon problems. With these assumptions, equation (3.1) becomes easily
+solvable,
+˙φ ≃ −∂φV (φ)
+3H
+≃ −∂φV (φ)
+MP
+�
+24πV (φ).
+(3.6)
+– 6 –
+
+For the particular potential V (φ) = 1
+2m2
+φφ2, the solution to (3.6) is
+φSR(t) = φ0 − MP
+√
+12πmφt,
+(3.7)
+where t < 0 corresponds to the inflationary period. Note that t = 0 and φ0 are the ending
+time of inflation and the value of the field at this instant, respectively. From here, it is
+straightforward to obtain an explicit expression for the Ricci scalar, introducing the solution
+into (3.3).
+The scale factor is obtained by integrating the Hubble rate, and in the slow-roll
+approximation it reads
+aSR(t) ≃ a0e
+−� φ(t)
+φ0
+dφ 8π
+M2
+P
+V (φ)
+∂φV (φ) ,
+(3.8)
+which for the quadratic potential becomes
+aSR(t) = a0e
+− 2π
+M2
+P [φ2
+SR(t)−φ2
+0].
+(3.9)
+Lastly, we need the relation between cosmological and conformal time in order to write
+both a(η) and R(η). This relation can be obtained numerically from η = η0 +
+� t
+0 dt/a(t).
+These are the necessary ingredients for determining the frequency of the mode equation in
+this region, under the slow-roll approximation.
+This regime is valid as long as the slow-roll parameter, ϵH = − ˙H/H2, is much smaller
+than one. When this no longer holds, at, say, t > t∗ with t∗ < 0, the equation of motion (3.1)
+has to be solved numerically. The field begins to exit the inflationary regime and t = η = 0
+marks both the end of inflation and the beginning of reheating. At this point, the scale
+factor reaches the value a0, which merely sets the scale and hence we take it to be a0 = 1.
+3.2
+Late reheating
+For late times, well into the reheating epoch (η∗ ≪ η ≲ ηf), one can find an approximate
+solution to (3.1) [24]. We do not use it for obtaining our results, but it will be important
+for the discussion in subsection 4.2. In this approximation, the Hubble rate reads
+H(t) ≃ 2
+3t
+�
+1 − sin (2mφt − 2ϕ)
+2mφt
++ O(m−2
+φ t−2)
+�−1
+,
+(3.10)
+whereas the inflaton field is given by the expression
+φ = Φ0
+t sin mφt
+�
+1 − cos 2mφt
+2mφt
++ O(m−2
+φ t−2)
+�
+,
+(3.11)
+with Φ0 ≡ MP /(
+√
+3πmφ). This solution is valid as long as mφt ≫ 1, condition which is
+fulfilled during reheating, since, as we will see, the scale factor behaves as that of a matter
+dominated universe. Indeed, we can integrate H(t) in order to approximately obtain the
+scale factor a(t),
+a(t) = Ct2/3 �
+1 + O(m−2
+φ t−2)
+�
+.
+(3.12)
+– 7 –
+
+The constant C is determined by requiring that the value of the scale factor at late times
+coincides with the one obtained from the numerical simulation in the previous region. One
+can now integrate the scale factor in order to obtain t(η) = (Cη/3)3.
+Now that we have a solution for the inflaton field and the scale factor valid for late
+times, we can obtain the Ricci scalar from (3.3) by taking the solution for φ(t) to first order
+in (mφt)−1. We end up with
+R = 8
+3t2
+�
+�2 sin2 mφt −
+�
+cos mφt − sin mφt
+mφt
+�2
++ O(m−3
+φ t−3)
+�
+� .
+(3.13)
+With this, we are able to describe the frequency of the mode equation until very late
+times, for which the approximations derived in this subsection behave even better. The
+density of produced particles will be calculated at a sufficiently large time ηf, such that the
+particle production is negligible from that point in time onwards. (3.13).
+4
+Particle production
+Once we have determined the behavior of the background geometry during inflation and
+reheating, we can solve the mode equation in order to extract the Bogoliubov coefficients
+after the evolution.
+4.1
+Solution to the mode equation
+In order to compute the gravitational production once reheating has ended, we need to
+solve equation (2.6) from the onset of inflation at ti until a time tf well inside the adiabatic
+regime at the end of reheating, with the frequency of the oscillator determined by the
+background geometry described in the previous section. In a similar way as we did for the
+background dynamics in section 3, the mode equation is solved in the regions
+η =
+�
+�
+�
+ηi ≤ η ≤ η∗,
+Slow-roll approximation,
+η∗ ≤ η ≤ ηf,
+Numerical solution.
+(4.1)
+Let us start with the slow-roll era. In a de Sitter geometry, the Hubble rate is exactly
+constant, H0, the Ricci scalar is R = 12H0, and the scale factor reads a(η) = 1/(1 − H0η).
+Therefore, the frequency (2.7) takes the form
+ω2
+k,dS = k2 +
+µ2
+(η − η0)2 ,
+with
+µ2 = m2/H2
+0 + 12(ξ − 1/6),
+(4.2)
+where H0 = H(ηi) = 1/η0 is the Hubble rate at the beginning of inflation. The solution to
+equation (2.6) in this simplified scenario which assimptotically at η → −∞ behaves as a
+positive frequency plane wave is given by
+vk,dS(η) =
+�
+π|η − η0|/2 eiπνH(1)
+ν
+(k|η − η0|) ,
+ν =
+�
+1/4 − µ2.
+(4.3)
+This is the so-called Bunch-Davies solution [42]. Note that there is a critical value µ2 = 1/4
+for which ν = 0, which separates the regimes of real and imaginary ν. In particular, for
+– 8 –
+
+m2/H2
+0 ≪ 1, we can approximately write µ2 ≈ 12 (ξ − 1/6), and therefore µ2 = 1/4 for
+ξ = 3/16. At this point, there is no gravitational pair production in a de Sitter geometry
+[41], and this fact will be important for the analysis in section 4.
+However, our background geometry is not exactly de Sitter, but given by the inflaton
+dynamics derived in section 3. Within the slow-roll approximation, valid from the start of
+inflation at ηi until η∗, the mode equation to solve is
+v′′
+k(η) + ω2
+k,SR(η)vk(η) = 0,
+(4.4)
+where the scale factor and the Ricci scalar in ωk,SR(η) correspond to the analysis in
+subsection 3.1. Nevertheless, in the slow-roll regime, and for a certain range in k, m,
+and ξ, we can approximate the solution satisfying Bunch-Davies initial conditions by (see
+subsection 4.3 for details)
+vk,SR(η) ≃
+�
+π|τk|/2eiπνH(1)
+ν
+(k|τk|) ,
+τk = ωk,SR(η)
+ωk,dS(η) (η − η∗,k) + η∗,k − η0,
+(4.5)
+where η∗,k marks the limit of validity of the approximation. From this point on, equation (2.6)
+has to be solved numerically, independently of the background dynamics being numerical or
+analytical, taking as initial condition solution (4.5) and its derivative at η∗,k. The frequency
+one has to use in this case is that in (2.7).
+4.2
+Choice of reference vacua
+The solution vk(η) to the mode equation is associated with a particular choice of vacuum:
+the one that behaves as a plane wave at η → −∞. The procedure in subsection 4.1 allows
+us to evaluate vk(ηf). However, in order to obtain the Bogoliubov coefficient βk, we also
+need uk(ηf), which is the solution to the mode equation associated with the vacuum at this
+point in time. Then, from the Bogoliubov coefficients αk and βk, we will be able to extract
+the number density of produced particles at ηf. This time is chosen such that particle
+production becomes negligible for later times, condition that is fulfilled in the adiabatic
+regime, i.e., when
+�����
+ω′
+k(ηf)
+ω2
+k(ηf)
+����� ≪ 1.
+(4.6)
+The value of ηf highly depends on the parameters of the scalar field, and in particular, it
+becomes larger as the mass m decreases. This is why, for certain regions in parameter space,
+it may be convenient to use the late-time approximation for the background dynamics
+described in 3.2, instead of solving numerically the equation of motion of the inflaton field.
+It is worth mentioning that at the same time, a smaller coupling ξ to the curvature implies
+that the Ricci scalar oscillations, which are the main source of non-adiabaticity, are less
+important, therefore resulting in an earlier ηf at which (4.6) holds true.
+As long as the background is not static, the meaning of vacuum will change in time.
+Nevertheless, if the evolution is adiabatic enough, namely condition (4.6) is fulfilled, one
+can use the so-called adiabatic prescription to define the instantaneous vacuum at a given
+– 9 –
+
+instant ηf,
+uk(ηf) =
+1
+�
+ωk(ηf)
+,
+u′
+k(ηf) = −
+1
+√ωk
+�
+iωk(ηf) + 1
+2
+ω′
+k(ηf)
+ωk(ηf)
+�
+.
+(4.7)
+In fact, it is this feature that allows us to extrapolate the results obtained at ηf to the
+present when considering fields that interact only gravitationally [22, 24].
+When the mass of the field ϕ is above mφ, particle production is governed by the mass
+term of the frequency (2.7), namely
+ω2
+k(η) ≃ k2 + a2(η)m2.
+(4.8)
+Since the scale factor at late times behaves as a(η) ∼ η2, condition (4.6) is fulfilled after few
+oscillations of the inflaton. In other words, in this case we have that ηf is small enough that
+we do not need to invoke the late-time solution for the background, since everything can be
+calculated numerically in an efficient way. This is not the case for masses smaller than the
+inflaton, for which production stabilizes after many, many oscillations. As a consequence, if
+we want to use the adiabatic vacuum description, we need to go up to a very large ηf, and
+therefore we need to use the analytic approximation for the inflaton dynamics described
+in (3.11).
+Alternatively, we can take a different definition for the vacuum that allows us to
+calculate the number density of produced particles at ¯η ≪ ηf, even for m ≪ mφ. Because
+the oscillating term in (2.7) becomes negligible at sufficiently large (numerically accessible) ¯η,
+we can define the frequency
+ω(avg) 2
+k
+(η) = k2 + a2(η)
+�
+m2 + (ξ − 1/6) ⟨R⟩ (η)
+�
+,
+(4.9)
+where the Ricci scalar oscillations are averaged. We can take this frequency to calculate the
+averaged vacuum
+u(avg)
+k
+(¯η) =
+1
+�
+ω(avg)
+k
+(¯η)
+,
+u(avg) ′
+k
+(¯η) = −
+1
+�
+ω(avg)
+k
+(η)
+�
+iω(avg)
+k
+(¯η) + 1
+2
+ω(avg) ′
+k
+(¯η)
+ω(avg)
+k
+(¯η)
+�
+.
+(4.10)
+This prescription of vacuum is such that the spectrum of produced particles obtained at ¯η
+essentially concides with the one given by the adiabatic vacuum at the time where we reach
+the adiabatic regime, ηf, namely
+n(avg)
+k
+���
+η=¯η ≃ n(ad)
+k
+���
+η=ηf
+.
+(4.11)
+The larger discrepancies will reside in low wavenumbers, for which k ∼ a2(η) ⟨R⟩, but this
+region of momentum space is supressed in the total density of produced particles by a factor
+k2 (for details see next subsection), since
+n(m, ξ) =
+�
+d3k
+(2π)3 ⟨0| ˆnk |0⟩ =
+�
+dk
+2π2 k2|βk|2.
+(4.12)
+– 10 –
+
+As a consequence, no differences are appreciated at the chosen ¯η.
+This procedure has a limitation: It is valid up to the smallest mass m for which
+the dynamics presented here remain the same until ηf. If reheating ends before ηf for
+a particular mass, the result provided by the averaged vacuum will not be the particles
+produced after this period is over. Nevertheless, a simple estimation shows that masses
+above the order of m ∼ 10−30 eV would reach adiabaticity early enough. This is many
+orders of magnitude below the mass of fuzzy cold dark matter, and hence all the interesting
+range of masses lie within the regime of validity of our method.
+4.3
+Slow-roll approximation for the solution to the mode equation
+During inflation, spacetime expands quasi-exponentially. More specifically, the number of
+e-folds
+a(t0)
+a(ti) = eN
+(4.13)
+is required to be such that N ≈ 50 − 60 [9–11].
+Because eq. (2.6) cannot be solved
+analytically, even considering a slowly rolling inflaton field, one would need to use numerical
+methods in order to find a solution. However, the large amount of e-folds to cover makes
+it more interesting and feasible to rely on an analytic approximation, such as (4.5). We
+dedicate this subsection to formally develop the approximation and to test its validity. For
+notational convenience, in the calculations that follow we will write η − η0 as η, and drop
+the mode index k. Let us start by defining the following small parameters for given values
+of k, m and ξ which will be useful in the following.
+• First, we have
+ϵ(m, ξ) = max
+η∈I1
+�����1 − ωSR(η; m, ξ)
+ωdS(η; m, ξ)
+�����,
+with
+I1 = (−∞, η1),
+(4.14)
+where η1 is chosen such that ϵ ≪ 1. Then, we can define f(η; m, ξ) by
+ωSR
+ωdS
+= 1 + ϵf.
+(4.15)
+By construction, |f(η)| ≤ 1 for η ∈ I1. Moreover, f′(η) ≥ 0.
+• It will also be convenient to define
+σ(m, ξ) = max
+η∈I2
+���f′(η; m, ξ)η
+���,
+with
+I2 = (−∞, η2),
+(4.16)
+and choose η2 such that σ ≤ ϵ. Then, we introduce g(η; m, ξ) as
+f′(η) = σg(η)
+η
+,
+(4.17)
+for which again we have that |gk(η)| ≤ 1 for η ∈ I2.
+– 11 –
+
+• Similarly, we define
+ρ(m, ξ) = max
+η∈I3
+�����
+ω′
+dS(η)
+ωdS(η)η
+�����,
+with
+I3 = (−∞, η3),
+(4.18)
+and choose η3 such that ρ ≤ ϵ.
+Now, we take η∗ = min(η1, η2, η3) and I = (−∞, η∗), where I is the interval for which
+the three parameters ϵ, σ, ρ are small. Note that η∗ < 0 since inflation ends at η = 0.
+• We also need |η∗/η0| > 1.
+The task is to solve equation (4.4), for which we define a new time coordinate ζ within
+the interval I,
+dζ = ωSR(η)
+ωdS(η) dη = [1 + ϵf(η)] dη.
+(4.19)
+After integration until η ∈ I and taking the absolute value, this becomes
+|(ζ − ζ∗) − (η − η∗)| = ϵ
+�����
+� η∗
+η
+f(t)dt
+����� = O(ϵ)(η − η∗).
+(4.20)
+Then, choosing ζ∗ = η∗, this can be expressed as
+ζ = η [1 + O(ϵ)] .
+(4.21)
+We change time coordinates η → ζ in the mode equation, which takes the form
+¨w(ζ) + ω2
+dS [η(ζ)] w(ζ) + ϵf′ [η(ζ)] ω2
+dS [η(ζ)]
+ω2
+SR [η(ζ)] ˙w(ζ) = 0,
+(4.22)
+where w(ζ) = v [η(ζ)] and the dot denotes here derivative with respect to ζ.
+Let us analyze the last term. With this aim, we introduce the dimensionless time
+¯ζ = ζ/η0. Then, in terms of ¯ζ, the equation above has the same form except for the last
+term that acquires an extra factor. Using the definition of f′ and σ above, the coefficient of
+this term is
+ϵf′ ω2
+dS
+ω2
+SR
+η0 = ϵσg(1 + ϵf)η0
+η = O(ϵ2)η0
+η
+(4.23)
+If we choose η∗ such that |η∗/η0| > 1, as mentioned above, this coefficient is of order O(ϵ2).
+Furthermore, the frequency in the second term of (4.22) is
+ω2
+dS(η(ζ)) = ω2
+dS (ζ [1 + O(ϵ)])
+(4.24)
+= ω2
+dS(ζ)
+�
+�1 + 2ω′
+dS
+ωdS
+�����
+ζ
+· ζ O(ϵ)
+�
+�
+(4.25)
+= ω2
+dS (ζ)
+�
+1 + O(ϵ2)
+�
+,
+(4.26)
+provided that |ζ ω′
+dS(ζ)/ωdS(ζ)| ≤ ρ = O(ϵ). This is satisfied for ζ = η [1 + O(ϵ)] < η∗, i.e.,
+for η < η∗. Thus, the equation for w can finally be written as
+¨w(ζ) + ω2
+dS(ζ)w(ζ) = O(ϵ2
+k).
+(4.27)
+– 12 –
+
+We can perturbatively solve the differential equation by writting w = w0 + ϵw1 + O(ϵ2).
+The solution to order ϵ0 is nothing but the de Sitter modes (4.3),
+w0(ζ) =
+�
+π|ζ| eiπνH(1)
+ν
+(k|ζ|) ,
+ν =
+�
+1/4 − µ2,
+(4.28)
+and as a consequence, wk,0 behaves asymptotically (ζ → −∞) as a plane wave. On the
+other hand, the coefficients of the solution to order ϵ1 will satisfy the same original equation
+but with the initial conditions that w1(−∞) = 0 and therefore w1 is identically zero. We
+can then write w as
+w(ζ) = w0(ζ)
+�
+1 + O(ϵ2)
+�
+=
+�
+π|ζ| eiπνH(1)
+ν
+(k|ζ|)
+�
+1 + O(ϵ2)
+�
+.
+(4.29)
+In order to undo the coordinate transformation ζ → η while keeping the error up to
+O(ϵ2), we need to consider the O(ϵ1) terms in ζ = η [1 + O(ϵ)]. For this, we note that
+����� (ζ − η∗) − ωSR(η)
+ωdS(η) (η − η∗)
+����� =
+����� (η − η∗) + ϵ
+� η
+η∗
+f(t)dt − [1 + ϵf(η)] (η − η∗)
+�����
+= ϵ
+�����
+� η
+η∗
+f(t)dt −
+� η
+η∗ f(η)dt
+�����
+≤ ϵ
+� η
+η∗
+|f(t) − f(η)|dt
+= ϵ
+� η
+η∗
+����f′(η)(t − η) + 1
+2!f′′(η)(t − η)2 + · · ·
+���� dt
+≤ ϵ
+�����
+1
+2f′(η)(η − η∗)2
+���� +
+����
+1
+3!f′′ (η − η∗)3
+���� + · · ·
+�
+.
+(4.30)
+This means that, as long as the terms in curly brackets are of order O(ϵ), we can write
+ζ = η∗ +
+�ωSR(η)
+ωdS(η) + O(ϵ2)
+�
+(η − η∗) = η∗ + ωSR(η)
+ωdS(η) (η − η∗)
+�
+1 + O(ϵ2)
+�
+.
+(4.31)
+The first term is equal to
+1
+2σ
+����g(η)η − η∗
+η
+���� = O(ϵ).
+(4.32)
+The next terms are of the form f(n) (η − η∗)n+1 /n!, which numerically can be seen to be
+smaller than the first one.
+Therefore, undoing the translation of η to η − η0 that we did at the beginning of this
+calculation, the solution to the mode equation can be written as (4.5) up to terms of order
+O(ϵ2). With fixed ξ, and choosing η∗ independent of k, the error ϵk increases with increasing
+m and decreasing k.
+When we numerically solve the mode equation (2.6) from η∗, the error in the initial
+condition coming from the slow-roll solution (4.5) carries through as
+vk(η) = vk,0(η)
+�
+1 + O(ϵ2
+k)
+�
+,
+(4.33)
+– 13 –
+
+0.01
+0.1
+1
+10
+100
+0.01
+0.1
+1
+10
+100
+0.01
+0.1
+1
+10
+100
+0.01
+0.1
+1
+10
+100
+-2.5
+-2.0
+-1.5
+-1.0
+-0.5
+0
+Figure 2. Maximum of the errors squared as function of the wave number k and the field mass m,
+for ξ = 0.2 (left) and ξ = 0.8 (right). We take η∗ = −500mφ for all values of k, m and ξ.
+0.01
+0.1
+1
+10
+100
+0.01
+0.1
+1
+10
+100
+0.01
+0.1
+1
+10
+100
+0.01
+0.1
+1
+10
+100
+-2.5
+-2.0
+-1.5
+-1.0
+-0.5
+0
+Figure 3. Maximum of the errors squared times k2 as function of the wave number k and the field
+mass m, for ξ = 0.2 (left) and ξ = 0.8 (right). We take η∗ = −500mφ for all values of k, m and ξ.
+such that vk(η) → vk,SR(η) as η → η∗. Therefore, we have for the total density defined
+in (4.12) that
+n(m, ξ) =
+� ∞
+0
+dk
+2π2 k2|βk|2 = n0
+�
+1 + 1
+n0
+� ∞
+0
+dk
+2π2 k2|βk,0|2O(ϵ2
+k)
+�
+,
+(4.34)
+where n0 =
+� ∞
+0
+dk
+2π2 k2|βk,0|2. Although the error ϵk increases as k decreases, the factor k2
+compensates this increase for low k. Essentially, although ϵ2
+k increases for k < mφ, the
+quantity k2ϵ2
+k remains small, whereas |βk,0|2 is roughly of the same order. More explicitly,
+for the calculations in this paper, we take η∗ = −500mφ, for which the maximum of the
+– 14 –
+
+-800
+-600
+-400
+10-5
+10-4
+0.001
+0.010
+-800
+-600
+-400
+10-8
+10-6
+10-4
+0.010
+Figure 4. Relative error in the absolute value (left panel) and the phase (right panel) of the
+numerical solution to the exact mode equation (2.6) compared to the analytical approximation (4.5),
+for wavenumbers ranging from k = 0.01mφ to k = 100mφ, and m = mφ, ξ = 0.5. Here, we take
+ηdS = −1000/mφ and η∗ = −500/mφ.
+three small parameters squared, ϵ2
+k, σ2
+k, ρ2
+k, as function of mass and wavenumber, for two
+different choices of coupling ξ, is shown in figure 2. For m ≤ mφ and k ≥ 0.1mφ, the
+error is of order O(0.01) or smaller for the various values of ξ considered, and thus the
+approximation is controlled in this regime. At the same time, we can observe in figure 3 that
+k2ϵ2
+k decreases as we move to the low-part of the momentum range. This guarantees that
+this region of the spectrum is robust against errors in the mode equation approximation we
+used.
+On the other hand, from figure 3 we observe that the quantity k2ϵ2
+k grows with k for
+k > mφ, since the decrease in ϵ2
+k (c.f. figure 2) can not compensate the power k2. However,
+gravitational production for high-momentum particles is very small, namely |βk|2 ≈ 0 for
+k ≫ mφ. As a consequence, n(m, ξ) ≈ n0 approximates well the total number density of
+particles produced, since the weight of wavenumbers k ≫ mφ is very small when compared
+to the rest of the spectrum.
+Furthermore, we can test the validity of (4.5) when compared to the numerical solution
+of (2.6) by putting ourselves in the following scenario: Let us assume that the geometry
+can be approximated by a de Sitter spacetime during the early stages of inflation, such
+that the solution (4.3) is valid for a region ηi ≤ η < ηdS. At ηdS, slow-roll starts to matter,
+and deviations from the de Sitter solution vk,dS(η) occur. In this scenario, we explore two
+different paths to continue continuing solving the equation:
+1. We assume slow-roll inflation is a good description for the background dynamics in
+the region ηdS ≤ η < η∗, and take as solution the approximation (4.5).
+2. We solve numerically the exact equation of motion for the inflaton, eq. (3.1), obtaining
+the frequency corresponding to (2.6), equation which we again solve numerically. This
+– 15 –
+
+solution, vk(η), will be valid even for η ≥ η∗.
+In figure 4, we compare the analytical slow-roll solution with the exact numerical solution
+by plotting the relative difference between their absolute values,
+∆rAbs [vk,SR(η)] ≡
+�����
+Abs [vk(η)] − Abs [vk,SR(η)]
+Abs [vk(η)]
+�����,
+(4.35)
+as well as their phase difference,
+∆rArg [vk,SR(η)] ≡
+�����
+Arg [vk(η)] − Arg [vk,SR(η)]
+π
+�����.
+(4.36)
+We do so for different wavenumbers, ranging from k = 0.01mφ to k = 100mφ, denoted by
+the different shapes in figure 4. We have taken ηdS = −1000/mφ as start of the slow-roll
+and η∗ = −500/mφ as the time when the slow-roll approximation breaks down. For k = mφ,
+the relative error is very small, of order ∼ 10−4 at η∗. For wavenumbers larger than the
+mass of the inflaton, k > mφ, the approximation is still good, although it worsens. On
+the other hand, the error for k = 0.01mφ starts becoming significant, and gets worse for
+k < 0.01mφ. However, the corresponding region of the spectrum of produced particles is
+highly suppressed, as discussed above, and therefore the contribution to the total density of
+particles is negligible. Similarly, particle production is very small for wavenumbers larger
+than k > 100mφ, and therefore the range of interest in k is under control. Hence, we can
+assume the approximation is valid in the region ηdS ≤ η < η∗.
+Note that if this solution behaves well in this region, it has to become an even better
+approximation before ηdS, since the further towards the past we go, the more de Sitter-like
+is the geometry. Thus, eq. (4.5) can be taken as well as a solution to the mode equation in
+the region ηi ≤ η < ηdS. Under this approximations, eq. (2.6) can be solved analytically
+from the start of inflation, ηi, until η∗, for which the slow-roll approximation starts to fail.
+From there, the mode equation is solved numerically.
+4.4
+Adiabaticity and oscillations
+In order to illustrate the importance of the choice of vacuum, we studied the evolution
+of spectra when calculated using prescription (4.7) before the dynamics has entered the
+adiabatic regime. As an example, we plotted in figure 5 the spectra of particles with mass
+m = 10−3mφ obtained at two different times. The dots correspond to η = 40/mφ, whereas
+the solid lines denote η = ηf = 100/mφ. For this particular choice of mass, the latter time
+lies within the adiabatic regime, and this is the reason why the non-adiabatic dots relax to
+their final value as we approach this limit. As expected, the effect is less noticeable the
+lower the coupling to the geometry is, as it is the main source of non-adiabaticity in the
+frequency.
+At the same time, we also characterized the importance of the first oscillations of the
+curvature scalar in the final spectrum of produced particles, obtained with the averaged
+vacuum defined in eq. (4.10). As can be seen in figure 6, even after several oscillations of
+– 16 –
+
+0.01
+0.1
+1
+10
+100
+0
+0.2
+0.4
+0.6
+0.8
+Figure 5. Spectra of produced particles of mass m = 10−3mφ and different values of ξ, obtained with
+the adiabatic prescription of the vacuum. The dots correspond to η = 40/mφ, before the adiabatic
+regime has been reached for this value of the mass. The solid lines correspond to η = ηf = 100/mφ,
+when most of the particles have been produced.
+R(η) (for example, at η = 2/mφ), the production changes greatly if one compares with the
+obtained spectra at ¯η. Even when looking only at the total number of produced particles in
+eq. (4.12), differences are still significant. We observe that the spectrum does not stabilize
+until η ≃ 5/mφ, which for our model means after hundreds of oscillations of the curvature
+scalar R(η). With this, we want to stress that obtaning the particle production after one or
+two oscillations does not account for the whole process.
+5
+Spectra of particles and total density
+Let us finally give the results for the spectra of produced particles as function of the
+parameters of the field, the mass m, and the coupling to the curvature ξ.
+We explore first the regime of masses below the inflaton mass. Represented by the solid
+line in figure 7, we have masses m ≤ 10−4mφ. For these values, the mass contribution to
+the frequency becomes negligible, and the dynamics is entirely given by the coupling to the
+geometry. The spectra lie on top of each other, with very small differences in the low values
+of k ∼ a(η)m. We observe, however, slight differences in the shape of the spectrum when
+increasing the mass, especially for small wavenumbers, as the rest of the curves in figure
+7 show. We can choose a mass in this regime, m = 10−1mφ, and explore the influence of
+the coupling ξ in the final result. This is shown in figure 8, where one observes increasing
+production of particles with larger values of the coupling. Lastly, let us come to the mass
+of the inflaton, whose corresponding spectra are shown in figure 9. In such a case, it is
+– 17 –
+
+0.01
+0.1
+1
+10
+100
+0
+2
+4
+6
+Figure 6. Spectra for m = 10−4mφ and ξ = 1, obtained with the averaged vacuum prescription,
+for different instants of time. The spectrum stabilises after very many oscillations of the curvature
+scalar.
+0.001
+0.01
+0.1
+1
+10
+100
+0
+0.1
+0.2
+Figure 7.
+Spectrum of particles for masses below the mass of the inflaton, with ξ = 0.26.
+For very small masses (m ≤ 10−4mφ), production is dominated by curvature.
+In the region
+10−3 ≤ m ≤ 10−1mφ, differences in production due to the mass can be noticed, especially for low
+values of k/mφ ≃ 0.1 − 1.
+harder to characterize the behavior with ξ. It is clear, nevertheless, that particle production
+decreases as the mass of particles becomes larger.
+– 18 –
+
+0.01
+0.1
+1
+10
+100
+0
+0.2
+0.4
+0.6
+Figure 8. Spectra for m = 10−1mφ and several values of the coupling ξ. Particle production
+increases when the curvature term becomes more important, and the maximum of the spectrum is
+shifted towards higher values of k.
+0.01
+0.1
+1
+10
+100
+0
+0.01
+0.02
+Figure 9. Spectra of particles with the mass of the inflaton, for different values of the coupling. In
+this particular case, increasing the coupling does not translate directly into an increase of particle
+production. This can be more clearly seen by examining the total density of particles.
+It is easier to characterize particle production in this regime using the total number
+density of particles (4.12), which we show in figure 10 as function of the two parameters of
+the field, m and ξ. Here, one clearly sees that the prediction is independent of the value of
+– 19 –
+
+Figure 10. Logarithm of the total density of produced particles for different values of m and ξ. In
+order to give the mass and density in units of GeV, we took mφ = 1.2 × 1013 GeV for the mass of
+the inflaton. We explore a wide range of masses in the left panel while we focus on a smaller region
+close to the mass of the inflaton on the right panel in order to appreciate the dependence of the
+total density with the coupling ξ.
+the mass as long as it is below m ∼ 10−2mφ, in particular for a sufficiently high value of
+the coupling, ξ ≳ 0.2. In this case, the mass is completely negligible when compared to
+the dynamics of the curvature scalar. Only when the coupling to the curvature is close
+to ξ ∼ 1/6, the production of particles is still sensible to m, up to m ∼ 10−7mφ. For this
+value, even in the conformal case, the relevant wavenumbers, k ∼ a(η)m, are too suppressed
+to make a difference. In all these regime of low masses, the number of produced particles
+increases with larger coupling ξ. Closer to the mass of the inflaton, 10−2mφ < m < mφ,
+the fact that a heavier particle translates into a lower production becomes apparent. Lastly,
+in the region around the mass of the inflaton, m ∼ mφ, the behavior with the coupling is
+different, and production may even decrease when raising the value of ξ. In fact, there
+appears to exist a critical value ξc ≃ 0.22 which separates two qualitatively different regimes.
+As we commented previously, this value is related to the parameter µ2 = 1/4 of the Hankel
+functions, which were a good approximation of the mode functions of our problem. For
+m < mφ, the number density drops very rapidly if ξ < ξc. For m ∼ mφ, ξc is the value below
+which production decreases with ξ, and above which it increases. This is also illustrated
+in figure 9, where production for ξ = 1/6 is larger than for ξ = 0.26, and from there it
+increases again with the coupling. Moreover, we observe the expected strong suppression in
+the number density of produced particles for masses above the mass of the inflaton. We
+can confirm this behaviour by calculating the spectra for even higher masses, provided we
+select a negative enough η∗ — and therefore leading to a very heavy computation — in
+this case, as explained in 4.3. Note that we took mφ = 1.2 × 1013 GeV for the mass of the
+inflaton, and as a consequence, the density in figure 10 is given in units of GeV3.
+– 20 –
+
+10-3
+101
+105
+109
+1013
+1/6
+0.2
+0.4
+0.6
+0.8
+1.0
+10-3
+101
+105
+109
+1013
+1/6
+0.2
+0.4
+0.6
+0.8
+1.0
+-7.5
+-3.0
+1.5
+6.0
+10.5
+15.0
+Figure 11. Logarithm of the predicted abundance of dark matter today for different values of m
+and ξ, and a reheating temperature of Treh = 1015 GeV (left) and Treh = 1013 GeV (right). In order
+to give the mass and density in units of GeV, we took mφ = 1.2 × 1013 GeV for the mass of the
+inflaton.
+Finally, one can consider these gravitationally produced scalar particles as dark matter.
+In this case, it is interesting to compare the resulting abundance with observations. Assuming
+that the scalar field is non-interacting, the evolution of the particle density showed in figure 10
+is only due to the expansion of the universe. Then, the predicted abundance can be written
+in terms of the background radiation temperature [24] as
+Ω(m, ξ) =
+8π
+3M2
+P H2
+today
+gtoday
+∗S
+grh
+∗S
+�Ttoday
+Trh
+�3
+m n(m, ξ),
+(5.1)
+where Ttoday and Trh are the radiation temperature today and at the end of reheating,
+respectively, and gtoday
+∗S
+and grh
+∗S are the corresponding relativistic degrees of freedom. This is
+represented in figure 11 for two different reheating temperatures, together with the observed
+abundance given by the dashed line. We observe that the proposed mechanism can explain
+observations if the dark matter candidate is light enough (m ≲ 108 GeV), independently of
+the value of the coupling ξ for the range that we considered. In addition, heavier particles
+can also reach the observed dark matter abundance since their production is strongly
+suppressed above the inflaton mass.
+6
+Conclusions
+Gravitational particle production is a very interesting process due to its universality. It
+only requires the studied field to interact with gravity. Even without a direct coupling to
+the inflaton, as it is the case of spectator fields such as the one we have studied, it can
+– 21 –
+
+give rise to a significant abundance for the species considered after the heavy expansion of
+spacetime during the early stages of the universe. However, predictions need for a definition
+of vacuum after reheating, since the non-static geometry leads to certain ambiguity in the
+meaning of particle.
+In this manuscript, we studied the production of massive, scalar particles whose
+dynamics is described by a non-minimally coupled to gravity action. However, the discussion
+on the validity of the definition of vacuum is pertinent when considering any other field
+as well. First, we have provided a method for solving in a complete form the background
+dynamics, governed by a single scalar inflaton field. For this, we did not have to assume a
+de Sitter geometry of spacetime, which would significantly change the amount of particles
+produced. Although we make a choice of potential, this procedure can be extended to other
+cases as well. We provided an analytic approximation to the solution of the slow-roll mode
+equation where the error is well under control in our parameter region of interest. More
+importantly, we showed that, for masses smaller than the inflaton mass, the commonly
+used adiabatic prescription for the vacuum determines correctly the production of particles
+after reheating only when calculated at very late times. Moreover, we define an alternative
+vacuum choice that allows one to obtain the right abundance when calculating particle
+production at a much earlier time. This allowed us to explore the contribution of the
+first oscillations to the total number of produced particles, revealing that the spectra only
+stabilizes after hundreds of periods. Lastly, after all these considerations have been taken
+into account, we analyzed both the spectra and the total density of particles for different
+values of the mass of the field and its coupling to the curvature scalar. When regarded as
+dark matter, the production of the spectator field can be directly related to the abundance
+that would be observed today if one assumes no couplings to any other fields also after
+reheating. In particular, we find agreement with the observed dark matter abundance for a
+certain range of masses and couplings of the spectator field. Moreover, this analysis can be
+used to constrain the values of the field parameters by demanding that the predicted dark
+matter abundance does not exceed observations.
+Acknowledgements
+This work was partially supported by the MICINN (Ministerio de Ciencia e Innovación,
+Spain) projects PID2019-107394GB-I00/AEI/10.13039/501100011033 (AEI/FEDER, UE)
+and PID2020-118159GBC44. Additionally, Á.P.-L. is supported by the MIU (Ministerio
+de Universidades, Spain) fellowship FPU20/0560. Finally, JARC acknowledges support by
+Institut Pascal at Université Paris-Saclay during the Paris-Saclay Astroparticle Symposium
+2022, with the support of the P2IO Laboratory of Excellence (program “Investissements
+d’avenir” ANR-11-IDEX-0003-01 Paris-Saclay and ANR-10-LABX-0038), the P2I axis of
+the Graduate School of Physics of Université Paris-Saclay, as well as IJCLab, CEA, APPEC,
+IAS, OSUPS, and the IN2P3 master projet UCMN.
+– 22 –
+
+A
+Parameters
+In the majority of the analyses, we have left all the quantities expressed in terms of the mass
+of the inflaton, mφ, which sets up the scale of the problem. When it has been necessary to
+assume a numerical value for such a mass, we have taken mφ = 1.2 × 1013 GeV. Accordingly,
+the Planck mass MP has the value MP = 1.02 × 106mφ.
+The initial value for the inflaton field, under the slow-roll assumption, is taken to
+be φSR(ti) = φi = 3MP . When inflation ends, at t = 0, the field value is φSR(t = 0) =
+φ0 = 0.5MP . The slow-roll approximation can then be used to extract ti ≃ −15.35/mφ as
+the time when inflation starts. Equation of motion (3.1) can also be solved numerically
+taking as initial conditions the same as for slow-roll, φ(ti) = φi, and the derivative of the
+approximate solution at this point, φ′(ti) = φ′
+SR(ti). Both solutions will be very close up to
+t∗, where the slow-roll approximation starts to break down. Then, φ(t = 0) slightly deviates
+from φ0. The scale factor is chosen such that a(t = 0) = a0 = 1. Slow-roll is a assumed to
+be a good approximation until η∗ = −500/mφ.
+Unless the contrary is expressly stated, particle production is calculated using the
+averaged vacuum prescription at ¯η = 16.33/mφ. The range of masses explored is 10−7mφ ≤
+m ≤ 100.5mφ, although for obtaining figure 11 it is assumed that production is the same
+for m ≤ 10−7mφ. On the other hand, the coupling ξ is such that 1/6 ≤ ξ ≤ 1.
+References
+[1] S.W. Hawking, Particle creation by black holes, Comm. Math. Phys. 43 (1975) 199.
+[2] W.G. Unruh, Experimental Black-Hole Evaporation?, Phys. Rev. Lett. 46 (1981) 1351.
+[3] G.W. Gibbons and S.W. Hawking, Cosmological event horizons, thermodynamics, and particle
+creation, Phys. Rev. D 15 (1977) 2738.
+[4] L. Bombelli, R.K. Koul, J. Lee and R.D. Sorkin, Quantum source of entropy for black holes,
+Phys. Rev. D 34 (1986) 373.
+[5] D.N. Page, Information in black hole radiation, Phys. Rev. Lett. 71 (1993) 3743.
+[6] M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666.
+[7] L. Parker, Quantized Fields and Particle Creation in Expanding Universes. I, Phys. Rev. 183
+(1969) 1057.
+[8] L.H. Ford, Gravitational particle creation and inflation, Phys. Rev. D 35 (1987) 2955.
+[9] A. Starobinsky, A new type of isotropic cosmological models without singularity, Physics
+Letters B 91 (1980) 99.
+[10] A.H. Guth, Inflationary universe: A possible solution to the horizon and flatness problems,
+Phys. Rev. D 23 (1981) 347.
+[11] A. Linde, A new inflationary universe scenario: A possible solution of the horizon, flatness,
+homogeneity, isotropy and primordial monopole problems, Physics Letters B 108 (1982) 389.
+[12] E.W. Kolb and M.S. Turner, The Early Universe, vol. 69 (1990), 10.1201/9780429492860.
+[13] L. Kofman, A.D. Linde and A.A. Starobinsky, Reheating after inflation, Phys. Rev. Lett. 73
+(1994) 3195 [hep-th/9405187].
+– 23 –
+
+[14] L. Kofman, A. Linde and A.A. Starobinsky, Towards the theory of reheating after inflation,
+Phys. Rev. D 56 (1997) 3258.
+[15] R. Allahverdi, R. Brandenberger, F.-Y. Cyr-Racine and A. Mazumdar, Reheating in
+inflationary cosmology: Theory and applications, Annual Review of Nuclear and Particle
+Science 60 (2010) 27.
+[16] D. Baumann and L. McAllister, Inflation and String Theory, Cambridge Monographs on
+Mathematical Physics, Cambridge University Press (5, 2015), 10.1017/CBO9781316105733,
+[1404.2601].
+[17] D.J.H. Chung, E.W. Kolb and A. Riotto, Superheavy dark matter, Phys. Rev. D 59 (1998)
+023501.
+[18] D.J.H. Chung, P. Crotty, E.W. Kolb and A. Riotto, Gravitational production of superheavy
+dark matter, Phys. Rev. D 64 (2001) 043503.
+[19] D.J.H. Chung, E.W. Kolb and A.J. Long, Gravitational production of super-hubble-mass
+particles: an analytic approach, Journal of High Energy Physics 2019 (2019) .
+[20] S. Hashiba and J. Yokoyama, Gravitational particle creation for dark matter and reheating,
+Phys. Rev. D 99 (2019) 043008.
+[21] Y. Ema, R. Jinno, K. Mukaida and K. Nakayama, Gravitational particle production in
+oscillating backgrounds and its cosmological implications, Phys. Rev. D 94 (2016) 063517.
+[22] Y. Ema, K. Nakayama and Y. Tang, Production of purely gravitational dark matter, Journal of
+High Energy Physics 2018 (2018) .
+[23] T. Markkanen and S. Nurmi, Dark matter from gravitational particle production at reheating,
+Journal of Cosmology and Astroparticle Physics 2017 (2017) 008.
+[24] J.A. Cembranos, L.J. Garay and J.M.S. Velázquez, Gravitational production of scalar dark
+matter, Journal of High Energy Physics 2020 (2020) .
+[25] Y. Ema, K. Nakayama and Y. Tang, Production of purely gravitational dark matter: the case
+of fermion and vector boson, Journal of High Energy Physics 2019 (2019) .
+[26] M. Bastero-Gil, J. Santiago, L. Ubaldi and R. Vega-Morales, Vector dark matter production at
+the end of inflation, Journal of Cosmology and Astroparticle Physics 2019 (2019) 015.
+[27] T. Markkanen, A. Rajantie and T. Tenkanen, Spectator dark matter, Phys. Rev. D 98 (2018)
+123532.
+[28] N. Herring, D. Boyanovsky and A.R. Zentner, Nonadiabatic cosmological production of
+ultralight dark matter, Phys. Rev. D 101 (2020) 083516.
+[29] T. Tenkanen, Dark matter from scalar field fluctuations, Physical Review Letters 123 (2019) .
+[30] Planck collaboration, Planck 2018 results. X. Constraints on inflation, Astron. Astrophys.
+641 (2020) A10 [1807.06211].
+[31] T. Markkanen, Vacuum stability in the early universe and the backreaction of classical gravity,
+Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering
+Sciences 376 (2018) 20170115.
+[32] M. Fairbairn, K. Kainulainen, T. Markkanen and S. Nurmi, Despicable dark relics: generated
+by gravity with unconstrained masses, Journal of Cosmology and Astroparticle Physics 2019
+(2019) 005.
+– 24 –
+
+[33] S.-J. Wang, M. Yamada and A. Vilenkin, Constraints on non-minimal coupling from quantum
+cosmology, Journal of Cosmology and Astroparticle Physics 2019 (2019) 025.
+[34] A. Friedman, Über die Krümmung des Raumes, Z. Phys. 10 (1922) 377.
+[35] A. Friedman, Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des
+Raumes, Z. Phys. 21 (1924) 326.
+[36] A.G. Lemaître, A Homogeneous Universe of Constant Mass and Increasing Radius accounting
+for the Radial Velocity of Extra-galactic Nebulæ, Mon. Not. R. Astron. Soc. 91 (1931) 483.
+[37] H.P. Robertson, Kinematics and World-Structure, Astrophys. J. 82 (1935) 284.
+[38] H.P. Robertson, Kinematics and World-Structure II, Astrophys. J. 83 (1936) 187.
+[39] H.P. Robertson, Kinematics and World-Structure III, Astrophys. J. 83 (1936) 257.
+[40] A.G. Walker, On Milne’s Theory of World-Structure*, Proc. London Math. Soc. s2-42 (1937)
+90.
+[41] D.E. Borrajo Gutiérrez, J.A.R. Cembranos, L.J. Garay and J.M. Sánchez Velázquez, Derivative
+couplings in gravitational production in the early universe, JHEP 09 (2020) 069 [2006.08546].
+[42] N.D. Birrell and P.C.W. Davies, Quantum Fields in Curved Space, Cambridge Monographs on
+Mathematical Physics, Cambridge University Press, Cambridge (1982),
+10.1017/CBO9780511622632.
+[43] V. Mukhanov and S. Winitzki, Introduction to Quantum Effects in Gravity, Cambridge
+University Press, Cambridge (2007), 10.1017/CBO9780511809149.
+[44] E.A. Calzetta and B.-L.B. Hu, Nonequilibrium Quantum Field Theory, Cambridge Monographs
+on Mathematical Physics, Cambridge University Press (2008), 10.1017/CBO9780511535123.
+[45] L.E. Parker and D. Toms, Quantum Field Theory in Curved Spacetime: Quantized Field and
+Gravity, Cambridge Monographs on Mathematical Physics, Cambridge University Press (8,
+2009), 10.1017/CBO9780511813924.
+[46] V. Mukhanov, Physical Foundations of Cosmology, Cambridge University Press, Oxford
+(2005), 10.1017/CBO9780511790553.
+[47] S. Weinberg, Cosmology, Oxford University Press, Oxford (2008).
+– 25 –
+

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+mRpostman: An IMAP Client for R
+Allan V. C. Quadros
+Department of Statistics
+Kansas State University
+Manhattan, KS 66506, United States
+Abstract
+Internet Message Access Protocol (IMAP) clients are a common feature in
+several programming languages. Despite having some packages for electronic
+messages retrieval, the R language, until recently, lacked a broader solution,
+capable of coping with different IMAP servers and providing a wide spec-
+trum of features. mRpostman covers most of the IMAP 4rev1 functionalities
+by implementing tools for message searching, selective fetching of message
+attributes, mailbox management, attachment extraction, and several other
+IMAP features that can be executed in virtually any mail provider. By doing
+so, it enables users to perform data analysis based on e-mail content. The
+goal of this article is to showcase the toolkit provided with the mRpostman
+package, to describe its key features and provide some application examples.
+Keywords:
+IMAP, e-mail, R
+1. Motivation and significance
+The acknowledgement of the R programming language[1] as having re-
+markable statistical capabilities is much due to the excellence brought by
+its statistical and data analysis packages. This reputation also stands on
+the capabilities of a myriad of utility packages, which extends the use of the
+language by facilitating the integration of the steps involved in data collec-
+tion, analysis, and communication. With that in mind, and considering the
+amount of data transmitted daily through e-mail, mRpostman was conceived
+to fill the absence of an Internet Message Access Protocol (IMAP) client in
+the R statistical environment; therefore, providing an appropriate toolkit for
+electronic messages retrieval, and paving the way for e-mail data analysis in
+R.
+Email address: [email protected] (Allan V. C. Quadros)
+Preprint submitted to SoftwareX
+January 10, 2023
+arXiv:2301.03350v1  [cs.NI]  11 Dec 2022
+
+The Comprehensive R Archive Network (CRAN) has at least seven pack-
+ages for sending emails (Table 1). Whereas some of these packages aim to
+provide a plain Simple Mail Transport Protocol (SMTP) client for R (e.g.
+sendmailR and emayili), others focus on more sophisticated implementations,
+using Application Program Interfaces (API), or providing seamless integra-
+tion between SMTP and other R features such as rmarkdown[2]. However,
+despite the surplus of available clients in R, the SMTP protocol is not suit-
+able for receiving e-mails. It only allows clients to communicate with servers
+to deliver their messages.
+For the purpose of message retrieval, there are the Post Office Protocol 3
+(POP3) and the Internet Message Access Protocol (IMAP). In comparison
+with IMAP, POP3 is a very limited protocol, working as a simple interface
+for clients to download e-mails from servers. IMAP, on the other hand, is
+a much more complex protocol, and can be considered as the evolution of
+POP3, with a very different and broader set of functionalities. In contrast to
+POP3, all the messages are kept on the IMAP server and not locally. This
+means that a user can access the same mail account using parallel connections
+from different clients[3]. Besides the mail folders structure and management,
+the capacity of issuing sophisticated search queries also contribute to the
+level of complexity of the IMAP protocol.
+Amid CRAN packages for e-mail communication, only gmailr and edeR
+have IMAP capabilities (Table 1). However, those capabilities are restricted
+to Gmail accounts and few IMAP functionalities. Although gmailr supports
+both protocols, the package is more SMTP-focused, which explains its low
+number of IMAP features. Therefore, R was clearly lacking a broader IMAP
+client solution. It was in that mainstay that mRpostman was conceived.
+2
+
+Features
+protocol
+mail
+providers
+search
+queries
+message
+fetch
+attachment
+extrac-
+tion
+mailbox
+manage-
+ment
+active
+develop-
+ment
+sendmailR[4]
+SMTP
+-
+-
+-
+-
+-
+-
+mailR[5]
+SMTP
+-
+-
+-
+-
+-
+-
+mail[6]
+SMTP
+-
+-
+-
+-
+-
+-
+blatr[7]
+SMTP
+-
+-
+-
+-
+-
+-
+gmailr[8]
+SMTP/IMAP Gmail
+no
+limited
+limited
+no
+yes
+blastula[9]
+SMTP
+-
+-
+-
+-
+-
+-
+emayili[10]
+SMTP
+-
+-
+-
+-
+-
+-
+edeR[11]
+IMAP
+Gmail
+no
+limited
+no
+no
+no
+mRpostman
+IMAP
+all
+yes
+yes
+yes
+yes
+yes
+Table 1: Comparison of the current available CRAN packages for e-mail communica-
+tion. The following attributes are evaluated: protocol - the supported protocol (SMTP
+or IMAP); mail providers - if the IMAP protocol is supported, which mail providers are
+supported by the package; Features - which type of IMAP features are available in the
+package; active development - if the package is currently under active development. If the
+package does not provide IMAP support, the remaining fields do not apply.
+In this article, we present a brief view of the main functionalities of the
+package and its applications.
+2. Software description
+mRpostman is conceived to be an easy-to-use session-based IMAP client
+for R. The package implements intuitive methods for executing the major-
+ity of the IMAP commands described in the Request for Comments 35011,
+such as mailbox management, and selectively search and fetch of message at-
+tributes. The package also implements complementary functions for decoding
+quoted-printable and base 64 content, following the MIME specification2.
+All these methods and functions play an important role in facilitating e-
+mail data analysis. We shall not overlook the amount of data analyses daily
+performed on e-mail content. The package has proved to be very useful as an
+1The RFC 3501[12] is a formal document from the Internet Engineering Task Force
+(IETF) specifying standards for the IMAP, Version 4rev1 (IMAP4rev1).
+2The RFC 2047[13] specifies rules for encoding and decoding non-ASCII characters in
+electronic messages.
+3
+
+additional feature in this workflow by, for instance, enabling the possibility
+of automating the attachments retrieval step. Also, by fetching other mes-
+sage contents, users are able to apply statistical techniques for analysing the
+frequency of e-mails with regard to some message aspect, running sentiment
+analysis on e-mail content, etc.
+Since mRpostman works as a session-based IMAP client, one can think
+of the provided methods following a natural order in which the steps shall be
+organised in the event of an IMAP session (Fig. 1). For instance, if the goal
+is to search messages within a specific period of time and/or containing a
+specific word, first we need to configure the connection to the IMAP server;
+then, choose a mail folder where the search is to be performed; and execute
+the single criteria (left) or the custom multi-criteria search (right). If the
+user intends to fetch the matched message(s) or its parts, additional fetch
+steps can be chained to the described schema.
+con <- configure imap()
+con$select folder()
+con$fetch *()
+con$search *()
+con$search()
+a connection
+object is configured
+a mailbox
+is
+selected
+a mailbox
+is
+selected
+return message ids
+return message ids
+Fig. 1: Basic schema for fetching the full content of a message or its parts after a search
+query.
+mRpostman is flexible in the sense that the aforementioned steps can be
+used either under the tidy framework, with pipes[14], or via the conventional
+base R approach.
+4
+
+3. Software architeture
+The software was designed following the object-oriented framework from
+the R6 package[15]. A class called ImapCon is implemented to retain and
+organize the necessary IMAP connection parameters. All the methods that
+derive from this class will serve one of the two following purposes: to issue a
+request toward the IMAP server (request methods) or re-configure an existing
+IMAP connection (reset methods).
+In order to execute IMAP commands, this package makes extensive use
+of the curl[16] R package3. All mRpostman’s request methods are built on
+top of the so-called curl handles. Under the hood, a curl handle consists
+of a C pointer variable that gathers the necessary parameters to execute a
+request to the server. As a matter of fact, the handle itself does not issue
+any command, but is used as a parameter inside a curl’s fetch function. This
+last object is the one that actually triggers the request to the server, ranging
+from mail folder selection to search queries, or message fetch requests.
+The object-oriented framework combined with the use of one curl handle
+per session enables mRpostman to elegantly run as a session based IMAP
+client, without demanding a connection reconfiguration between commands.
+For example, if a mail folder is selected on the current session, all requests
+using the same connection token will be performed on the selected folder,
+unless the user re-selects a different one.
+3.1. Software functionalities
+3.1.1. Configuring an IMAP connection
+As we demonstrated in Fig. 1, the first step for using mRpostman is to
+configure an IMAP connection. It consists of creating a connection token
+object of class ImapCon that will retain all the relevant information to issue
+requests toward the server.
+configure imap is the function used to configure and create a new IMAP
+connection.
+The mandatory arguments are three character strings: url,
+username, and password for plain authentication; or url, username, and
+xoauth2 bearer for OAuth2.0 authentication4.
+The following example illustrates how to configure a connection to a Mi-
+crosoft Exchange IMAP 4 server; more specifically, to an Office 365 Outlook
+account using plain authentication.
+library("mRpostman")
+3The curl package is a binding for the libcurl[17] C library.
+4Please refer to the “IMAP OAuth2.0 authentication in mRpostman” vignette in [18].
+5
+
+con <- configure_imap(url = "imaps://outlook.office365.com",
+username = "[email protected]",
+password = rstudioapi::askForPassword())
+We opted for using an Outlook Office 365 account as an example in order
+to highlight the difference between mRpostman and the other two CRAN
+packages which, although also capable of receiving e-mails, are restricted to
+Gmail accounts and fewer IMAP functionalities. Although mRpostman is
+able to theoretically connect to any mail provider5, the Outlook Office 365
+service is broadly used by universities and companies. This enriches the range
+of data analyses applications of this package, thus justifying our choice.
+In a hypothetical situation where the user needs to simultaneously con-
+nect to more than one e-mail account (in different providers or not) in the
+same R session, it can be easily attained by creating and configuring multiple
+connection tokens, such as con1, con2, and so on.
+3.1.2. Selecting a mail folder
+Mailboxes are structured as folders in the IMAP protocol. This allows us
+to replicate many of the operations done in a local folder such as creating,
+renaming or deleting folders. As messages are kept inside the mail folders,
+users need to select one of them whenever they intend to execute a search,
+fetch or other message-related operation, as presented in Fig. 1.
+In this sense, the select folder method is one of the key features of
+this package. It selects a mail folder for the current IMAP section. The
+mandatory argument is a character string containing the name of the folder
+to be selected.
+Supposing that we want to select the "INBOX" folder and considering that
+we are going to use the same connection object (con) that has been previously
+created, the command would be:
+con$select_folder(name = "INBOX")
+Further details on other important mailbox management features are pro-
+vided in [18].
+3.1.3. Message search
+The IMAP protocol is designed to allow the execution of single or multi-
+criteria queries on the mailboxes. This package implements a vast range of
+5Besides Outlook Office 365, the package has been already successfully tested with
+Gmail, Yahoo, Yandex, AOL, and Hotmail accounts.
+6
+
+IMAP search commands, which consist of a critical feature for performing
+data analysis on email content.
+As of its version 1.0.0, mRpostman has five types of single-criterion
+search methods implemented: by date; string; flag, size; and span of time
+(WITHIN extension)6.
+The custom-search, on the other hand, enables the
+execution of multi-criteria queries by allowing the combination of two or
+more types of search. However, in this article, we will focus on the single-
+criterion search-by-string type.
+The search string method searches messages that contain a specific
+string or expression. One or more specific sections of a message, such as the
+TEXT section or the TO header field, for example, must be specified.
+In the following code snippet, we search for messages from senders whose
+mail domain is "@ksu.edu".
+ids <- con$search_string(expr = "@ksu.edu", where = "FROM")
+The resulting object is a vector containing the matched unique ids (UID)
+or the message sequence numbers7 such as presented below:
+[1]
+60 145 147 159 332 333 336 338 341 428
+Further details on the other single-search methods and the custom-search
+method available in this package are provided in [18].
+3.1.4. Message fetch
+After executing a search query, users may be interested in fetching the
+full content or some part of the messages indicated in the search results. In
+this regard, mRpostman implements six types of fetch features:
+fetch body Fetches the message body (message’s full content), or an
+specified MIME level, which can refer to the text or the attachments if there
+are any.
+fetch header Fetches the message header, which comprises all the com-
+ponents of the HEADER section of a message. Besides the traditional ones
+(from, to, cc, subject), it may include several more fields.
+fetch metadata Fetches the message metadata, which consists of some
+message’s attributes such as the internal date, and the envelope (from, to,
+cc, and subject fields).
+6The WITHIN extension is not supported by all IMAP servers. A call to the list -
+server capabilities method will present all the IMAP extensions supported by the
+mail provider[18].
+7More details on the message identification methodology deployed by the IMAP pro-
+tocol are provided in [19, 12, 18].
+7
+
+fetch text Fetches the message text section, which can comprise attach-
+ment MIME levels if applicable.
+Each of these methods can be seamlessly integrated into a previous search
+operation so that the returned ids are used as input for the fetch method.
+Above all, these methods consist of a powerful source of information for
+performing data analysis on e-mail content. Here, we mimic the extraction
+of the TEXT portion of a message. Although there is a fetch text method,
+the recommended approach is to use fetch body(..., mime level = 1L)
+because the former may collect attachment parts along with the message
+text.
+out <- ids %>%
+fetch_body(mime_level = 1L)
+Once the messages are fetched, the text can be cleaned and decoded with
+the clean msg text helper function. A subsequent call to the writeLines
+base R function produces a clean printing of the fetched text:
+cleaned_text <- clean_msg_text(msg_list = out)
+writeLines(cleaned_text[[1]])
+Receipt Number: XXXXXXX
+Customer: Vieira de Castro Quadros, Allan
+Kansas State University
+Current Date: 04/15/2020
+Description
+Amount
+--------------------------------------------------------------------------------
+HOUSING & DINING
+$30.00
+User Number: XXXXXXXXX
+Total
+$30.00
+Payments Received
+Amount
+--------------------------------------------------------------------------------
+07 CREDIT CARD PAYMENTS
+$30.00
+Visa XXXXXXXXXXXX8437
+Authorization # XXXXXX
+Total
+$30.00
+Thank you for the payment.
+Besides other applications, the exported function clean msg text can be
+used to decode hexadecimal and base 64 characters in the text and other
+parts of the message. In some locales such as French, German or Portuguese
+speaking countries, message parts may contain non-ASCII characters. SMTP
+servers, then, encode it using the RFC 2047 specifications when sending the
+e-mail. In these cases, clean msg text is capable of correctly decoding the
+non-ASCII characters.
+8
+
+3.1.5. Attachment extraction
+In its pretension to be an IMAP client for R, mRpostman provides meth-
+ods that enable users to list and download message payloads. This feature
+can be particularly critical for automating the analysis of attachment data
+files, for instance.
+Attachments can be downloaded using two different approaches in this
+package: extending the fetch text/body operation by adding an attach-
+ment extraction step at the end of the workflow with get attachments; or
+directly fetching attachment parts via the fetch attachments method. In
+this article, we focus on the first type of attachment methods, adding a step
+to our previous workflow.
+The get attachments method extracts attachment files from the fetched
+messages and saves these files to the disk. In the following code excerpt, we
+extract attachments in a unique pipeline that gathers fetching and search
+steps.
+con$search_string(expr = "@ksu.edu", where = "FROM") %>%
+con$fetch_text() %>%
+con$get_attachments()
+During the execution, the software locally saves the extracted attach-
+ments into sub-folders inside the user’s working directory. These sub-folders
+are named following the messages’ ids.
+The attachments are placed into
+their respective messages’ sub-folders as demonstrated in Fig. 2. Note that
+the parent levels are named after the informed username and the selected
+mail folder.
+For more information on the other attachment-related methods, the reader
+should refer to the documentation in [18].
+4. Illustrative Examples
+To demonstrate the capabilities of the proposed software, we explore two
+use cases of this package in support of data analysis tasks: a simple study
+of the frequency of e-mails grouped by senders; and a sentiment analysis
+run on a set of e-mails received during a period. The R scripts needed for
+reproducing these examples are provided in the appendixes. Although the
+results cannot be exactly reproduced once it reflects the author’s mailbox
+contents, they can be easily adapted to the reader’s context.
+9
+
+. (working directory)
+[email protected]
+INBOX
+141
+final.zip
+prob plot.svg
+staa2072.pdf
+144
+app.R
+image001.png
+recording.mp4
+Fig. 2: Local directory tree for the extracted attachment files
+4.1. Frequency analysis of e-mail data
+In the first example, we run a simple analysis of the e-mail frequency with
+regard to senders. This can be especially useful in professional fields, such
+as marketing and customer service offices. A period of analysis was defined,
+and a search-by-date is performed using the search period method. Then,
+senders’ information for the returned ids are fetched via fetch metadata,
+using the ENVELOPE attribute. After some basic manipulation with regular
+expressions, the data is ready to be plotted as shown in Fig. 3.
+10
+
+omitted@tbs−education.fr
+[email protected]
+[email protected]
+[email protected]
+no−[email protected]
+E−mail Frequency (by sender)
+count
+0
+2
+4
+6
+8
+10
+12
+14
+ResearchGate
+Cortana
+Claudio Piga
+Chen, Daqing
+MANTOVANI Andrea
+Period: 01−Nov to 01−Dec−2020
+Fig. 3: An example of e-mail frequency analysis grouped by sender
+The same kind of analysis can be replicated for the messages’ subjects
+with only a few modifications in the regular expressions code chunks. Con-
+sidering that some companies/users deal with subject-standardized e-mails,
+this approach can be useful to analyze the frequency of e-mails with regard
+to different categories of subjects.
+4.2. Sentiment analysis on e-mail data
+For the sentiment analysis example, we also define a period of analysis and
+run a search period query. Then, we retrieve the text part of the messages
+by fetching the first MIME level with fetch body(..., mime level = 1L).
+The texts go trough a first cleaning step with a call to the clean msg text
+function.
+After further cleaning procedures, we use a lexicon[20] via the
+syuzhet package[21] to evaluate the sentiment of each e-mail. The output
+below is a subset of the resulting data frame. The last two columns indicate,
+respectively, the counts of negative and positive words for each message.
+The other columns provide counts related to detailed emotions, which are
+not necessarily positive nor negative.
+anger anticipation disgust fear joy sadness surprise trust negative positive
+body91
+1
+5
+1
+1
+2
+2
+0
+9
+1
+13
+body92
+0
+1
+0
+0
+1
+0
+0
+3
+0
+1
+body93
+0
+3
+0
+2
+0
+1
+2
+2
+1
+3
+body94
+0
+1
+0
+1
+0
+0
+1
+4
+1
+4
+body95
+0
+5
+0
+0
+3
+0
+2
+8
+0
+13
+body96
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+body97
+4
+20
+4
+11
+13
+11
+4
+25
+16
+51
+11
+
+body98
+0
+3
+0
+0
+2
+0
+1
+4
+0
+6
+body99
+3
+9
+1
+6
+1
+5
+2
+16
+14
+24
+body100
+4
+15
+1
+13
+6
+7
+6
+15
+16
+31
+5. Impact
+As we have demonstrated, mRpostman clearly fills an existent gap of a
+broad, complete, and, at the same time, easy-to-use IMAP client for the
+R language. The package has consolidated itself as an important tool for
+collecting massive e-mail content, thus contributing to data analysis tasks in
+R.
+Although all sort of users have been taking advantage of this package,
+we are inclined to think that its use has been prevailing amid companies.
+We have received a considerable number of feedback from enterprise users
+who deploy mRpostman as an additional feature for automatically produc-
+ing daily reports based on attachment data files.
+Besides this, there are
+important applications for marketing and post-sales departments, for exam-
+ple. They can also deploy this package to collect e-mail data for analyzing
+e-mail frequency, or performing sentiment analysis, as we have demonstrated
+in Section 4.
+6. Conclusions
+mRpostman aims to provide an easy-to-use IMAP client for R. Its design
+allows the efficient, elegant, and intuitive execution of several IMAP com-
+mands on a wide range of mail providers. Consequently, users cannot only
+manage their mailboxes but also conduct e-mail data analysis from inside R.
+Finally, because IMAP is such a complex protocol, this package is in con-
+stant development, which means that new features are to be implemented in
+future versions.
+7. Conflict of Interest
+No conflict of interest exists: We wish to confirm that there are no known
+conflicts of interest associated with this publication and there has been no
+significant financial support for this work that could have influenced its out-
+come.
+Acknowledgements
+The author would like to acknowledge the Department of Statistics at
+Kansas State University (K-State) for the assistantship provided for his doc-
+torate studies. He wants to especially thank Dr. Christopher Vahl and Dr.
+12
+
+Michael Higgins for the academic support. The author also acknowledges the
+academic guidance of Dr. George von Borries at the University of Brasilia
+(UnB). The contents of this article are the responsibility of the author and
+do not reflect the views of K-State or UnB.
+Appendix A. Code for example 1
+library(mRpostman)
+con <- configure_imap(
+url="imaps://outlook.office365.com",
+username="[email protected]",
+password=rstudioapi::askForPassword()
+)
+con$select_folder(name = "INBOX")
+meta_res <- con$search_period(since_date_char = "01-Nov-2020",
+before_date_char = "01-Dec-2020") %>%
+con$fetch_metadata(attribute = "ENVELOPE")
+# cleaning
+# step 1
+clean_meta <- lapply(meta_res, function(x){
+regmatches(x, regexpr(pattern = "\\(\\(.*\"(.*?)\"\\)\\)", x, perl = TRUE))
+})
+# step 2
+# cleaning Ccs
+senders1 <- lapply(clean_meta, function(x){
+gsub(")) NIL .*$|)) .*$|))$", "", x)
+})
+# step 3
+senders1 <- lapply(senders1, function(x){
+gsub(’^\\(\\(|\"+’, "", x)
+})
+# splitting
+name <- c()
+email <- c()
+for (i in seq_along(senders1)){
+# i = 1
+out <- unlist(strsplit(senders1[[i]], " NIL "))
+name <- c(name, out[1])
+email <- c(email, gsub(" ", "@", out[2]))
+}
+df <- data.frame(name, email)
+df$name <- decode_mime_header(string = as.character(df$name))
+df2 <- as.data.frame(table(df$email))
+colnames(df2) <- c("email", "count")
+df2 <- df2[order(-df2[,2]), ][1:5,]
+df2$name <- unique(df$name[df$email %in% df2$email])
+par(mar=c(5,13,4,1)+.1)
+pal_cols <- c(’#3B4992FF’, ’#EE0000FF’, ’#008B45FF’, ’#631879FF’, ’#008280FF’)
+barplot(rev(df2$count),
+main = "E-mail Frequency (by sender)",
+xlab = "count",
+names.arg = rev(df2$email),
+las = 1,
+col = pal_cols,
+horiz = TRUE)
+mysubtitle <- "Period: 01-Nov to 01-Dec-2020"
+legend(x = "bottomright", legend = df2$name, fill = rev(pal_cols), bty = "n", y.intersp = 1)
+mtext(side=3, line=0.3, at=-0.07, adj=0, cex=0.9, mysubtitle)
+13
+
+Appendix B. Code for example2
+library(mRpostman)
+con <- configure_imap(url="imaps://outlook.office365.com",
+username="[email protected]",
+password=rstudioapi::askForPassword(),
+timeout_ms = 20000
+)
+con$select_folder("INBOX")
+ids <- con$search_period(since_date_char = "10-Oct-2020", before_date_char = "20-Dec-2020")
+fetch_res2 <- ids %>%
+con$fetch_body(mime_level = 1L)
+cleaned_text_list <- clean_msg_text(msg_list = fetch_res2)
+cleaned_text_list[[4]]
+for (i in seq_along(cleaned_text_list)) {
+clean_text <- gsub("\r\n", " ", cleaned_text_list[[i]])
+clean_text <- unlist(strsplit(clean_text, " "))
+words <- clean_text[!grepl("\\d|_|http|www|nbsp|@|(?<=[[:lower:]])(?=[[:upper:]])",
+clean_text, perl = TRUE)]
+words <- tolower(gsub("\\W+", "", words))
+words <- gsub(’[^a-zA-Z|[:blank:]]’, "", words)
+cleaned_text_list[[i]] <- paste(words, collapse = " ")
+}
+cleaned_text_df <- do.call(’rbind’, cleaned_text_list)
+library(syuzhet)
+email_sentiment_df <-get_nrc_sentiment(cleaned_text_df)
+rownames(email_sentiment_df) <- rownames(cleaned_text_df)
+head(email_sentiment_df,10)
+References
+[1] R Core Team, R: A Language and Environment for Statistical Comput-
+ing, R Foundation for Statistical Computing, Vienna, Austria (2020).
+URL https://www.R-project.org/
+[2] J. Allaire, Y. Xie, J. McPherson, J. Luraschi, K. Ushey, A. Atkins,
+H. Wickham, J. Cheng, W. Chang, R. Iannone, rmarkdown: Dynamic
+Documents for R, r package version 2.5 (2020).
+URL https://github.com/rstudio/rmarkdown
+[3] P. Heinlein, P. Hartleben, The Book of IMAP: Building a Mail Server
+with Courier and Cyrus, No Starch Press, 2008.
+[4] O. Mersmann, sendmailR: send email using R, r package version 1.2-1
+(2014).
+URL https://CRAN.R-project.org/package=sendmailR
+[5] R. Premraj, mailR: A Utility to Send Emails from R, r package version
+0.4.1 (2015).
+URL https://CRAN.R-project.org/package=mailR
+14
+
+[6] L. Himmelmann, mail: Sending Email Notifications from R, r package
+version 1.0 (2011).
+URL https://CRAN.R-project.org/package=mail
+[7] S. M. Bache, blatr: Send Emails Using ’Blat’ for Windows, r package
+version 1.0.1 (2015).
+URL https://CRAN.R-project.org/package=blatr
+[8] J. Hester, gmailr: Access the ’Gmail’ ’RESTful’ API, r package version
+1.0.0 (2019).
+URL https://CRAN.R-project.org/package=gmailr
+[9] R. Iannone, J. Cheng, blastula: Easily Send HTML Email Messages, r
+package version 0.3.2 (2020).
+URL https://CRAN.R-project.org/package=blastula
+[10] A. B. Collier, emayili: Send Email Messages, r package version 0.4.4
+(2020).
+URL https://CRAN.R-project.org/package=emayili
+[11] A. B. Collier, edeR: Email Data Extraction Using R, r package version
+1.0.0 (2014).
+URL https://CRAN.R-project.org/package=edeR
+[12] M. Crispin, Internet message access protocol - version 4rev1, request for
+Comments 3501 (RFC 3501), Internet Engineering Task Force (IETF)
+(2003).
+URL https://tools.ietf.org/html/rfc3501
+[13] K. Moore, Multipurpose Internet Mail Extensions (MIME), part three:
+Message header extensions for non-ascii text, request for Comments 2047
+(RFC 2047), Internet Engineering Task Force (IETF) (1996).
+URL https://tools.ietf.org/html/rfc2047
+[14] S. M. Bache, H. Wickham, magrittr: A Forward-Pipe Operator for R, r
+package version 1.5 (2014).
+URL https://CRAN.R-project.org/package=magrittr
+[15] W. Chang, R6: Encapsulated Classes with Reference Semantics, r pack-
+age version 2.5.0 (2020).
+URL https://CRAN.R-project.org/package=R6
+[16] J. Ooms, curl: A Modern and Flexible Web Client for R, r package
+version 4.3 (2020).
+URL https://CRAN.R-project.org/package=curl
+15
+
+[17] D. Stenberg, libcurl - the multiprotocol file transfer library, version
+7.69.1 (2020).
+URL https://curl.haxx.se/
+[18] A. Quadros, mRpostman: An IMAP Client for R, r package version
+1.0.0 (2020).
+URL https://allanvc.github.io/
+[19] P. Resnick, Internet message format, request for Comments 5322 (RFC
+5322), Internet Engineering Task Force (IETF) (2008).
+URL https://tools.ietf.org/html/rfc5322
+[20] S. Mohammad, P. Turney, Emotions evoked by common words and
+phrases:
+Using mechanical turk to create an emotion lexicon, in:
+CAAGET ’10: Proceedings of the NAACL HLT 2010 Workshop on
+Computational Approaches to Analysis and Generation of Emotion in
+Text, Los Angeles, California, 2010, p. 26–34, june, 2010.
+URL http://saifmohammad.com/WebPages/lexicons.html
+[21] M. L. Jockers, Syuzhet: Extract Sentiment and Plot Arcs from Text, r
+package version 1.0.4 (2015).
+URL https://github.com/mjockers/syuzhet
+16
+

GdE1T4oBgHgl3EQfrAWz/content/tmp_files/load_file.txt

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+page_content='mRpostman: An IMAP Client for R  Allan V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Quadros  Department of Statistics  Kansas State University  Manhattan, KS 66506, United States  Abstract  Internet Message Access Protocol (IMAP) clients are a common feature in  several programming languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Despite having some packages for electronic  messages retrieval, the R language, until recently, lacked a broader solution,  capable of coping with different IMAP servers and providing a wide spec-  trum of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' mRpostman covers most of the IMAP 4rev1 functionalities  by implementing tools for message searching, selective fetching of message  attributes, mailbox management, attachment extraction, and several other  IMAP features that can be executed in virtually any mail provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' By doing  so, it enables users to perform data analysis based on e-mail content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' The  goal of this article is to showcase the toolkit provided with the mRpostman  package, to describe its key features and provide some application examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  Keywords:  IMAP, e-mail, R  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Motivation and significance  The acknowledgement of the R programming language[1] as having re-  markable statistical capabilities is much due to the excellence brought by  its statistical and data analysis packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' This reputation also stands on  the capabilities of a myriad of utility packages, which extends the use of the  language by facilitating the integration of the steps involved in data collec-  tion, analysis, and communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' With that in mind, and considering the  amount of data transmitted daily through e-mail, mRpostman was conceived  to fill the absence of an Internet Message Access Protocol (IMAP) client in  the R statistical environment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' therefore, providing an appropriate toolkit for  electronic messages retrieval, and paving the way for e-mail data analysis in  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  Email address: quadros@k-state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='edu (Allan V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Quadros)  Preprint submitted to SoftwareX  January 10, 2023  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='03350v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='NI]  11 Dec 2022  The Comprehensive R Archive Network (CRAN) has at least seven pack-  ages for sending emails (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Whereas some of these packages aim to  provide a plain Simple Mail Transport Protocol (SMTP) client for R (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  sendmailR and emayili), others focus on more sophisticated implementations,  using Application Program Interfaces (API), or providing seamless integra-  tion between SMTP and other R features such as rmarkdown[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' However,  despite the surplus of available clients in R, the SMTP protocol is not suit-  able for receiving e-mails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' It only allows clients to communicate with servers  to deliver their messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  For the purpose of message retrieval, there are the Post Office Protocol 3  (POP3) and the Internet Message Access Protocol (IMAP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' In comparison  with IMAP, POP3 is a very limited protocol, working as a simple interface  for clients to download e-mails from servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' IMAP, on the other hand, is  a much more complex protocol, and can be considered as the evolution of  POP3, with a very different and broader set of functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' In contrast to  POP3, all the messages are kept on the IMAP server and not locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' This  means that a user can access the same mail account using parallel connections  from different clients[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Besides the mail folders structure and management,  the capacity of issuing sophisticated search queries also contribute to the  level of complexity of the IMAP protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  Amid CRAN packages for e-mail communication, only gmailr and edeR  have IMAP capabilities (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' However, those capabilities are restricted  to Gmail accounts and few IMAP functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Although gmailr supports  both protocols, the package is more SMTP-focused, which explains its low  number of IMAP features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Therefore, R was clearly lacking a broader IMAP  client solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' It was in that mainstay that mRpostman was conceived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  2  Features  protocol  mail  providers  search  queries  message  fetch  attachment  extrac-  tion  mailbox  manage-  ment  active  develop-  ment  sendmailR[4]  SMTP mailR[5]  SMTP mail[6]  SMTP blatr[7]  SMTP gmailr[8]  SMTP/IMAP Gmail  no  limited  limited  no  yes  blastula[9]  SMTP emayili[10]  SMTP edeR[11]  IMAP  Gmail  no  limited  no  no  no  mRpostman  IMAP  all  yes  yes  yes  yes  yes  Table 1: Comparison of the current available CRAN packages for e-mail communica-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' The following attributes are evaluated: protocol - the supported protocol (SMTP  or IMAP);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' mail providers - if the IMAP protocol is supported, which mail providers are  supported by the package;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Features - which type of IMAP features are available in the  package;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' active development - if the package is currently under active development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' If the  package does not provide IMAP support, the remaining fields do not apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  In this article, we present a brief view of the main functionalities of the  package and its applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Software description  mRpostman is conceived to be an easy-to-use session-based IMAP client  for R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' The package implements intuitive methods for executing the major-  ity of the IMAP commands described in the Request for Comments 35011,  such as mailbox management, and selectively search and fetch of message at-  tributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' The package also implements complementary functions for decoding  quoted-printable and base 64 content, following the MIME specification2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  All these methods and functions play an important role in facilitating e-  mail data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' We shall not overlook the amount of data analyses daily  performed on e-mail content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' The package has proved to be very useful as an  1The RFC 3501[12] is a formal document from the Internet Engineering Task Force  (IETF) specifying standards for the IMAP, Version 4rev1 (IMAP4rev1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  2The RFC 2047[13] specifies rules for encoding and decoding non-ASCII characters in  electronic messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  3  additional feature in this workflow by, for instance, enabling the possibility  of automating the attachments retrieval step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Also, by fetching other mes-  sage contents, users are able to apply statistical techniques for analysing the  frequency of e-mails with regard to some message aspect, running sentiment  analysis on e-mail content, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  Since mRpostman works as a session-based IMAP client, one can think  of the provided methods following a natural order in which the steps shall be  organised in the event of an IMAP session (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' For instance, if the goal  is to search messages within a specific period of time and/or containing a  specific word, first we need to configure the connection to the IMAP server;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  then, choose a mail folder where the search is to be performed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' and execute  the single criteria (left) or the custom multi-criteria search (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' If the  user intends to fetch the matched message(s) or its parts, additional fetch  steps can be chained to the described schema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  con <- configure imap()  con$select folder()  con$fetch *()  con$search *()  con$search()  a connection  object is configured  a mailbox  is  selected  a mailbox  is  selected  return message ids  return message ids  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' 1: Basic schema for fetching the full content of a message or its parts after a search  query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  mRpostman is flexible in the sense that the aforementioned steps can be  used either under the tidy framework, with pipes[14], or via the conventional  base R approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Software architeture  The software was designed following the object-oriented framework from  the R6 package[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' A class called ImapCon is implemented to retain and  organize the necessary IMAP connection parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' All the methods that  derive from this class will serve one of the two following purposes: to issue a  request toward the IMAP server (request methods) or re-configure an existing  IMAP connection (reset methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  In order to execute IMAP commands, this package makes extensive use  of the curl[16] R package3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' All mRpostman’s request methods are built on  top of the so-called curl handles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Under the hood, a curl handle consists  of a C pointer variable that gathers the necessary parameters to execute a  request to the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' As a matter of fact, the handle itself does not issue  any command, but is used as a parameter inside a curl’s fetch function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' This  last object is the one that actually triggers the request to the server, ranging  from mail folder selection to search queries, or message fetch requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  The object-oriented framework combined with the use of one curl handle  per session enables mRpostman to elegantly run as a session based IMAP  client, without demanding a connection reconfiguration between commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  For example, if a mail folder is selected on the current session, all requests  using the same connection token will be performed on the selected folder,  unless the user re-selects a different one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Software functionalities  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Configuring an IMAP connection  As we demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' 1, the first step for using mRpostman is to  configure an IMAP connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' It consists of creating a connection token  object of class ImapCon that will retain all the relevant information to issue  requests toward the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  configure imap is the function used to configure and create a new IMAP  connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  The mandatory arguments are three character strings: url,  username, and password for plain authentication;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' or url, username, and  xoauth2 bearer for OAuth2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='0 authentication4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  The following example illustrates how to configure a connection to a Mi-  crosoft Exchange IMAP 4 server;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' more specifically, to an Office 365 Outlook  account using plain authentication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  library("mRpostman")  3The curl package is a binding for the libcurl[17] C library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  4Please refer to the “IMAP OAuth2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='0 authentication in mRpostman” vignette in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  5  con <- configure_imap(url = "imaps://outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='office365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='com",  username = "user@agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='gov",  password = rstudioapi::askForPassword())  We opted for using an Outlook Office 365 account as an example in order  to highlight the difference between mRpostman and the other two CRAN  packages which, although also capable of receiving e-mails, are restricted to  Gmail accounts and fewer IMAP functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Although mRpostman is  able to theoretically connect to any mail provider5, the Outlook Office 365  service is broadly used by universities and companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' This enriches the range  of data analyses applications of this package, thus justifying our choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  In a hypothetical situation where the user needs to simultaneously con-  nect to more than one e-mail account (in different providers or not) in the  same R session, it can be easily attained by creating and configuring multiple  connection tokens, such as con1, con2, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Selecting a mail folder  Mailboxes are structured as folders in the IMAP protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' This allows us  to replicate many of the operations done in a local folder such as creating,  renaming or deleting folders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' As messages are kept inside the mail folders,  users need to select one of them whenever they intend to execute a search,  fetch or other message-related operation, as presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  In this sense, the select folder method is one of the key features of  this package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' It selects a mail folder for the current IMAP section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' The  mandatory argument is a character string containing the name of the folder  to be selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  Supposing that we want to select the "INBOX" folder and considering that  we are going to use the same connection object (con) that has been previously  created, the command would be:  con$select_folder(name = "INBOX")  Further details on other important mailbox management features are pro-  vided in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Message search  The IMAP protocol is designed to allow the execution of single or multi-  criteria queries on the mailboxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' This package implements a vast range of  5Besides Outlook Office 365, the package has been already successfully tested with  Gmail, Yahoo, Yandex, AOL, and Hotmail accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  6  IMAP search commands, which consist of a critical feature for performing  data analysis on email content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  As of its version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='0, mRpostman has five types of single-criterion  search methods implemented: by date;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' string;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' flag, size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' and span of time  (WITHIN extension)6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  The custom-search, on the other hand, enables the  execution of multi-criteria queries by allowing the combination of two or  more types of search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' However, in this article, we will focus on the single-  criterion search-by-string type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  The search string method searches messages that contain a specific  string or expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' One or more specific sections of a message, such as the  TEXT section or the TO header field, for example, must be specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  In the following code snippet, we search for messages from senders whose  mail domain is "@ksu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='edu".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  ids <- con$search_string(expr = "@ksu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='edu", where = "FROM")  The resulting object is a vector containing the matched unique ids (UID)  or the message sequence numbers7 such as presented below:  [1]  60 145 147 159 332 333 336 338 341 428  Further details on the other single-search methods and the custom-search  method available in this package are provided in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Message fetch  After executing a search query, users may be interested in fetching the  full content or some part of the messages indicated in the search results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' In  this regard, mRpostman implements six types of fetch features:  fetch body Fetches the message body (message’s full content), or an  specified MIME level, which can refer to the text or the attachments if there  are any.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  fetch header Fetches the message header, which comprises all the com-  ponents of the HEADER section of a message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Besides the traditional ones  (from, to, cc, subject), it may include several more fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  fetch metadata Fetches the message metadata, which consists of some  message’s attributes such as the internal date, and the envelope (from, to,  cc, and subject fields).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  6The WITHIN extension is not supported by all IMAP servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' A call to the list -  server capabilities method will present all the IMAP extensions supported by the  mail provider[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  7More details on the message identification methodology deployed by the IMAP pro-  tocol are provided in [19, 12, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  7  fetch text Fetches the message text section, which can comprise attach-  ment MIME levels if applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  Each of these methods can be seamlessly integrated into a previous search  operation so that the returned ids are used as input for the fetch method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  Above all, these methods consist of a powerful source of information for  performing data analysis on e-mail content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Here, we mimic the extraction  of the TEXT portion of a message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Although there is a fetch text method,  the recommended approach is to use fetch body(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=', mime level = 1L)  because the former may collect attachment parts along with the message  text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  out <- ids %>%  fetch_body(mime_level = 1L)  Once the messages are fetched, the text can be cleaned and decoded with  the clean msg text helper function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' A subsequent call to the writeLines  base R function produces a clean printing of the fetched text:  cleaned_text <- clean_msg_text(msg_list = out)  writeLines(cleaned_text[[1]])  Receipt Number: XXXXXXX  Customer: Vieira de Castro Quadros, Allan  Kansas State University  Current Date: 04/15/2020  Description  Amount  --------------------------------------------------------------------------------  HOUSING & DINING  $30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='00  User Number: XXXXXXXXX  Total  $30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='00  Payments Received  Amount  --------------------------------------------------------------------------------  07 CREDIT CARD PAYMENTS  $30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='00  Visa XXXXXXXXXXXX8437  Authorization # XXXXXX  Total  $30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='00  Thank you for the payment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  Besides other applications, the exported function clean msg text can be  used to decode hexadecimal and base 64 characters in the text and other  parts of the message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' In some locales such as French, German or Portuguese  speaking countries, message parts may contain non-ASCII characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' SMTP  servers, then, encode it using the RFC 2047 specifications when sending the  e-mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' In these cases, clean msg text is capable of correctly decoding the  non-ASCII characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  8  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Attachment extraction  In its pretension to be an IMAP client for R, mRpostman provides meth-  ods that enable users to list and download message payloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' This feature  can be particularly critical for automating the analysis of attachment data  files, for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  Attachments can be downloaded using two different approaches in this  package: extending the fetch text/body operation by adding an attach-  ment extraction step at the end of the workflow with get attachments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' or  directly fetching attachment parts via the fetch attachments method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' In  this article, we focus on the first type of attachment methods, adding a step  to our previous workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  The get attachments method extracts attachment files from the fetched  messages and saves these files to the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' In the following code excerpt, we  extract attachments in a unique pipeline that gathers fetching and search  steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  con$search_string(expr = "@ksu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='edu", where = "FROM") %>%  con$fetch_text() %>%  con$get_attachments()  During the execution, the software locally saves the extracted attach-  ments into sub-folders inside the user’s working directory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' These sub-folders  are named following the messages’ ids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  The attachments are placed into  their respective messages’ sub-folders as demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Note that  the parent levels are named after the informed username and the selected  mail folder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  For more information on the other attachment-related methods, the reader  should refer to the documentation in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Illustrative Examples  To demonstrate the capabilities of the proposed software, we explore two  use cases of this package in support of data analysis tasks: a simple study  of the frequency of e-mails grouped by senders;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' and a sentiment analysis  run on a set of e-mails received during a period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' The R scripts needed for  reproducing these examples are provided in the appendixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Although the  results cannot be exactly reproduced once it reflects the author’s mailbox  contents, they can be easily adapted to the reader’s context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  9  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' (working directory)  user@company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='com  INBOX  141  final.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='zip  prob plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='svg  staa2072.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='pdf  144  app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='R  image001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='png  recording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='mp4  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' 2: Local directory tree for the extracted attachment files  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Frequency analysis of e-mail data  In the first example, we run a simple analysis of the e-mail frequency with  regard to senders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' This can be especially useful in professional fields, such  as marketing and customer service offices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' A period of analysis was defined,  and a search-by-date is performed using the search period method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Then,  senders’ information for the returned ids are fetched via fetch metadata,  using the ENVELOPE attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' After some basic manipulation with regular  expressions, the data is ready to be plotted as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  10  omitted@tbs−education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='fr  omitted@lsbu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='uk  omitted@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='com  cortana@microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='com  no−reply@researchgatemail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='net  E−mail Frequency (by sender)  count  0  2  4  6  8  10  12  14  ResearchGate  Cortana  Claudio Piga  Chen, Daqing  MANTOVANI Andrea  Period: 01−Nov to 01−Dec−2020  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' 3: An example of e-mail frequency analysis grouped by sender  The same kind of analysis can be replicated for the messages’ subjects  with only a few modifications in the regular expressions code chunks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Con-  sidering that some companies/users deal with subject-standardized e-mails,  this approach can be useful to analyze the frequency of e-mails with regard  to different categories of subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Sentiment analysis on e-mail data  For the sentiment analysis example, we also define a period of analysis and  run a search period query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Then, we retrieve the text part of the messages  by fetching the first MIME level with fetch body(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=', mime level = 1L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  The texts go trough a first cleaning step with a call to the clean msg text  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  After further cleaning procedures, we use a lexicon[20] via the  syuzhet package[21] to evaluate the sentiment of each e-mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' The output  below is a subset of the resulting data frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' The last two columns indicate,  respectively, the counts of negative and positive words for each message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  The other columns provide counts related to detailed emotions, which are  not necessarily positive nor negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  anger anticipation disgust fear joy sadness surprise trust negative positive  body91  1  5  1  1  2  2  0  9  1  13  body92  0  1  0  0  1  0  0  3  0  1  body93  0  3  0  2  0  1  2  2  1  3  body94  0  1  0  1  0  0  1  4  1  4  body95  0  5  0  0  3  0  2  8  0  13  body96  0  0  0  0  0  0  0  0  0  0  body97  4  20  4  11  13  11  4  25  16  51  11  body98  0  3  0  0  2  0  1  4  0  6  body99  3  9  1  6  1  5  2  16  14  24  body100  4  15  1  13  6  7  6  15  16  31  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Impact  As we have demonstrated, mRpostman clearly fills an existent gap of a  broad, complete, and, at the same time, easy-to-use IMAP client for the  R language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' The package has consolidated itself as an important tool for  collecting massive e-mail content, thus contributing to data analysis tasks in  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  Although all sort of users have been taking advantage of this package,  we are inclined to think that its use has been prevailing amid companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  We have received a considerable number of feedback from enterprise users  who deploy mRpostman as an additional feature for automatically produc-  ing daily reports based on attachment data files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  Besides this, there are  important applications for marketing and post-sales departments, for exam-  ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' They can also deploy this package to collect e-mail data for analyzing  e-mail frequency, or performing sentiment analysis, as we have demonstrated  in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Conclusions  mRpostman aims to provide an easy-to-use IMAP client for R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Its design  allows the efficient, elegant, and intuitive execution of several IMAP com-  mands on a wide range of mail providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Consequently, users cannot only  manage their mailboxes but also conduct e-mail data analysis from inside R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  Finally, because IMAP is such a complex protocol, this package is in con-  stant development, which means that new features are to be implemented in  future versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Conflict of Interest  No conflict of interest exists: We wish to confirm that there are no known  conflicts of interest associated with this publication and there has been no  significant financial support for this work that could have influenced its out-  come.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  Acknowledgements  The author would like to acknowledge the Department of Statistics at  Kansas State University (K-State) for the assistantship provided for his doc-  torate studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' He wants to especially thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Christopher Vahl and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  12  Michael Higgins for the academic support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' The author also acknowledges the  academic guidance of Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' George von Borries at the University of Brasilia  (UnB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' The contents of this article are the responsibility of the author and  do not reflect the views of K-State or UnB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Code for example 1  library(mRpostman)  con <- configure_imap(  url="imaps://outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='office365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='com",  username="user@company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='com",  password=rstudioapi::askForPassword()  )  con$select_folder(name = "INBOX")  meta_res <- con$search_period(since_date_char = "01-Nov-2020",  before_date_char = "01-Dec-2020") %>%  con$fetch_metadata(attribute = "ENVELOPE")  # cleaning  # step 1  clean_meta <- lapply(meta_res, function(x){  regmatches(x, regexpr(pattern = "\\\\(\\\\(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='*\\"(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='*?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' )\\"\\\\)\\\\)", x, perl = TRUE))  })  # step 2  # cleaning Ccs  senders1 <- lapply(clean_meta, function(x){  gsub(")) NIL .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' *$|)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' *$|))$", "", x)  })  # step 3  senders1 <- lapply(senders1, function(x){  gsub(’^\\\\(\\\\(|\\"+’, "", x)  })  # splitting  name <- c()  email <- c()  for (i in seq_along(senders1)){  # i = 1  out <- unlist(strsplit(senders1[[i]], " NIL "))  name <- c(name, out[1])  email <- c(email, gsub(" ", "@", out[2]))  }  df <- data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='frame(name, email)  df$name <- decode_mime_header(string = as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='character(df$name))  df2 <- as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='frame(table(df$email))  colnames(df2) <- c("email", "count")  df2 <- df2[order(-df2[,2]), ][1:5,]  df2$name <- unique(df$name[df$email %in% df2$email])  par(mar=c(5,13,4,1)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='1)  pal_cols <- c(’#3B4992FF’, ’#EE0000FF’, ’#008B45FF’, ’#631879FF’, ’#008280FF’)  barplot(rev(df2$count),  main = "E-mail Frequency (by sender)",  xlab = "count",  names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='arg = rev(df2$email),  las = 1,  col = pal_cols,  horiz = TRUE)  mysubtitle <- "Period: 01-Nov to 01-Dec-2020"  legend(x = "bottomright", legend = df2$name, fill = rev(pal_cols), bty = "n", y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='intersp = 1)  mtext(side=3, line=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='3, at=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='07, adj=0, cex=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='9, mysubtitle)  13  Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Code for example2  library(mRpostman)  con <- configure_imap(url="imaps://outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='office365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='com",  username="user@company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='com",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  password=rstudioapi::askForPassword(),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  timeout_ms = 20000  )  con$select_folder("INBOX")  ids <- con$search_period(since_date_char = "10-Oct-2020",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' before_date_char = "20-Dec-2020")  fetch_res2 <- ids %>%  con$fetch_body(mime_level = 1L)  cleaned_text_list <- clean_msg_text(msg_list = fetch_res2)  cleaned_text_list[[4]]  for (i in seq_along(cleaned_text_list)) {  clean_text <- gsub("\\r\\n",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' " ",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' cleaned_text_list[[i]])  clean_text <- unlist(strsplit(clean_text,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' " "))  words <- clean_text[!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='grepl("\\\\d|_|http|www|nbsp|@|(?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='<=[[:lower:]])(?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='=[[:upper:]])",  clean_text, perl = TRUE)]  words <- tolower(gsub("\\\\W+", "", words))  words <- gsub(’[^a-zA-Z|[:blank:]]’, "", words)  cleaned_text_list[[i]] <- paste(words, collapse = " ")  }  cleaned_text_df <- do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='call(’rbind’, cleaned_text_list)  library(syuzhet)  email_sentiment_df <-get_nrc_sentiment(cleaned_text_df)  rownames(email_sentiment_df) <- rownames(cleaned_text_df)  head(email_sentiment_df,10)  References  [1] R Core Team, R: A Language and Environment for Statistical Comput-  ing, R Foundation for Statistical Computing, Vienna, Austria (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='R-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='org/  [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Allaire, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Xie, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' McPherson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Luraschi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Ushey, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Atkins,  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Wickham, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Cheng, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Chang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Iannone, rmarkdown: Dynamic  Documents for R, r package version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='5 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  URL https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='com/rstudio/rmarkdown  [3] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Heinlein, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Hartleben, The Book of IMAP: Building a Mail Server  with Courier and Cyrus, No Starch Press, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  [4] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Mersmann, sendmailR: send email using R, r package version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='2-1  (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  URL https://CRAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='R-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='org/package=sendmailR  [5] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Premraj, mailR: A Utility to Send Emails from R, r package version  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='1 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  URL https://CRAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='R-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='org/package=mailR  14  [6] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Himmelmann, mail: Sending Email Notifications from R, r package  version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='0 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  URL https://CRAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='R-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='org/package=mail  [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Bache, blatr: Send Emails Using ’Blat’ for Windows, r package  version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='1 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  URL https://CRAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='R-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='org/package=blatr  [8] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Hester, gmailr: Access the ’Gmail’ ’RESTful’ API, r package version  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='0 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  URL https://CRAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='R-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='org/package=gmailr  [9] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Iannone, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Cheng, blastula: Easily Send HTML Email Messages, r  package version 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='2 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  URL https://CRAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='R-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='org/package=blastula  [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Collier, emayili: Send Email Messages, r package version 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='4  (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  URL https://CRAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='R-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='org/package=emayili  [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Collier, edeR: Email Data Extraction Using R, r package version  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='0 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  URL https://CRAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='R-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='org/package=edeR  [12] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Crispin, Internet message access protocol - version 4rev1, request for  Comments 3501 (RFC 3501), Internet Engineering Task Force (IETF)  (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  URL https://tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='ietf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='org/html/rfc3501  [13] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Moore, Multipurpose Internet Mail Extensions (MIME), part three:  Message header extensions for non-ascii text, request for Comments 2047  (RFC 2047), Internet Engineering Task Force (IETF) (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  URL https://tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='ietf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='org/html/rfc2047  [14] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Bache, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Wickham, magrittr: A Forward-Pipe Operator for R, r  package version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='5 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  URL https://CRAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='R-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='org/package=magrittr  [15] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Chang, R6: Encapsulated Classes with Reference Semantics, r pack-  age version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='0 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  URL https://CRAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
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+page_content='org/package=R6  [16] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Ooms, curl: A Modern and Flexible Web Client for R, r package  version 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='3 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  URL https://CRAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='R-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='org/package=curl  15  [17] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Stenberg, libcurl - the multiprotocol file transfer library, version  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='1 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  URL https://curl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='haxx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='se/  [18] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Quadros, mRpostman: An IMAP Client for R, r package version  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='0 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  URL https://allanvc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='io/  [19] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Resnick, Internet message format, request for Comments 5322 (RFC  5322), Internet Engineering Task Force (IETF) (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  URL https://tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='ietf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='org/html/rfc5322  [20] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Mohammad, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Turney, Emotions evoked by common words and  phrases:  Using mechanical turk to create an emotion lexicon, in:  CAAGET ’10: Proceedings of the NAACL HLT 2010 Workshop on  Computational Approaches to Analysis and Generation of Emotion in  Text, Los Angeles, California, 2010, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' 26–34, june, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  URL http://saifmohammad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='com/WebPages/lexicons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='html  [21] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content=' Jockers, Syuzhet: Extract Sentiment and Plot Arcs from Text, r  package version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
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+page_content='4 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='  URL https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}
+page_content='com/mjockers/syuzhet  16' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'}

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+RiskProp: Account Risk Rating on Ethereum via De-anonymous
+Score and Network Propagation
+Dan Lin
+School of Software Engineering,
+Sun Yat-sen University
+Zhuhai, China
+[email protected]
+Jiajing Wu∗
+School of Computer Science and
+Engineering, Sun Yat-sen University
+Guangzhou, China
+[email protected]
+Qishuang Fu
+School of Computer Science and
+Engineering, Sun Yat-sen University
+Guangzhou, China
+[email protected]
+Zibin Zheng
+School of Software Engineering,
+Sun Yat-sen University
+Zhuhai, China
+[email protected]
+Ting Chen
+University of Electronic Science and
+Technology of China
+Guangzhou, China
+[email protected]
+ABSTRACT
+As one of the most popular blockchain platforms supporting smart
+contracts, Ethereum has caught the interest of both investors and
+criminals. Differently from traditional financial scenarios, executing
+Know Your Customer verification on Ethereum is rather difficult
+due to the pseudonymous nature of the blockchain. Fortunately,
+as the transaction records stored in the Ethereum blockchain are
+publicly accessible, we can understand the behavior of accounts or
+detect illicit activities via transaction mining. Existing risk control
+techniques have primarily been developed from the perspectives of
+de-anonymizing address clustering and illicit account classification.
+However, these techniques cannot be used to ascertain the potential
+risks for all accounts and are limited by specific heuristic strate-
+gies or insufficient label information. These constraints motivate
+us to seek an effective rating method for quantifying the spread
+of risk in a transaction network. To the best of our knowledge,
+we are the first to address the problem of account risk rating on
+Ethereum by proposing a novel model called RiskProp, which in-
+cludes a de-anonymous score to measure transaction anonymity
+and a network propagation mechanism to formulate the relation-
+ships between accounts and transactions. We demonstrate the ef-
+fectiveness of RiskProp in overcoming the limitations of existing
+models by conducting experiments on real-world datasets from
+Ethereum. Through case studies on the detected high-risk accounts,
+we demonstrate that the risk assessment by RiskProp can be used
+to provide warnings for investors and protect them from possible
+financial losses, and the superior performance of risk score-based
+account classification experiments further verifies the effectiveness
+of our rating method.
+KEYWORDS
+Abnormal detection, network propagation, Ethereum, risk control,
+de-anonymization
+1
+INTRODUCTION
+Ethereum [30] has the second-largest market cap in the blockchain
+ecosystem. The account model is adopted on Ethereum, and the
+native cryptocurrency on Ethereum is named Ether (abbreviated as
+∗Corresponding author.
+“ETH”), which is widely accepted as payments and transferred from
+one account to another. It is known that Ethereum accounts are
+indexed according to pseudonyms, and the creation of accounts is
+almost cost-free. This anonymous nature and the lack of regulation
+result in the bad reputation of Ethereum and other blockchain sys-
+tems for breeding malicious behaviors and enabling fraud, thereby
+resulting in large property losses for investors. As reported in a
+Chainalysis Crime Report, the illicit share of all cryptocurrency
+activities was valued at nearly USD 2.7 billion in 2020. These losses
+illustrate that Know-Your-Customer (KYC) and risk control of ac-
+counts are critical and necessary. Risk control [23] not only helps
+wallet customers identify risky accounts and avoid losses but also
+plays a vital role in the anti-money laundering of virtual asset
+service providers, such as cryptocurrency exchanges.
+Therefore, a wealth of efforts have been expended in risk con-
+trol on Ethereum in recent years. In September 2020, the Financial
+Action Task Force (FATF) published a recommendation report on
+virtual assets and released information on Red Flag Indicators [11]
+related to transactions, anonymity, senders or recipients, the source
+of funds, and geographical risks. In addition, researchers in the aca-
+demic community have proposed various techniques from the per-
+spectives of address clustering and illicit account classification. Ad-
+dress clustering techniques perform entity identification of anony-
+mous accounts. For example, Victor [27] proposes several clustering
+heuristics for Ethereum accounts and clusters 17.9% of all active ex-
+ternally owned accounts. Illicit account detection techniques focus
+on training classifiers based on well-designed features extracted
+from transactions [8, 10, 32]. Moreover, some researchers have de-
+veloped methods for automatic feature extraction incorporating
+structural information [19, 21, 29, 33].
+However, there are still some limitations (L) associated with
+these techniques. L1: Account clustering techniques can only be
+applied to part of accounts and therefore have limited applicability,
+and most accounts beyond heuristic rules cannot thus be identified.
+L2: In the existing methods for illicit account detection, binary
+classifiers are usually trained via supervised learning. However, as
+only a very small percentage of risky nodes have clear labels, which
+are required for these methods, the vast majority of accounts that
+may be involved in malicious events are unlabeled. In particular,
+arXiv:2301.00354v1  [cs.SI]  1 Jan 2023
+
+WWW ’23, April 30–May 4, 2023, Austin, TX, US
+Anonymous author(s)
+TxnHash: 0x0bf742... 
+From: 0x3da2b... 
+To: 0x41b53... 
+Timestamp: 1525153486 
+Value: 0.1 Ether 
+TxnFee: 0.000861 Ether
+Customer
+...
+...
+Scammer
+Scammer
+Exchange
+Txns
+Txns
+Txns
+...
+Account risk rating
+Public transactions
+Ethereum blockchain
+0x3da2b...
+0x41b53...
+Figure 1: The procedure of ETH transfer in Ethereum.
+“From” denotes the sender, “To” denotes the receiver, and
+“Txn” denotes “Transaction”.
+risky accounts with few transactions or unseen patterns are likely
+to be misidentified in practical use.
+To address the limitations presented above, we explore risk con-
+trol on Ethereum from a new perspective: Account risk rating. In
+traditional financial scenarios, credit scoring is usually conducted
+by authorized financial institutions, which perform audits on their
+customers to fully understand their identity, background, and fi-
+nancial credit standing. Similarly to credit scoring, risk rating on
+Ethereum can help us quantify the latent risk of a transaction or
+account with a quantitative score, thereby combating money laun-
+dering and identifying potential scams before new victims emerge.
+In terms of the abovementioned L1, in contrast to the traditional
+account clustering method, which can only de-anonymize a small
+number of accounts, the account risk method proposed in this paper
+can obtain quantitative risk indicators for all accounts. Regarding
+L2, the proposed risk rating method can achieve decent perfor-
+mance in an unsupervised manner without feeding labels. The
+output of the proposed method is risk values, which are provided
+continuously and allow evaluation of the severity of risk.
+Compared with traditional financial scenarios, several unique
+challenges (C) are encountered in the task of account risk rating on
+Ethereum. C1: Nature of anonymity. Transactions on Ethereum
+do not require real-name verification. Even worse, perpetrators of
+some malicious activities deliberately enhance their anonymity to
+counter the impact of de-anonymizing clustering techniques [29].
+C2: Complex transaction relationship. Compared with tradi-
+tional financial scenarios, a user or entity on Ethereum may control
+a large number of accounts at almost no cost, and the transaction
+relationship between accounts is also more complex. How to quan-
+tify the impact of trading behavior between accounts on account
+risk is a challenging core problem.
+To overcome the challenges mentioned above, we propose a
+novel approach called Risk Propagation (RiskProp) for Ethereum ac-
+count rating. It comprises two core designs, namely de-anonymous
+score and a network propagation mechanism. To resolve C1, de-
+anonymous score measures the degree to which transactions remain
+anonymous. For example, both the payer and the payee of an illicit
+transaction prefer to have a small number of transactions to en-
+sure anonymity-preserving protection. In contrast, both sides of a
+licit transaction may participate in numerous interactions without
+evading the impact of the de-anonymized clustering algorithm. Af-
+terward, to resolve C2, we model the massive transaction records
+as a directed bipartite graph and introduce a network propagation
+mechanism with three interdependent metrics, namely Confidence
+of the de-anonymous score, Trustiness of the payee, and Reliability
+of the payer. Intuitively, payees with higher trustiness receive trans-
+actions with higher de-anonymous scores, and payers with higher
+reliability will send transactions with higher confidence. Clearly,
+reliability, trustiness, and confidence are related to each other, so
+we define five items of prior knowledge that these metrics should
+satisfy and propose three mutually recursive equations to estimate
+the values of these metrics. To verify the effectiveness of the pro-
+posed risk rating method and further illustrate the significance of
+rating for risk control on Ethereum, we evaluate the effect of the
+risk rating system via experiments from two aspects, i.e., analysis of
+risk rating results and rating score-based illicit/licit classification.
+Overall, our contributions are summarized as follows:
+• A new perspective for Ethereum risk control. This paper is
+the first to propose tackling the problem of Ethereum risk control
+via the perspective of account risk rating.
+• A novel risk metric for transactions. We creatively develop a
+metric called de-anonymous score for transactions, which measures
+the degree of de-anonymization to quantify the risk of a transaction.
+• An effective method and interesting insights. We implement
+a novel risk rating method called RiskProp and demonstrate its su-
+perior effectiveness and efficiency via experiments on a real-world
+Ethereum transaction dataset together with theoretical analysis. By
+analyzing the rating results and case studies on high-risk accounts,
+we obtain interesting insights into the Ethereum ecosystem and
+further show how our method could prevent financial losses ahead
+of blacklisting malicious accounts.
+2
+PRELIMINARY
+2.1
+Ethereum Financial Background
+Ether is the native “currency” on Ethereum and plays a fundamen-
+tal part in the Ethereum payment system. Ether can be paid or
+received in financial activities, just like currency in real life. In
+conventional financial scenarios, a Know Your Customer (KYC)
+check is the mandatory process to identify and verify a customer’s
+identity when opening an account and to periodically understand
+the legitimacy of the involved funds over time. However, unlike
+traditional transaction systems, where customers’ identity informa-
+tion is required and obtained in KYC checks, Ethereum accounts
+are designed as pseudonymous addresses identified by 20 bytes of
+public key information generated by cryptographic algorithms, for
+example, “0x99f154f6a393b088a7041f1f5d0a7cbfa795d301”.
+Figure 1 depicts the risky scenario of Ether transfer in aspects of
+data acquisition. It includes three layers: 1) Ethereum blockchain.
+The Ethereum historical data are irreversible and publicly trace-
+able on the chain. 2) Public transactions. The transaction denotes
+a signed data package from an account to another account, in-
+cluding the sending address, receiver address, transferred Ether
+amount, etc. 3) Account risk rating. Usually, the identities who con-
+trol the accounts are not labeled. Customers may become involved
+in suspicious financial crimes or be vulnerable to frauds and scams.
+Furthermore, the illicit funds can be laundered and cashed out via
+exchanges. In this procedure, our proposed RiskProp is implemented
+to measure the risk of unlabeled accounts that may have ill inten-
+tions and alert customers when engaging in suspicious, potentially
+illegal transactions.
+
+RiskProp
+WWW ’23, April 30–May 4, 2023, Austin, TX, US
+2.2
+The Nature of Blockchain: Anonymity
+It is known that the Ethereum account is identified as a pseudony-
+mous address. However, if customers repeatedly use the same ad-
+dress as on-chain identification, the relationship between addresses
+becomes linkable via public transaction records. Accounts that
+participate in more transactions and connect with more accounts
+experience degrading anonymity [29]. To reduce the likelihood
+of exposure, criminals naturally tend to initiate transactions with
+fewer accounts. Here is an example on Ethereum: The two accounts
+of transaction 0x9a9d have only three transactions and became in-
+active thereafter. These two accounts are considered suspicious and
+reported as relevant accounts of Upbit exchange hack. On the con-
+trary, entities who do not deliberately take anonymity-preserving
+measures are likely to be normal [29]. Thus, the transaction is
+scored based on the fact of whether the accounts are trying to hide
+or not, which is the de-anonymous score.
+Definition 1 (De-anonymous score, abbreviated as “score”).
+The de-anonymous score of a transaction from account𝑢 to 𝑣 where
+there is no intention to hide is defined as
+𝑆𝑐𝑜𝑟𝑒(𝑢, 𝑣) =1
+2 ( 2 log |𝑂𝑢𝑡𝑇𝑥𝑛(𝑢)| − log𝑚𝑎𝑥𝑂𝑢𝑡
+log𝑚𝑎𝑥𝑂𝑢𝑡
++ 2 log |𝐼𝑛𝑇𝑥𝑛(𝑣)| − log𝑚𝑎𝑥𝐼𝑛
+log𝑚𝑎𝑥𝐼𝑛
+),
+(1)
+where 𝑂𝑢𝑡𝑇𝑥𝑛(𝑢) represents the outgoing transactions (payments)
+of payer 𝑢, 𝐼𝑛𝑇𝑥𝑛(𝑣) represents the incoming transactions (recep-
+tions) of payee 𝑣, and | × | denotes the size of a set. The minimum
+value of |𝑂𝑢𝑡𝑇𝑥𝑛(𝑢)| and |𝐼𝑛𝑇𝑥𝑛(𝑣)| is 1. Let 𝑚𝑎𝑥𝑂𝑢𝑡 and 𝑚𝑎𝑥𝐼𝑛
+be the largest number of payments and receptions, respectively. The
+de-anonymous scores of a transaction (𝑢, 𝑣) range from −1 (very
+high anonymity, abnormal) to 1 (very low anonymity, normal).
+Intuitively, the score of (𝑢, 𝑣) increases as the transaction num-
+bers of either payer or payee grow. Note that tricky criminals may
+camouflage themselves by deliberately conducting low-anonymity
+transactions [20].
+2.3
+Transaction Network Construction
+First, each transaction on Ethereum has one payer (i.e., sender) and
+one payee (i.e., receiver). Any account can be the role of payer or
+payee, just as a person in real life has different roles. The payee is a
+passive role and, therefore, we consider the incoming transactions
+to indicate the trustiness of an account. For instance, exchange
+accounts that receive more transactions are considered to be more
+trustworthy. In contrast, the payer is an active role and, thus, the
+outgoing transactions embody the intention of an account. For ex-
+ample, a scam account subjectively wants to transfer stolen money
+to its partners.
+Next, the transaction records are modeled as a directed bipartite
+graph 𝐺 = (𝑈,𝑉,𝑆), where 𝑈 , 𝑉 , and 𝑆 represent the set of all
+payers, payees, and scores, respectively. A weighted edge (𝑢, 𝑣)
+denotes the transfer of Ethers from account 𝑢 ∈ 𝑈 to account 𝑣 ∈ 𝑉
+with 𝑆𝑐𝑜𝑟𝑒(𝑢, 𝑣) ∈ 𝑆. The graph construction procedure is shown
+in Figure 2.
+Then, the ego network of a payer 𝑢 is introduced. It is formed by
+its outgoing scores and corresponding payee neighbors, formulated
+Money transfer
+Accounts
+Tnx Score
+Accounts
+Payer 
+Payee 
+Tnx Score
+Accounts
+(A) Ethereum
+transaction records
+(B) De-anonymous
+score calculation
+(C) Payer-payee graph
+Figure 2: The transformation from the raw transaction
+records to the directed bipartite graph. “Txn” denotes
+“Transaction”.
+Payee set
+V
+Payer A
+Payer B
+Payee X
+Payee Y
+Score(A, X)
+Score(B, Y)
+Payer set
+U
+Score(B, X)
+In(      ) = { Score(A, X), Score(B, X) }
+Payee X
+Out(      ) = { Score(B, X), Score(B, Y) }
+Payer B
+Figure 3: A toy example of the directed bipartite graph estab-
+lished from transactions and the illustration of functions 𝐼𝑛
+and 𝑂𝑢𝑡.
+as 𝑂𝑢𝑡(𝑢) ∪ {𝑣|(𝑢, 𝑣) ∈ 𝑂𝑢𝑡(𝑢)}, where 𝑂𝑢𝑡(𝑢) is the set of scores
+connected with 𝑢. It is similar for the ego network of a payee,
+formulated as 𝐼𝑛(𝑣)∪{𝑢|(𝑢, 𝑣) ∈ 𝐼𝑛(𝑣)}. Figure 3 shows an example
+in which there are two payers, two payees, and three transactions.
+3
+MODEL
+In this section, we describe the prior knowledge that establishes the
+relationships among accounts and transactions and then propose
+risk propagation formulations that satisfy the prior knowledge. It
+is worth noticing that the proposed algorithm does not require
+handcraft feature engineering.
+3.1
+Problem Definition and Model Overview
+Given raw transaction records of Ethereum, we model the transac-
+tion relationships between accounts as a directed bipartite graph
+𝐺 = (𝑈,𝑉,𝑆) with payers and payees as nodes and prepossessed
+de-anonymous scores as weights of edges. We believe that accounts
+have intrinsic metrics to quantify their reliability and trustworthi-
+ness and transactions have intrinsic metrics to measure the con-
+fidence of their calculated de-anonymous scores. Naturally, those
+metrics are interdependent and interplay with each other via the
+risk propagation mechanism:
+• Payers vary in terms of their Reliability, which indicates how
+motivated they are. A licit payer without malicious intent usually
+does not hide himself or disguise its intentions during transactions.
+Specifically, a reliable payer has harmless intentions regardless of
+whether it is transferring money to an exchange or to a scammer
+account (being gypped). In contrast, a perpetrator (e.g., a scammer)
+hopes to cover up its traces [27]. The reliability metric 𝑅(𝑢) of a
+payer 𝑢 lies in [0, 1], ∀𝑢 ∈ 𝑈 . A value of 1 denotes a 100% reliable
+payer and 0 denotes a 0% reliable payer.
+
+WWW ’23, April 30–May 4, 2023, Austin, TX, US
+Anonymous author(s)
+Table 1: An example of propagation. 𝑅0 is initial value, 𝑅𝑓 𝑖𝑛𝑎𝑙
+and Risk𝑓 𝑖𝑛𝑎𝑙 are the results after convergence.
+Account
+Label
+𝑅0
+𝑅𝑓 𝑖𝑛𝑎𝑙
+Risk𝑓 𝑖𝑛𝑎𝑙
+0xa768
+Contract-deployer
+0.7
+0.8575
+1.425
+0x8271
+Exchange
+0.7
+0.9526
+0.474
+0xebdc
+Phish-hack
+0.7
+0.1195
+8.805
+0xfe34
+Phish-hack
+0.7
+0.2330
+7.670
+• Payees vary in their trustworthiness level, measured by a metric
+called Trustiness, which indicates how trustworthy they are. Intu-
+itively, a cryptocurrency service provider with a better reputation
+will receive more licit transactions (with higher scores) from well-
+motivated payers. Trustiness of a payee 𝑇 (𝑣) ranges from 0 (very
+untrustworthy) to 1 (very trustworthy) ∀𝑣 ∈ 𝑉 .
+• De-anonymous scores vary in terms of Confidence, which re-
+flects the confidence in the estimated risk probability of a trans-
+action. The confidence metric 𝐶𝑜𝑛𝑓 (𝑢, 𝑣) ranges from 0 (lack of
+confidence) to 1 (very confident).
+The connection between the reliability and risk of accounts: We
+define Reliability to characterize the risk rating of accounts because
+an account’s intention can be inferred by its (active) sending be-
+havior, rather than by its (passive) receiving behavior. A scammer
+transferring stolen money to its gang is a better reflection of its
+evil intention than the receipt of stolen money from victims. In the
+later section, we calculate the risk rating of accounts based on the
+Reliability of payer roles.
+3.2
+Network Propagation Mechanism
+Given a cryptocurrency payer–payee graph, all intrinsic metrics
+are unknown but are interdependent. Here, we introduce five items
+of prior knowledge that establish the relationships and how the net-
+work propagation mechanism is specially designed for our problem.
+The first two items of prior knowledge reflect the interdependency
+between a payee and the de-anonymous scores that they receive.
+[Prior knowledge 1] Payees with higher trustiness receive trans-
+actions with higher de-anonymous scores. Intuitively, a payee
+receiving transactions with high de-anonymous scores is more
+likely to be trustworthy. Formally, if two payees 𝑣1 and 𝑣2 have
+a one-to-one mapping, ℎ : 𝐼𝑛(𝑣1) → 𝐼𝑛(𝑣2) and 𝑆𝑐𝑜𝑟𝑒(𝑢, 𝑣1) >
+𝑆𝑐𝑜𝑟𝑒(ℎ(𝑢), 𝑣2) ∀(𝑢, 𝑣1) ∈ 𝐼𝑛(𝑣1), then 𝑇 (𝑣1) > 𝑇 (𝑣2).
+[Prior knowledge 2] Payees with higher trustiness receive trans-
+actions with more positive confident scores. For two payees 𝑣1 and
+𝑣2 with identical de-anonymous score networks, if the confidence
+of the in-transactions of payee 𝑣1 is higher than that of payee 𝑣2,
+the trustiness of payee 𝑣1 should be higher. Formally, if two pay-
+ees 𝑣1 and 𝑣2 have a one-to-one mapping, ℎ : 𝐼𝑛(𝑣1) → 𝐼𝑛(𝑣2)
+and 𝐶𝑜𝑛𝑓 (𝑢, 𝑣1) > 𝐶𝑜𝑛𝑓 (ℎ(𝑢), 𝑣2) ∀(𝑢, 𝑣1) ∈ 𝐼𝑛(𝑣1), then 𝑇 (𝑣1) >
+𝑇 (𝑣2).
+According to the above prior knowledge, we develop the Trusti-
+ness formulation for ∀𝑣 ∈ 𝑉 of our RiskProp algorithm:
+𝑇 (𝑣) =
+�
+(𝑢,𝑣) ∈𝐼𝑛(𝑣) 𝑆𝑐𝑜𝑟𝑒(𝑢, 𝑣) × 𝐶𝑜𝑛𝑓 (𝑢, 𝑣)
+|𝐼𝑛(𝑣)|
+.
+(2)
+The next item of prior knowledge defines the relationship be-
+tween the score of a transaction and the connected payer–payee
+pair using the anonymous nature of cryptocurrency.
+Algorithm 1 RiskProp Algorithm
+1: Input: Directed Bipartite Graph 𝐺 = (𝑈,𝑉,𝑆)
+2: Output: Risk of accounts
+3: Initialize 𝑇 0 = 0.5, 𝑅0 = 0.7,𝐶𝑜𝑛𝑓 0 = 0.5,𝑡 = 0, Δ = 1
+4: while Δ ≥ 0.01 do
+5:
+𝑡 = 𝑡 + 1
+6:
+Update 𝑡𝑟𝑢𝑠𝑡𝑖𝑛𝑒𝑠𝑠 of payees using Equation 2
+7:
+Update 𝑟𝑒𝑙𝑖𝑎𝑏𝑙𝑖𝑡𝑦 of payers using Equation 4
+8:
+Update 𝑐𝑜𝑛𝑓 𝑖𝑑𝑒𝑛𝑐𝑒 of transactions using Equation 3
+9:
+Δ𝑇 = �
+𝑣∈𝑉 |𝑇 𝑡 (𝑣) −𝑇 𝑡−1(𝑣) |
+10:
+Δ𝑅 = �
+𝑢∈𝑈 |𝑅𝑡 (𝑢) − 𝑅𝑡−1(𝑢) |
+11:
+Δ𝐶 = �
+(𝑢,𝑣)∈𝑆 |𝐶𝑜𝑛𝑓 𝑡 (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡−1(𝑢, 𝑣) |
+12:
+Δ = max{Δ𝑇 , Δ𝑅, Δ𝐶 }
+13: end while
+14: 𝑅𝑖𝑠𝑘 (𝑢) = (1 − 𝑅(𝑢)) × 10, ∀𝑢 ∈ 𝑈
+15: return
+[Prior knowledge 3] Confident de-anonymous scores of transac-
+tions are closely linked with the connected payee’s trustiness. For-
+mally, if two scores (𝑢1, 𝑣1) and (𝑢2, 𝑣2) are such that𝑆𝑐𝑜𝑟𝑒(𝑢1, 𝑣1) =
+𝑆𝑐𝑜𝑟𝑒(𝑢2, 𝑣2),𝑅(𝑢1) = 𝑅(𝑢2), and |𝑆𝑐𝑜𝑟𝑒(𝑢1, 𝑣1) −𝑇 (𝑣1)| ⩽ |𝑆𝑐𝑜𝑟𝑒(𝑢2, 𝑣2)
+−𝑇 (𝑣2)|, then 𝐶𝑜𝑛𝑓 (𝑢1, 𝑣1) ⩾ 𝐶𝑜𝑛𝑓 (𝑢2, 𝑣2).
+We imply that different transactions sent by the same payers
+can have different intentions and anonymity. Even scammers on
+Ethereum can have transactions that seem normal.
+[Prior knowledge 4] Transactions with higher confidence de-
+anonymous scores are sent by more reliable payers. Formally, if two
+scores (𝑢1, 𝑣1) and (𝑢2, 𝑣2) are such that𝑆𝑐𝑜𝑟𝑒(𝑢1, 𝑣1) = 𝑆𝑐𝑜𝑟𝑒(𝑢2, 𝑣2),
+𝑇 (𝑣1) = 𝑇 (𝑣2), and𝑅(𝑢1) ⩾ 𝑅(𝑢2), then𝐶𝑜𝑛𝑓 (𝑢1, 𝑣1) ⩾ 𝐶𝑜𝑛𝑓 (𝑢2, 𝑣2).
+This prior knowledge incorporates the payer’s intention in mea-
+suring the confidence of transaction scores. In this way, payees
+may have different confidence in receiving transactions with the
+same anonymity. For instance, exchanges on Ethereum receive
+funds from payers with different motivations—some are ordinary
+investors and some are suspicious accounts.
+Below, we propose the Confidence formulation that satisfies the
+above items of prior knowledge:
+𝐶𝑜𝑛𝑓 (𝑢, 𝑣) = 𝑅(𝑢) + (1 − |𝑆𝑐𝑜𝑟𝑒(𝑢, 𝑣) −𝑇 (𝑣)|)
+2
+.
+(3)
+Then, we describe how to quantify the Reliability metric of a
+payer by the transactions it sends.
+[Prior knowledge 5] Payers with higher reliability send transac-
+tions with higher confidence. For two payers 𝑢1 and 𝑢2 with equal
+scores, if payer 𝑢1 has higher confidence for all out transaction
+scores than𝑢2, then payer𝑢1 has a higher reliability. Formally, if two
+payers 𝑢1 and 𝑢2 have ℎ : 𝑂𝑢𝑡(𝑢1) → 𝑂𝑢𝑡(𝑢2) and 𝐶𝑜𝑛𝑓 (𝑢1, 𝑣1) >
+𝐶𝑜𝑛𝑓 (𝑢2,ℎ(𝑣)) ∀(𝑢1, 𝑣) ∈ 𝑂𝑢𝑡(𝑢1), then 𝑅(𝑢1) > 𝑅(𝑢2). The corre-
+sponding formulation of Reliability metric for ∀𝑢 ∈ 𝑈 is defined
+as
+𝑅(𝑢) =
+�
+(𝑢,𝑣) ∈𝑂𝑢𝑡 (𝑢) 𝐶𝑜𝑛𝑓 (𝑢, 𝑣)
+|𝑂𝑢𝑡(𝑢)|
+.
+(4)
+Finally, the risk rating of an account is calculated by 𝑅𝑖𝑠𝑘(𝑢) =
+(1 − 𝑅(𝑢)) × 10. The pseudo-code of RiskProp network propagation
+is described in Algorithm 1. Let 𝑇 0, 𝐶𝑜𝑛𝑓 0, 𝑅0 be initial values
+and 𝑡 be the number of interactions. In the beginning, we have
+initial reliability 𝑅0 ∀𝑢 ∈ 𝑈 , initial trustiness 𝑇 0 = 0.5 ∀𝑣 ∈ 𝑉 , and
+
+RiskProp
+WWW ’23, April 30–May 4, 2023, Austin, TX, US
+Account Risk Rating
+Trustiness
+Confidence
+Risk
+Reliability
+Update
+Propagation Mechanism
+ Risk Rating
+Results Analysis
+Ablation
+Study
+Risk
+Threshold
+Guarantees  
+for Practice
+Comparative
+Evaluation
+ Further 
+ 
+  Analysis 
+Results Analysis
+Data Acquisition
+Labeled data
+Etherscan
+Ethereum
+Transactions
+Data Pre-processing
+Directed Bipartite Graph
+Construction
+De-anonymous Score
+Calculation
+Figure 4: The workflow of account risk rating on Ethereum.
+initial confidence 𝐶𝑜𝑛𝑓 0 = 0.5 for all transactions. Then, we keep
+updating metrics using Equations 2–4 until Δ is less than 0.01.
+RiskProp+: A Semi-supervised Version. Sometimes, we have
+partial information about the labels of fraudulent accounts (verified,
+phishing scams, etc.) and licit accounts. We can take advantage of
+such prior information and incorporate them into our approach
+in a semi-supervised manner. In the semi-supervised RiskProp+,
+we initialize the Reliability metrics only for the training accounts.
+According to the risk levels of services reported by Chainalysis [26],
+we set 𝑅0 = 0.9 for ICO wallet, Converter, and Mining, 𝑅0 = 0.7 for
+Exchange, 𝑅0 = 0.4 for Gambling, 𝑅0 = 0 for Phish/Hack, and set
+𝑅0 = 0.7 for testing accounts. The reliability values of labeled illicit
+accounts are unchanged during the training procedure.
+Example. Here, we use a small real-world dataset on Ethereum
+to intuitively show the results of RiskProp+ after interactions. We
+collect transactions of 10 accounts (6 for training and 4 for testing),
+including 28,598 accounts and 52,733 transactions in total. Table 1
+shows how the reliability of the 4 testing accounts varies over
+interactions (we omit trustiness and confidence for brevity). These
+testing accounts have the same reliability values at the beginning
+(𝑅0 = 0.7). After convergence, accounts labeled as “phish/hack”
+get a lower value of reliability, and other licit accounts get higher
+reliability. Confirming our intuition, RiskProp learns that accounts
+0xebdc and 0xfe34 are high-risk accounts that investors need to be
+aware of.
+Workflow for Account Risk Rating. Figure 4 shows the work-
+flow of account risk rating on Ethereum, which contains four mod-
+ules: (i) Data acquisition collects accounts, transactions, and la-
+bels from Ethereum and Etherscan. Only a few labels are provided,
+and these labels are not available in the unsupervised setting. (ii)
+Data pre-processing of raw transaction data described in Fig-
+ure 1 is conducted in two steps: de-anonymous score calculation
+and directed bipartite graph construction (i.e., payer–payee net-
+work). (iii) Account risk rating recursively calculates the Relia-
+bility, Trustiness of accounts, and Confidence of transaction scores
+until convergence, updated by the propagation mechanism. (iv) Re-
+sults analysis contains risk rating results analysis, comparative
+evaluation, and further analysis.
+4
+EXPERIMENTS
+To investigate the effectiveness of RiskProp, we conduct experi-
+ments on a real-world Ethereum transaction dataset. As risk rating
+is an issue without any ground truth, we verify the effectiveness
+and significance of the risk rating results of RiskProp via three tasks:
+1) risk rating analysis, which includes distribution of risk rating
+results and case studies of transaction pattern; 2) comparative
+evaluation, which reports on the classification performance of
+labeled accounts compared with various baselines; and 3) further
+analysis, which contains ablation study, impact of risk threshold,
+and guarantees for practical use. RiskProp is open source and repro-
+ducible, and the code and dataset are publicly available after the
+paper is accepted.
+4.1
+Data Collection
+We first obtain 803 ground truth account labels from an official
+Ethereum explorer and then include all the accounts and transac-
+tions that are within the one-hop and two-hop neighborhood of
+each labeled account. Next, we filter out the zero-ETH transactions
+and construct the records into a graph, retaining the largest weakly
+connected component for experiments. As a result, there are 1.19
+million accounts and 4.13 million transactions in the network. In the
+dataset, 0.02 percent (243) are labeled illicit (e.g., phishing scam),
+whereas 0.05 percent (560) are labeled licit (e.g., exchanges). The
+remaining unknown accounts are not labeled with regards to licit
+versus illicit.
+4.2
+Effectiveness of De-anonymous Score
+We use one-way analysis of variance (ANOVA) to assess whether
+there is a significant difference between illicit and licit transactions
+in the proposed de-anonymous score in Equation (1). We consider
+a transaction as illicit (versus licit) if its payer is marked as illicit
+(versus licit). Table 2 shows that compared with the random score,
+our proposed score achieves a larger mean square (MS) between
+groups and smaller MS within groups; in addition, our proposed
+score has a higher F value, and the 𝑝-value equals 0. These results
+suggest that the de-anonymous score is a useful metric for assessing
+the quality of transactions.
+Table 2: ANOVA of random scores and de-anonymous
+scores.
+Random scores
+De-anonymous score
+Src of var.
+MS
+F
+𝑝-value
+MS
+F
+𝑝-value
+Between groups
+8.8 × 10−1
+2.6 × 101
+1.0 × 10−1
+7.8 × 102
+7.7 × 103
+0
+Within groups
+3.3 × 10−1
+-
+-
+1.0 × 10−1
+-
+-
+4.3
+Analysis of Risk Rating Results
+The principal task of RiskProp is to rate Ethereum accounts based on
+how ill-disposed they are. Given the account risk rating obtained by
+RiskProp, we first review the results and investigate the capability
+of RiskProp in discovering new risky accounts. Then, we dig deeper
+into the predicted high-risk accounts and obtain some insights.
+4.3.1
+Distribution of risk rating results. The risk value of an account
+ranges from 0 (low risk) to 10 (high risk). The distribution of the
+predicted risk scores is as follows: 33.58% are located at (0,2], 63.45%
+
+WWW ’23, April 30–May 4, 2023, Austin, TX, US
+Anonymous author(s)
+(b)
+(a) 
+(c)
+(d)  
+Exchange
+(e)  
+Phish_contract
+Victims
+Victims
+Exchange
+Scammer
+Scammers
+(f)  
+16 ETH
+16 ETH
+37 ETH
+37 ETH
+8 ETH
+8 ETH
+8 ETH
+Create
+0.29 ETH
+0.29 ETH
+Figure 5: Visualization showing some typical transaction
+patterns of risky accounts (in red circles).
+are located at (2,4], 2.03% are located at (4,6], 0.78% are located
+at (6, 8], and 0.19% are located at (8, 10]. This is consistent with
+expectations: The risk value of the Ethereum transaction network
+meets the power distribution law, indicating that the overwhelming
+majority of accounts act normally, and only very few accounts have
+abnormal behaviors. We are interested in whether the high-risk
+accounts predicted by RiskProp are actually questionable. Thus, we
+first manually check the top 150 accounts with the highest risk
+(with both in-coming and out-going transactions). The finding is
+that 119 out of 150 (approximately 80%) accounts have abnormal
+behaviors. Among these 119 illicit accounts, 43 accounts are already
+labeled as “phish/hack” by Etherscan, whereas the remaining 76
+are newly discovered suspicious accounts that are not marked in
+the existing label library. This result indicates the capabilities of
+RiskProp in predicting undiscovered risky accounts and reducing
+financial losses.
+4.3.2
+Case studies of transaction pattern. We then manually veri-
+fied the predicted risky accounts by investigating their abnormal
+behaviors and find that there are many suspicious transaction pat-
+terns in the network. In order to save space, we show 6 typical
+patterns in Figure 5. These patterns are summarized from the real-
+world Ethereum transaction data and guided by current research
+and recommendation reports.
+(a) Hacking scammers are a list of addresses related to phish-
+ing and hacks. Figure 5(a) shows a pattern of phishing accounts
+reported by users who suffered financial loss. A typical phishing
+scam on Ethereum is the “Bee Token ICO Scam” attack, in which
+the phishers sent fake emails to the investors of an ICO with a fake
+Ethereum address to deposit their contributions into. For example,
+account 0xe336 has been confirmed to be part of this “Bee Token”
+scam, and 243 ETH has been sent to this address by 165 victims.
+(b) Fund source of hacking scammers are the upstream ac-
+counts of the known illicit accounts, which are collusion scam ac-
+counts to attract victims or provide money for hacking. As shown
+in Figure 5(b), the behaviors of collusion scam accounts may look
+similar to victims. Nevertheless, we find that the upstream collusion
+accounts appear to participate in fewer transactions with shorter
+time intervals, and there are attempts to transfer the entire ETH
+balance of the scammers according to the Red Flag Indicators of
+FATF [11].
+(c) Money laundering of scammers are the downstream ac-
+counts of the known illicit accounts, which are collusion scam
+accounts to accept and transfer the stolen money, obfuscating the
+true sources. As shown in Figure 5(c), account 0x78f1 received
+stolen funds from several known hacking scammers, appearing
+to be the account used in the “placement” stage of money laun-
+dering. Another example is 0xcfdd, which receives stolen funds
+from the Fake Starbase Crowdsale Contribution account 0x122c. (d)
+Zero-out middle accounts are the middle accounts that serve as
+a bridge defined by Li et al. [20]. As shown in Figure 5(d), most of
+the received funds will be transferred out in short succession (such
+as within 24 hours). See 0x126e for an example.
+(e) Round transfers among exchanges denote a pattern that
+an account withdraws ETH without additional activity to a pri-
+vate wallet and then deposits back to the exchange, as shown in
+Figure 5(e). Account 0x886e withdraws 0.4 ETH from Cryptopia
+exchange and then deposits the same amount of ETH back to Cryp-
+topia, which is an unnecessary step and incurs transaction fees [11].
+Such a phenomenon indicates that the exchange is misused as a
+money-laundering mixer or is conducting wash trading [28].
+(f) Creators of illicit contracts are often the manipulators
+behind the scenes. The Origin Protocol phishing scam contact ac-
+count 0x9819 was created by account 0xff1a. After victims deposited
+money into the phishing contract, the creator transfers the stolen
+funds back to himself via internal transactions, which deliberately
+enhances anonymity.
+We observe that many illicit accounts are outside the label li-
+brary and are still considered risk-free. Based on the results, we
+infer that our RiskProp is able to expose unlabeled illicit accounts.
+This is crucial on Ethereum, which lacks authorized and effective
+regulation. In addition, the newly identified illicit accounts can
+complete the current label collection for additional analysis.
+4.4
+Comparative Evaluation Settings
+To further evaluate the performance of our method and show the
+potential application, we employ the rating scores to conduct clas-
+sification experiments that divide Ethereum accounts into illicit
+and licit accounts, and we compare the results with the existing
+baseline methods for further verification. We wish to investigate if
+RiskProp can give a higher risk rating for the known illicit accounts
+and a lower rating for known licit accounts.
+4.4.1
+Compared Methods. As mentioned earlier, RiskProp is the
+first algorithm that explores the risk rating of blockchain accounts.
+We chose a variety of methods (unsupervised and supervised) as
+baselines, which are similar to the problem we want to solve. We
+compare unsupervised RiskProp with (i) web page ranking, such
+as PageRank [6], and (ii) bipartite graph-based fraud detection,
+such as FraudEagle [2], BIRDNEST [12], and REV2 [16], which are
+also unsupervised methods.
+The (semi-)supervised approaches are as follows. (i) Machine
+learning methods, e.g., logistic regression (LR), naïve Bayes (NB),
+decision tree (DT), support vector machine (SVM), random for-
+est (RF), extreme gradient boosting (XGBoost), and LightGBM.
+These methods are used by [1, 4, 8, 19] for detection of abnor-
+mal Ethereum accounts. (ii) Traditional graph neural network,
+including DeepWalk, Node2Vec, and graph convolutional network
+
+RiskProp
+WWW ’23, April 30–May 4, 2023, Austin, TX, US
+50 100 150 200 250 300 350 400
+Top k
+0
+20
+40
+60
+80
+100
+Precision@k (%)
+(a)
+50 100 150 200 250 300 350 400
+Top k
+0
+20
+40
+60
+80
+100
+Recall@k (%)
+(b)
+RiskProp
+PageRank
+Birdnest
+FraudEagle
+REV2
+Figure 6: The 𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛@𝑘 and 𝑅𝑒𝑐𝑎𝑙𝑙@𝑘 of illicit account pre-
+diction with different rating methods.
+(GCN) were conducted by Chen et al. [7] for detection of Ethereum
+phishing scams. (iii) Graph neural network for graphs with
+heterophily, such as CPGNN [34]. The application of this type of
+algorithms is a recent research advancement in the task of Ethereum
+account classification [14].
+4.4.2
+Evaluation Metrics. To evaluate the performance of the mod-
+els, we calculate the following metrics: 𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛,𝑅𝑒𝑐𝑎𝑙𝑙, 𝐹1,𝐴𝑐𝑐𝑢𝑟𝑎𝑐𝑦,
+and 𝐴𝑈𝐶. As we know, there are only 6 out of 10,000 (0.067 percent)
+accounts labeled in the entire dataset. To measure the order of the
+risk rating, we employ 𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛@𝑘 and 𝑅𝑒𝑐𝑎𝑙𝑙@𝑘 to evaluate the
+ranking order of the algorithm (@𝑘 means the top 𝑘 accounts). All
+baseline methods are tested using the original codes published by
+the authors. We repeat experiments 10 times and report the average
+results.
+4.4.3
+Implementation Details. We evaluate the methods with bi-
+nary labeled accounts (illicit verse licit) and, thus, we assume ac-
+counts in the top 1% to be the illicit accounts (corresponding thresh-
+old: 6 for RiskProp). The reason for this threshold and percentage
+setting is discussed in Section 4.7. The split of the dataset in the
+(semi-)supervised setting is 𝑡𝑟𝑎𝑖𝑛𝑖𝑛𝑔 : 𝑡𝑒𝑠𝑡 = 8 : 2.
+4.5
+Comparative Evaluation Results
+We report the 𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛@𝑘 and 𝑅𝑒𝑐𝑎𝑙𝑙@𝑘 curves of the compared
+algorithms, as shown in Figure 6. We observe that RiskProp obtains
+superior precision and recall than that of baseline with different
+𝑘. Up to 𝑘 = 100, the precision of RiskProp is almost 1 for illicit
+account prediction, which is surprising for an unsupervised setting.
+The 𝑅𝑒𝑐𝑎𝑙𝑙@𝑘 curve of RiskProp is significantly higher than the
+compared methods, and also increases steadily with 𝑘. Table 3
+shows the performance of unsupervised and supervised methods
+separately. We observe that RiskProp remarkably outperforms the
+unsupervised graph rating baselines in terms of accuracy and AUC,
+improving by 38.90% and 34.16%, respectively. Meanwhile, for the
+licit account prediction, we observe that RiskProp beats the best
+baseline (FraudEagle) with a 10.48% improvement in its F1-score.
+These demonstrate the effectiveness of our account risk rating
+method without labeling information.
+Next, we turn our attention to the results of the semi-supervised
+RiskProp+ compared with the existing (semi-)supervised classifica-
+tion in Table 3, from which we derive the following conclusions:
+1) RiskProp+ outperforms all baseline methods by 12.32% in terms
+of F1-score, 13.62% in terms of AUC, and 10.56% in terms of accu-
+racy. 2) The precision of licit accounts prediction is improved from
+Table 3: The classification results (%) of unsupervised and
+(semi-)supervised methods.
+Illicit account
+Licit account
+Total
+Methods
+P
+R
+F1
+P
+R
+F1
+Acc.
+AUC
+PageRank
+29.13
+67.49
+40.69
+67.08
+28.75
+40.25
+40.47
+48.12
+FraudEagle
+12.28
+2.88
+4.670
+68.36
+91.07
+78.10
+64.38
+46.98
+BIRDNEST
+22.24
+47.32
+30.26
+55.24
+28.21
+37.35
+34.00
+37.77
+REV2
+14.10
+4.527
+6.854
+68.00
+88.04
+76.73
+62.76
+46.28
+RiskProp
+71.48
+71.48
+76.15
+91.44
+85.89
+88.58
+84.56
+83.69
+Illicit account
+Licit account
+Total
+Methods
+P
+R
+F1
+P
+R
+F1
+Acc.
+AUC
+LR
+65.67
+74.58
+69.84
+83.87
+77.23
+80.41
+76.25
+75.90
+NB
+59.79
+98.31
+74.36
+98.41
+61.39
+75.61
+75.00
+79.85
+DT
+62.66
+54.07
+58.04
+75.79
+81.19
+78.40
+71.75
+68.39
+SVM
+90.00
+45.76
+60.67
+75.38
+97.03
+84.85
+78.12
+71.40
+RF
+71.52
+53.39
+61.14
+75.55
+86.93
+80.84
+74.00
+69.40
+XGBoost
+67.35
+55.95
+61.11
+76.58
+84.16
+80.19
+70.05
+73.75
+LightGBM
+75.77
+65.19
+69.93
+84.23
+92.57
+88.21
+81.86
+77.75
+DeepWalk
+66.85
+66.30
+66.54
+86.48
+86.75
+86.61
+83.13
+81.03
+Node2Vec
+62.36
+63.26
+62.76
+85.10
+84.56
+84.82
+78.13
+72.78
+GCN
+20.83
+27.78
+23.81
+79.46
+68.99
+73.86
+60.63
+47.40
+CPGNN
+52.17
+61.54
+56.47
+86.84
+81.82
+84.26
+76.88
+71.68
+RiskProp+
+70.91
+84.78
+77.23
+93.33
+85.96
+89.49
+85.63
+85.37
+Table 4: Illicit account prediction of ablation studies.
+Methods
+Precision
+Recall
+F1-score
+RiskProp+
+0.7091
+0.8478
+0.7723
+RiskProp+ (w/o label)
+0.7148
+0.8148
+0.7615
+RiskProp+ (w/o NP)
+0.3811
+0.9959
+0.5513
+RiskProp+ (w/o DS)
+0.4737
+0.1957
+0.2769
+82.52% (i.e., the average precision in baselines) to 93.33%, which
+means more licit accounts can be correctly identified. 3) The supe-
+rior performance of RiskProp is more significant in the prediction of
+illicit accounts. The recall of illicit accounts prediction is improved
+from 60.56% (i.e., the average recall of illicit accounts prediction in
+baselines) to 84.78%. This shows the effectiveness of our framework
+in the prediction of both illicit and licit accounts.
+4.6
+Ablation Study
+To further validate the contribution of each component of the pro-
+posed RiskProp+, we conduct an ablation study as follows.
+• RiskProp+ (Full model): All components of the model and label
+data are included.
+• w/o label: Labels are unavailable in the learning procedure, and
+the model is trained in an unsupervised manner.
+• w/o network propagation (NP): Remove the NP procedure
+and calculate the average de-anonymous scores (𝐴𝐷𝑆) for each ac-
+counts’ outgoing transactions (payer role). An account is predicted
+as abnormal if its 𝐴𝐷𝑆 ⩽ 0.
+• w/o de-anonymous score (DS): Replace DS with random scores,
+ranging from −1 to 1.
+We derive the following findings from Table 4: 1) Without the
+labels, the F1-score drops only slightly, indicating that our RiskProp
+does not rely on label data and can obtain good results in an un-
+supervised manner. To our surprise, the full model outperforms
+the RiskProp (w/o label), with a 3.3% increase in recall and 0.51%
+
+WWW ’23, April 30–May 4, 2023, Austin, TX, US
+Anonymous author(s)
+decrease in precision. This may be possibly explained by the re-
+liability values of labeled illicit accounts remaining unchanged
+during training in the supervised setting. 2) RiskProp (w/o NP)
+has a only lower precision but a greatly improved recall, revealing
+that most of the illicit accounts are correctly predicted as illicit
+but that some licit accounts are misjudged to be illicit. This result
+demonstrates that de-anonymous score is an effective indicator of
+illicit transactions but their confidence varies among transactions.
+This result also confirms why we need to consider the confidence
+of the score in the propagation mechanism. 3) RiskProp (w/o DS)
+yields low precision (47.37%) and severely low recall (19.57%). This
+result demonstrates that, even if the network propagation model is
+retained, the wrong scores of transactions will be spread through-
+out the entire Ethereum transaction network, resulting in poor
+prediction results.
+4.7
+Risk Threshold of RiskProp
+Given accounts in the reversed order of their risk ratings, a natural
+question is how to classify licit or illicit accounts according to their
+risk ratings for the classification task? One possible option is to
+determine the percentage of known illicit labels in the dataset
+and set the risk value of this percentage as a demarcation line
+for account classification. However, the percentage is imprecise
+because some of the illicit accounts remain unrevealed according
+to the experimental results in Section 4.3.2. Therefore, we try to
+establish a suitable risk threshold (𝑅𝑇𝐻) of RiskProp by conducting
+classification experiments. Figure 7(a) demonstrates the results of
+illicit account prediction with different risk thresholds, ranging
+from 1 to 10. As expected, the precision increases while the recall
+decreases with increasing 𝑅𝑇𝐻. In addition, F1, accuracy, and AUC
+first increase and then decrease with the increase in 𝑅𝑇𝐻. The best
+performance for F1 and AUC is when 𝑅𝑇𝐻 = 6. Thus, we set the
+risk threshold 𝑅𝑇𝐻 = 6 for RiskProp.
+4.8
+Guarantees for Practical Use
+Here, we present guarantees for RiskProp in practical use regard-
+ing the following aspects: 1) guarantee of convergence; 2) time
+complexity; and 3) linear scalability.
+Convergence and Uniqueness. We present the theoretical prop-
+erties of RiskProp, including the proofs of prior knowledge, con-
+vergence, and uniqueness of the proposed metrics, i.e., reliability,
+trustiness, and confidence. Proofs are shown in the Appendix due to
+lack of space.
+Time complexity. In each interaction, the RiskProp updates the
+reliability, Trustiness metrics of accounts and Confidence metric of
+transactions. Therefore, the complexity of each iteration is O(|𝑈 | +
+|𝑆|) = O(|𝑆|), |𝑆| is the total edges in the payer–payee network.
+Thus, for 𝑘 iterations, the total running time is O(𝑘|𝑆|).
+Linear scalability. We have shown that RiskProp is linear in run-
+ning time in the number of nodes. To show this experimentally as
+well, we create random networks of an increasing number of nodes
+and edges and compute the running time of the algorithm until
+convergence. Figure 7(b) shows that the running time increases lin-
+early with the number of nodes in the network. Therefore, we can
+conclude that RiskProp is a scalable rating method that is suitable
+for applications on large-scale transaction networks.
+1 2 3 4 5 6 7 8 9 10
+RTH
+0
+20
+40
+60
+80
+100
+Performance(%)
+Precision
+Recall
+F1
+Accuracy
+AUC
+(a) Impact of different RTH
+103
+104
+105
+106
+107
+Number of nodes
+10
+1
+100
+101
+102
+103
+104
+Run time (seconds)
+(b) Scalability of RiskProp
+Figure 7: Further analysis of RiskProp.
+Analysis of incorrect predictions. Furthermore, the results of the
+RiskProp+ experiment showed that 39 out of 46 (85%) phishing ac-
+counts were correctly predicted as illicit accounts. To understand
+why the remaining accounts failed to be detected as illicit by our
+model, we manually checked their transactions and neighbors and
+obtained the following results: (i) For one of the accounts, we have
+a risk score of 5.72, and in practice, the system will also warn about
+such accounts that are close to the risk threshold (𝑅𝑇𝐻 = 6). (ii)
+One account is set as the default risk value because the phishing
+account has no outgoing transactions for the time being, and in
+practice, we can make correct predictions as soon as the phish-
+ing account starts laundering money. (iii) Among the remaining
+five accounts, one account has a high transaction volume of 154.
+The remaining four accounts have a high volume of transactions
+along with withdrawals of ETH from exchanges, which directly
+contributed to the high de-anonymity score of transactions. How-
+ever, there are two sides to the story: regulation and fraud are a
+game of confrontation. For hackers, reusing accounts reduces the
+probability of being identified as high risk and, at the same time,
+reusing accounts and withdrawing money from exchanges increase
+the risk of exposure and fund freezing.
+5
+RELATED WORK
+Risk control studies in cryptocurrency. In recent years, there
+has been growing interest in account clustering and detecting il-
+licit activities (e.g., financial scams, money laundering) in cryp-
+tocurrency transaction networks [31]. Victor [27] is the first to
+propose clustering heuristics for the Ethereum’s account model,
+including deposit address reuse, airdrop multi-participation, and
+self-authorization. A recent review of the literature on cryptocur-
+rency scams [5] showed that the existing methods (e.g., [9], [10],
+and [15]) are mainly based on supervised classifiers fed with hand-
+crafted features. Many attempts have been made [25, 29, 32] to
+incorporate structural information by learning the latent repre-
+sentations of accounts. Some researchers have investigated and
+modeled the money flow from a network perspective [18, 21] to
+better identify illicit activities. After all, there is still a black area
+regarding the estimation of the risk value of Ethereum accounts,
+which is the key task in alerting about suspicious accounts and
+transactions on the chain.
+Rating and ranking on graph data. The aim of ratings and
+rankings on graph data is to provide a score or an order for each
+node in a graph. Currently, the main solutions are based on link
+analysis technique [24], Bayesian model [12], and iterative learn-
+ing [13], etc. Similarly to the proposed RiskProp algorithm, [16]
+
+RiskProp
+WWW ’23, April 30–May 4, 2023, Austin, TX, US
+proposed axioms and iterative formulations to establish the rela-
+tionship between ratings. In [22], the authors measured the bias
+and prestige of nodes in networks based on trust scores. In [17], the
+authors highlighted that graph-based approaches provide unique
+solution opportunities for financial crime and fraud detection. A
+review on this topic [3] described the problems in current studies:
+lack of ground truths, imbalanced class, and large-scale network.
+These challenges also exist in our risk rating problem on Ethereum
+transaction networks.
+6
+CONCLUSIONS AND FUTURE WORK
+In this paper, we present the first systematic study to assess the
+account risk via a rating system named RiskProp. In RiskProp, we
+modeled transaction records of Ethereum as a bipartite graph, pro-
+posed a novel metric called de-anonymous score to quantify the
+transaction risk, and designed a network propagation mechanism
+based on transaction semantics. By analyzing the rating results and
+manually checking the accounts with high risk, we evaluated the
+performance of RiskProp and obtained new insights about transac-
+tion risks on Ethereum. In addition, we employed the obtained risk
+scores to conduct illicit/licit account classification experiments on
+labeled data, and the superiority of this method over baseline meth-
+ods further verified the effectiveness of RiskProp in risk estimation.
+For future work, we plan to integrate the transaction amounts and
+temporal information in our model, develop a web page or online
+tool for querying risk values of accounts, and share the details of
+risky cases with the Ethereum community.
+REFERENCES
+[1] Rachit Agarwal, Shikhar Barve, and Sandeep Kumar Shukla. 2021. Detecting ma-
+licious accounts in permissionless blockchains using temporal graph properties.
+Applied Network Science 6, 1 (Dec. 2021), 9.
+[2] Leman Akoglu, Rishi Chandy, and Christos Faloutsos. 2013. Opinion fraud
+detection in online reviews by network effects. In Proceedings of the International
+AAAI Conference on Web and Social Media, Vol. 7. 2–11.
+[3] Leman Akoglu, Hanghang Tong, and Danai Koutra. 2015. Graph based anomaly
+detection and description: A survey. Data Mining and Knowledge Discovery 29, 3
+(2015), 626–688.
+[4] Salam Al-E’mari, Mohammed Anbar, Yousef Sanjalawe, and Selvakumar Man-
+ickam. 2021. A Labeled Transactions-Based Dataset on the Ethereum Network.
+Communications in Computer and Information Science 1347 (2021), 61–79.
+[5] Massimo Bartoletti, Stefano Lande, Andrea Loddo, Livio Pompianu, and Sergio
+Serusi. 2021. Cryptocurrency scams: Analysis and perspectives. IEEE Access 9
+(2021), 148353–148373.
+[6] Sergey Brin and Lawrence Page. 1998. The anatomy of a large-scale hypertextual
+Web search engine. Computer Networks and ISDN Systems 30, 1 (1998), 107–117.
+[7] Liang Chen, Jiaying Peng, Yang Liu, Jintang Li, Fenfang Xie, and Zibin Zheng.
+2021. Phishing Scams Detection in Ethereum Transaction Network. ACM Trans-
+actions on Internet Technology 21, 1 (Feb. 2021), 1–16.
+[8] Weili Chen, Xiongfeng Guo, Zhiguang Chen, Zibin Zheng, and Yutong Lu. 2020.
+Phishing scam detection on Ethereum: Towards financial security for blockchain
+ecosystem. In IJCAI. 4506–4512.
+[9] Weili Chen, Zibin Zheng, Jiahui Cui, Edith Ngai, Peilin Zheng, and Yuren Zhou.
+2018. Detecting ponzi schemes on Ethereum: Towards healthier blockchain
+technology. In WWW. ACM, 1409–1418.
+[10] Steven Farrugia, Joshua Ellul, and George Azzopardi. 2020. Detection of illicit
+accounts over the Ethereum blockchain. Expert Systems with Applications 150
+(Jul. 2020), 113318.
+[11] FATF.
+2022.
+Money
+Laundering
+and
+Terrorist
+Financing
+Red
+Flag
+Indicators
+Associated
+with
+Virtual
+Assets.
+http://www.fatf-
+gafi.org/publications/fatfrecommendations/documents/Virtual-Assets-Red-
+Flag-Indicators.html.
+[12] Bryan Hooi, Neil Shah, Alex Beutel, Stephan Günnemann, Leman Akoglu, Mohit
+Kumar, Disha Makhija, and Christos Faloutsos. 2016. Birdnest: Bayesian inference
+for ratings-fraud detection. In ICDM. SIAM, 495–503.
+[13] Bryan Hooi, Hyun Ah Song, Alex Beutel, Neil Shah, Kijung Shin, and Christos
+Faloutsos. 2016. FRAUDAR: Bounding graph fraud in the face of camouflage. In
+SIGKDD. ACM, 895–904.
+[14] Tao Huang, Dan Lin, and Jiajing Wu. 2022. Ethereum Account Classification
+Based on Graph Convolutional Network. IEEE Transactions on Circuits and
+Systems II: Express Briefs 69, 5 (2022), 2528–2532.
+[15] Rahmeh Fawaz Ibrahim, Aseel Mohammad Elian, and Mohammed Ababneh. 2021.
+Illicit account detection in the Ethereum blockchain using machine learning. In
+ICIT. IEEE, 488–493.
+[16] Srijan Kumar, Bryan Hooi, Disha Makhija, Mohit Kumar, Christos Faloutsos, and
+V.S. Subrahmanian. 2018. REV2: Fraudulent user prediction in rating platforms.
+In WSDM. ACM, 333–341.
+[17] Eren Kurshan and Hongda Shen. 2020. Graph computing for financial crime
+and fraud detection: Trends, challenges and outlook. International Journal of
+Semantic Computing 14, 4 (2020), 565–589.
+[18] Banwari Lal, Rachit Agarwal, and Sandeep Kumar Shukla. 2021. Understanding
+money trails of suspicious activities in a cryptocurrency-based blockchain. arXiv
+preprint arXiv:2108.11818 (2021).
+[19] Sijia Li, Gaopeng Gou, Chang Liu, Chengshang Hou, Zhenzhen Li, and Gang
+Xiong. 2022. TTAGN: Temporal Transaction Aggregation Graph Network for
+Ethereum Phishing Scams Detection. In Proceedings of the ACM Web Conference.
+ACM, 661–669.
+[20] Xiangfeng Li, Shenghua Liu, Zifeng Li, Xiaotian Han, Chuan Shi, Bryan Hooi, He
+Huang, and Xueqi Cheng. 2020. Flowscope: Spotting money laundering based
+on graphs. In AAAI, Vol. 34. 4731–4738.
+[21] Dan Lin, Jiajing Wu, Qi Yuan, and Zibin Zheng. 2020. Modeling and understanding
+Ethereum transaction records via a complex network approach. IEEE Transactions
+on Circuits and Systems II: Express Briefs 67, 11 (2020), 2737–2741.
+[22] Abhinav Mishra and Arnab Bhattacharya. 2011. Finding the bias and prestige of
+nodes in networks based on trust scores. In WWW. ACM, 567–576.
+[23] Malte Möser, Rainer Böhme, and Dominic Breuker. 2014. Towards risk scoring
+of Bitcoin transactions. In FC. Springer, 16–32.
+[24] Alex Sangers, Maran van Heesch, Thomas Attema, Thijs Veugen, Mark Wigger-
+man, Jan Veldsink, Oscar Bloemen, and Daniël Worm. 2019. Secure multiparty
+pageRank algorithm for collaborative fraud detection. In FC. Springer, 605–623.
+[25] Da Sun Handason Tam, Wing Cheong Lau, Bin Hu, Qiu Fang Ying, Dah Ming Chiu,
+and Hong Liu. 2019. Identifying illicit accounts in large scale e-payment networks
+– A graph representation learning approach. arXiv preprint arXiv:1906.05546
+(2019).
+[26] Chainalysis Team. 2021.
+Report: Key Players of the Cryptocurrency Ecosys-
+tem. Retrieved January 11, 2022 from https://go.chainalysis.com/rs/503-FAP-
+074/images/Key-players-in-crypto-report.pdf
+[27] Friedhelm Victor. 2020.
+Address clustering heuristics for Ethereum. In FC.
+Springer, 617–633.
+[28] Friedhelm Victor and Andrea Marie Weintraud. 2021. Detecting and quantifying
+wash trading on decentralized cryptocurrency exchanges. In WWW. ACM, 23–
+32.
+[29] Mark Weber, Giacomo Domeniconi, Jie Chen, Daniel Karl I. Weidele, Claudio
+Bellei, Tom Robinson, and Charles E. Leiserson. 2019. Anti-money laundering in
+Bitcoin: Experimenting with graph convolutional networks for financial forensics.
+arXiv preprint arXiv:1908.02591 (2019).
+[30] Gavin Wood. 2014. Ethereum: A secure decentralised generalised transaction
+ledger. Ethereum Project Yellow Paper 151 (2014), 1–32.
+[31] Jiajing Wu, Jieli Liu, Yijing Zhao, and Zibin Zheng. 2021. Analysis of cryptocur-
+rency transactions from a network perspective: An overview. Journal of Network
+and Computer Applications 190 (2021), 103139.
+[32] Jiajing Wu, Qi Yuan, Dan Lin, Wei You, Weili Chen, Chuan Chen, and Zibin
+Zheng. 2022. Who are the phishers? Phishing scam detection on Ethereum via
+network embedding. IEEE Transactions on Systems, Man, and Cybernetics: Systems
+52, 2 (2022), 1156–1166.
+[33] Zihao Yuan, Qi Yuan, and Jiajing Wu. 2020. Phishing detection on ethereum via
+learning representation of transaction subgraphs. In Blocksys. Springer, 178–191.
+[34] Jiong Zhu, Ryan A Rossi, Anup Rao, Tung Mai, Nedim Lipka, Nesreen K Ahmed,
+and Danai Koutra. 2021. Graph neural networks with heterophily. In Proceedings
+of the AAAI Conference on Artificial Intelligence, Vol. 35. 11168–11176.
+
+WWW ’23, April 30–May 4, 2023, Austin, TX, US
+Anonymous author(s)
+A
+APPENDIX
+A.1
+Proof for Prior Knowledge
+In this part, we provide proofs that the proposed metrics, i.e., Re-
+liability, Trustiness, and Confidence satisfy Prior knowledge 1 - 5.
+Prior knowledge 1 in the main paper is the following:
+[Prior knowledge 1] Payees with higher trustiness receive trans-
+actions with higher de-anonymous scores. Formally, if two pay-
+ees 𝑣1 and 𝑣2 have a one-to-one mapping, ℎ : 𝐼𝑛(𝑣1) → 𝐼𝑛(𝑣2)
+and 𝑆𝑐𝑜𝑟𝑒(𝑢, 𝑣1) > 𝑆𝑐𝑜𝑟𝑒(ℎ(𝑢), 𝑣2) ∀(𝑢, 𝑣1) ∈ 𝐼𝑛(𝑣1), then 𝑇 (𝑣1) >
+𝑇 (𝑣2).
+The formulation to be used to show that the prior knowledge is
+satisfied is Equations 2, 3, and 4 in the main paper.
+𝑇 (𝑣) =
+�
+(𝑢,𝑣)∈𝐼𝑛(𝑣) 𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) × 𝐶𝑜𝑛𝑓 (𝑢, 𝑣)
+|𝐼𝑛(𝑣) |
+𝑅(𝑢) =
+�
+(𝑢,𝑣)∈𝑂𝑢𝑡 (𝑢) 𝐶𝑜𝑛𝑓 (𝑢, 𝑣)
+|𝑂𝑢𝑡 (𝑢) |
+𝐶𝑜𝑛𝑓 (𝑢, 𝑣) = 𝑅(𝑢) + (1 − |𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) −𝑇 (𝑣) |)
+2
+Proof. To prove the Prior Knowledge 1, let us take two payees 𝑣1
+and 𝑣2 that have identically ego networks and a one-to-one mapping
+ℎ, such that |𝐼𝑛(𝑣1)| = |𝐼𝑛(𝑣2)|, 𝐶𝑜𝑛𝑓 (𝑢, 𝑣1) = 𝐶𝑜��𝑓 (ℎ(𝑢), 𝑣2), and
+𝑆𝑐𝑜𝑟𝑒(𝑢, 𝑣1) > 𝑆𝑐𝑜𝑟𝑒(ℎ(𝑢), 𝑣2) ∀(𝑢, 𝑣1) ∈ 𝐼𝑛(𝑣1).
+According to Equation 2, we have
+𝑇 (𝑣1) −𝑇 (𝑣2) =
+�
+(𝑢,𝑣1)∈𝐼𝑛(𝑣1) 𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣1) × 𝐶𝑜𝑛𝑓 (𝑢, 𝑣1)
+|𝐼𝑛(𝑣1) |
+−
+�
+(𝑢,𝑣2)∈𝐼𝑛(𝑣2) 𝑆𝑐𝑜𝑟𝑒 (ℎ(𝑢), 𝑣2) × 𝐶𝑜𝑛𝑓 (ℎ(𝑢), 𝑣2)
+|𝐼𝑛(𝑣2) |
+=
+�
+(𝑢,𝑣1)∈𝐼𝑛(𝑣1) (𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣1) − 𝑆𝑐𝑜𝑟𝑒 (ℎ(𝑢), 𝑣2)) × 𝐶𝑜𝑛𝑓 (𝑢, 𝑣1)
+|𝐼𝑛(𝑣1) |
+As 𝑆𝑐𝑜𝑟𝑒(𝑢, 𝑣1) > 𝑆𝑐𝑜𝑟𝑒(ℎ(𝑢), 𝑣2), so
+𝑇 (𝑣1) −𝑇 (𝑣2) >
+�
+(𝑢,𝑣1)∈𝐼𝑛(𝑣1) 𝐶𝑜𝑛𝑓 (𝑢, 𝑣1)
+|𝐼𝑛(𝑣1) |
+As 𝐶𝑜𝑛𝑓 (𝑢, 𝑣1) ≥ 0 because Confidence are non-negative,
+𝑇 (𝑣1) −𝑇 (𝑣2) > 0 ⇒ 𝑇 (𝑣1) > 𝑇 (𝑣2)
+The other items of prior knowledge have very similar and straight-
+forward proof.
+□
+A.2
+Proof for Convergence
+Before the proof of convergence, we first discuss the boundary of
+proposed metrics. At the end of iteration 𝑡 of Algorithm 1, and by
+equation 2, 3, and 4, we get,
+𝑇 𝑡 (𝑣) =
+�
+(𝑢,𝑣)∈𝐼𝑛(𝑣) 𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) × 𝐶𝑜𝑛𝑓 𝑡−1(𝑢, 𝑣)
+|𝐼𝑛(𝑣) |
+𝑅𝑡 (𝑢) =
+�
+(𝑢,𝑣)∈𝑂𝑢𝑡 (𝑢) 𝐶𝑜𝑛𝑓 𝑡−1(𝑢, 𝑣)
+|𝑂𝑢𝑡 (𝑢) |
+𝐶𝑜𝑛𝑓 𝑡 (𝑢, 𝑣) = 𝑅𝑡 (𝑢) + (1 − |𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) −𝑇 𝑡 (𝑣) |
+2
+)
+𝑇 ∞(𝑣), 𝑅(∞) (𝑢),𝐶𝑜𝑛𝑓 (∞) (𝑢, 𝑣) are their final values after con-
+vergence.
+Lemma A.1. (Boundary discussion) Set the maximum score in the
+transaction network as 𝑀, namely:
+𝑀 = max
+(𝑢,𝑣) 𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣)
+then |𝑀| < 1.
+The difference between a payee𝑣’s final Trustiness and its Trustiness
+after the first iteration is
+|𝑇 ∞(𝑢) −𝑇 1(𝑢) | ≤ |𝑀 |
+Similarly,
+|𝑅∞(𝑢) − 𝑅1(𝑢) | ≤ 1
+|𝐶𝑜𝑛𝑓 ∞(𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 1(𝑢, 𝑣) | ≤ 1 + |𝑀 |
+2
+= 𝛼 (𝛼 ≤ 1)
+Proof. First, we state that |𝑀| is strictly less than 1 in prac-
+tice. According to the formulation of 𝑆𝑐𝑜𝑟𝑒 of the main paper, we
+can see that 𝑆𝑐𝑜𝑟𝑒(𝑢, 𝑣) = 1 when 𝑂𝑢𝑡𝑇𝑥𝑛(𝑢) = 𝑚𝑎𝑥𝑂𝑢𝑡 and
+𝐼𝑛𝑇𝑥𝑛(𝑣) = 𝑚𝑎𝑥𝐼𝑛, it is an extreme situation where the largest
+number of payments and receptions of the entire network appears
+in one transaction. The other case is 𝑆𝑐𝑜𝑟𝑒(𝑢, 𝑣) = −1 when that
+𝑂𝑢𝑡𝑇𝑥𝑛(𝑢) = 1 and 𝐼𝑛𝑇𝑥𝑛(𝑣) = 1 at the same time. This situation
+presents to be some isolated transactions, however, they do not
+propagate risk and thus do not influence convergence. These situa-
+tions are out of our consideration. So we get |𝑀| is strictly smaller
+than 1.
+Then, we prove that 𝑇 (𝑣) is bounded during the iterations:
+|𝑇 ∞(𝑣) −𝑇 1(𝑣) | = |
+�
+(𝑢,𝑣)∈𝐼𝑛(𝑣) 𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) × 𝐶𝑜𝑛𝑓 ∞(𝑢, 𝑣)
+|𝐼𝑛(𝑣) |
+−
+�
+(𝑢,𝑣)∈𝐼𝑛(𝑣) 𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) × 𝐶𝑜𝑛𝑓 0(𝑢, 𝑣)
+|𝐼𝑛(𝑣) |
+|
+Since |𝑥 + 𝑦| ≤ |𝑥| + |𝑦|, we get,
+|𝑇 ∞ (𝑣) −𝑇 1 (𝑣) | ≤
+�
+(𝑢,𝑣)∈𝐼𝑛(𝑣) |𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) × (𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 0 (𝑢, 𝑣)) |
+|𝐼𝑛(𝑣) |
+Since |𝑥 × 𝑦| = |𝑥| × |𝑦|, we have,
+|𝑇 ∞ (𝑣) −𝑇 1 (𝑣) | ≤
+�
+(𝑢,𝑣)∈𝐼𝑛(𝑣) |𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) | × (𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 0 (𝑢, 𝑣)) |
+|𝐼𝑛(𝑣) |
+(5)
+Since |𝑆𝑐𝑜𝑟𝑒(𝑢, 𝑣)| ≤ |𝑀| ≤ 1, and |(𝐶𝑜𝑛𝑓 ∞(𝑢, 𝑣)−𝐶𝑜𝑛𝑓 0(𝑢, 𝑣))| ≤
+1, we get,
+|𝑇 ∞ (𝑣) −𝑇 1 (𝑣) | ≤ |𝑀 | × |𝐼𝑛(𝑣) |
+|𝐼𝑛(𝑣) | = |𝑀 |
+Next, we conduct the proof on 𝑅(𝑢):
+|𝑅∞ (𝑢) − 𝑅1 (𝑢) | =
+| �
+(𝑢,𝑣)∈𝑂𝑢𝑡 (𝑢) 𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − �
+(𝑢,𝑣)∈𝑂𝑢𝑡 (𝑢) 𝐶𝑜𝑛𝑓 0 (𝑢, 𝑣) |
+|𝑂𝑢𝑡 (𝑢) |
+Again, since |𝑥 × 𝑦| = |𝑥| × |𝑦|, we get,
+|𝑅∞ (𝑢) − 𝑅1 (𝑢) | ≤
+�
+(𝑢,𝑣)∈𝑂𝑢𝑡 (𝑢) |𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 0 (𝑢, 𝑣) |
+|𝑂𝑢𝑡 (𝑢) |
+(6)
+Similarly, since |(𝐶𝑜𝑛𝑓 ∞(𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 0(𝑢, 𝑣))| ≤ 1, we have,
+|𝑅∞ (𝑢) − 𝑅1 (𝑢) | ≤ |𝑂𝑢𝑡 (𝑢) |
+|𝑂𝑢𝑡 (𝑢) | = 1
+
+RiskProp
+WWW ’23, April 30–May 4, 2023, Austin, TX, US
+Finally, we calculate the bound of 𝐶𝑜𝑛𝑓 (𝑢, 𝑣):
+|𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 1 (𝑢, 𝑣) | =
+|𝑅∞ (𝑢) − 𝑅1 (𝑢) + |𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) −𝑇 1 (𝑣) | − |𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) −𝑇 ∞ (𝑣) ||
+2
+Since |𝑥 + 𝑦| ≤ |𝑥| + |𝑦|, we have
+|𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 1 (𝑢, 𝑣) | ≤
+|𝑅∞ (𝑢) − 𝑅1 (𝑢) | + ( ||𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) −𝑇 1 (𝑣) | − |𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) −𝑇 ∞ (𝑣) ||)
+2
+Since ||𝑥| − |𝑦|| ≤ |𝑥 − 𝑦|, it follows that,
+|𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 1 (𝑢, 𝑣) | ≤ |𝑅∞ (𝑢) − 𝑅1 (𝑢) | + |𝑇 ∞ (𝑣) −𝑇 1 (𝑣) |
+2
+(7)
+Since |𝑅∞(𝑢) − 𝑅1(𝑢)| ≤ 1, and|𝑇 ∞(𝑣) −𝑇 1(𝑣)| ≤ |𝑀|, we get,
+|𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 1 (𝑢, 𝑣) | ≤ 1 + |𝑀 |
+2
+For convenience, we let 1+|𝑀 |
+2
+= 𝛼. Since |𝑀| < 1, then 𝛼 < 1.
+□
+Theorem A.2. Convergence of Propagation: The difference during
+iterations is bounded as as |𝐶𝑜𝑛𝑓 ∞(𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡 (𝑢, 𝑣)| ≤ 𝛼𝑡 (𝛼 =
+1+|𝑀 |
+2
+< 1), ∀(𝑢, 𝑣) ∈ 𝑆. As 𝑡 increases, the difference decreases and
+𝐶𝑜𝑛𝑓 𝑡 (𝑢, 𝑣) converges to |𝐶𝑜𝑛𝑓 ∞(𝑢, 𝑣). Similarly, |𝑇 ∞(𝑣) −𝑇𝑡 (𝑣)| ≤
+𝛼𝑡−1, ∀𝑣 ∈ 𝑉 , |𝑅∞(𝑢) − 𝑅𝑡 (𝑢)| ≤ 𝛼𝑡−1, ∀𝑢 ∈ 𝑈 .
+Proof. Similar to Equations 5, 6, and 7, we have,
+|𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡 (𝑢, 𝑣) | ≤ |𝑅∞ (𝑢) − 𝑅𝑡 (𝑢) | + |𝑇 ∞ (𝑣) −𝑇𝑡 (𝑣) |
+2
+(8)
+|𝑅∞ (𝑢) − 𝑅𝑡 (𝑢) | ≤
+�
+(𝑢,𝑣,)∈𝑂𝑢𝑡 (𝑢) |𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡−1 (𝑢, 𝑣) |
+|𝑂𝑢𝑡 (𝑢) |
+(9)
+|𝑇 ∞ (𝑣) −𝑇𝑡 (𝑣) |
+≤
+�
+(𝑢,𝑣)∈𝐼𝑛(𝑣) |𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) | × |(𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡−1 (𝑢, 𝑣)) |
+|𝐼𝑛(𝑣) |
+(10)
+First, we will prove the convergence of Confidence using mathe-
+matical induction.
+Base case of induction.
+When 𝑡 = 1, as we proved in Lemma A.1, we get:
+|𝐶𝑜𝑛𝑓 ∞(𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 1(𝑢, 𝑣) | ≤ 𝛼1
+Induction step.
+We assume by hypothesis that
+|𝐶𝑜𝑛𝑓 ∞(𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡−1(𝑢, 𝑣) | ≤ 𝛼𝑡−1,
+which is consistent with the base case already.
+Then, by substituting Equations 9 and 10 into Equation 8, for the
+case in the next iteration where time is 𝑡, we have,
+|𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡 (𝑢, 𝑣) |
+≤
+�
+(𝑢,𝑣)∈𝑂𝑢𝑡 (𝑢) |𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡−1 (𝑢, 𝑣) |
+2 × |𝑂𝑢𝑡 (𝑢) |
+)
++
+�
+(𝑢,𝑣)∈𝐼𝑛(𝑣) |𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) | × |(𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡−1 (𝑢, 𝑣) |
+2 × |𝐼𝑛(𝑣) |
+≤ 1
+2 ×
+�
+( 1 + |𝑀 |
+2
+)𝑡−1 +
+|𝑀 | × �
+(𝑢,𝑣)∈𝐼𝑛(𝑣) |𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡−1 (𝑢, 𝑣) |
+|𝐼𝑛(𝑣) |
+�
+≤ 1
+2 ×
+�
+( 1 + |𝑀 |
+2
+)𝑡−1 + |𝑀 | × ( 1 + |𝑀 |
+2
+)𝑡−1
+�
+≤ ( 1 + |𝑀 |
+2
+)𝑡 = 𝛼𝑡
+Therefore, |𝐶𝑜𝑛𝑓 ∞(𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡 (𝑢, 𝑣)| ≤ 𝛼𝑡.
+|𝑅∞ (𝑢) − 𝑅𝑡 (𝑢) | ≤
+�
+(𝑢,𝑣)∈𝑂𝑢𝑡 (𝑢) |𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡−1 (𝑢, 𝑣) |
+|𝑂𝑢𝑡 (𝑢) |
+≤
+�
+(𝑢,𝑣)∈𝑂𝑢𝑡 (𝑢) ( 1+|𝑀|
+2
+)𝑡−1
+|𝑂𝑢𝑡 (𝑢) |
+≤ ( 1 + |𝑀 |
+2
+)𝑡−1 = 𝛼𝑡−1
+|𝑇 ∞ (𝑣) −𝑇 (𝑣)𝑡 |
+≤
+�
+(𝑢,𝑣)∈𝐼𝑛(𝑣) |𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) | × |(𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡−1 (𝑢, 𝑣)) |
+|𝐼𝑛(𝑣) |
+≤
+�
+(𝑢,𝑣)∈𝐼𝑛(𝑣) |(𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡−1 (𝑢, 𝑣)) |
+|𝐼𝑛(𝑣) |
+≤
+�
+(𝑢,𝑣)∈𝐼𝑛(𝑣) ( 1+|𝑀|
+2
+)𝑡−1
+|𝐼𝑛(𝑣) |
+≤ ( 1 + |𝑀 |
+2
+)𝑡−1 = 𝛼𝑡−1
+As discussed in the Lemma A.1. we know that |𝑀| is strictly
+smaller than 1, then we have 𝛼 < 1. As 𝑡 increases, 𝛼𝑡−1 → 0
+and 𝛼𝑡 → 0, so after t iterations, 𝐶𝑜𝑛𝑓 (𝑢, 𝑣)𝑡 → 𝐶𝑜𝑛𝑓 ∞(𝑢, 𝑣),
+𝑅(𝑢)𝑡 → 𝑅∞(𝑢), and 𝑇 (𝑣)𝑡 → 𝑇 ∞(𝑣), the algorithm converges.
+□
+A.3
+Proof for Uniqueness
+In this part, we provides proofs that Reliability, Trustiness, and
+Confidence are unique.
+Theorem A.3. Confidence, Reliability, and Trustiness converge to
+the unique value.
+Proof. First, we consider the uniqueness of Confidence using
+mathematical contradiction.
+Let the 𝐶𝑜𝑛𝑓 (𝑢, 𝑣) converges to different values. So, let (���, 𝑣)
+be the transaction with maximum Confidence difference, 𝐷 (with
+𝐷 ≥ 0), between its two possible 𝐶𝑜𝑛𝑓1(𝑢, 𝑣) and 𝐶𝑜𝑛𝑓2(𝑢, 𝑣).
+According to Equation 8, we get,
+𝐷 = |𝐶𝑜𝑛𝑓 ∞
+1 (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 ∞
+2 (𝑢, 𝑣) |
+≤ |𝑅∞
+1 (𝑢) − 𝑅∞
+2 (𝑢) | + |𝑇 ∞
+1 (𝑣) −𝑇 ∞
+2 (𝑣) |
+2
+(11)
+Then, according to Equation 9 and 10, we have,
+|𝑅∞
+1 (𝑢) − 𝑅∞
+2 (𝑢) | ≤
+�
+(𝑢,𝑣)∈𝑂𝑢𝑡 (𝑢) |𝐶𝑜𝑛𝑓 ∞
+1 (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 ∞
+2 (𝑢, 𝑣) |
+|𝑂𝑢𝑡 (𝑢) |
+≤ 𝐷
+(12)
+
+WWW ’23, April 30–May 4, 2023, Austin, TX, US
+Anonymous author(s)
+|𝑇 ∞
+1 (𝑣) −𝑇 ∞
+2 (𝑣) | ≤
+�
+(𝑢,𝑣)∈𝐼𝑛(𝑣) |𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) | × |𝐶𝑜𝑛𝑓 ∞
+1 (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 ∞
+2 (𝑢, 𝑣) |
+|𝐼𝑛(𝑣) |
+≤ |𝑀 | × 𝐷
+(13)
+We substitute Equation 12 and 13 into Equation (11), and get,
+𝐷 = |𝐶𝑜𝑛𝑓 ∞
+1 (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 ∞
+2 (𝑢, 𝑣) |
+≤ 1
+2 × (
+�
+(𝑢,𝑣)∈𝑂𝑢𝑡 (𝑢) |𝐶𝑜𝑛𝑓 ∞
+1 (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 ∞
+2 (𝑢, 𝑣) |
+|𝑂𝑢𝑡 (𝑢) |
++
+�
+(𝑢,𝑣)∈𝐼𝑛(𝑣) |𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) | × |𝐶𝑜𝑛𝑓 ∞
+1 (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 ∞
+2 (𝑢, 𝑣) |
+|𝐼𝑛(𝑣) |
+≤ 1
+2 ×
+�
+𝐷 +
+|𝑀 | × �
+(𝑢,𝑣)∈𝐼𝑛(𝑣) |𝐶𝑜𝑛𝑓 ∞(𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 ∞(𝑢, 𝑣) |
+|𝐼𝑛(𝑣) |
+�
+≤ 1
+2 × (𝐷 + |𝑀 | × 𝐷)
+≤ ( 1 + |𝑀 |
+2
+) × 𝐷
+= 𝛼 × 𝐷
+Thus, by solving 𝐷 ≤ 𝛼 × 𝐷(𝛼 ≠ 0) and with the condition that
+𝐷 ≥ 0, we obtain 𝐷 = 0. Then, |𝐶𝑜𝑛𝑓 ∞
+1 (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 ∞
+2 (𝑢, 𝑣)| = 0
+and converge value of Confidence is unique. The uniqueness of
+Trustiness and Reliability have similar proof.
+□
+

QNE0T4oBgHgl3EQfkQEN/content/tmp_files/2301.02469v1.pdf.txt

@@ -0,0 +1,795 @@
+1
+Cox Point Processes for Multi-Altitude LEO
+Satellite Networks
+Chang-Sik Choi and Franc¸ois Baccelli
+Abstract—We propose a simple analytical approach to describe
+the locations of low earth orbit (LEO) satellites based on a
+Cox point process. We develop a variable-altitude Poisson orbit
+process by accounting for the fact that satellites are always
+located on circular orbits and these orbits may have different
+altitudes. Then, the satellites on these orbits are modeled as
+the Poisson point processes conditionally on the orbit process.
+For this model, we derive the distribution of the distance to
+the nearest visible satellite, the outage probability, the Laplace
+functional of the proposed satellite Cox point process, and the
+Laplace transform of the interference under a general fading. The
+derived statistics allow one to evaluate the performance of such
+LEO satellite communication systems as functions of network
+parameters.
+Index Terms—LEO satellite communications, Stochastic geom-
+etry, Cox point process, Nearest distance, Total interference
+I. INTRODUCTION
+A. Motivation and Background
+LEO satellites provide global connectivity to millions of
+devices on earth [1]–[5]. The applications of LEO satellite net-
+works are numerous [1]: they provide Internet connections to
+devices where ground infrastructure is unavailable [2]; local-
+ization and emergency communications of aerial and ground
+devices can be enabled by LEO satellites [3]; LEO satellite
+networks provide cheaper Internet connections to developing
+countries [4]. LEO satellite networks can even be integrated
+with terrestrial networks to enable reliable connections to
+devices in a small area [5]. To support these applications, LEO
+satellite networks will have a very large number of satellites.
+The viability and performance of LEO satellite communi-
+cations are significantly determined by the way satellites are
+distributed in space. Various evaluation methodologies have
+been proposed to obtain the performance of LEO satellite
+communication networks. For satellite layout, some studies
+used probabilistic approaches including a binomial point pro-
+cess [6]–[9]. In contrast to the simulation-based approach,
+the benefits of employing such analytical models lie in the
+fact that they presents large-scale behaviors as functions of
+network key parameters such as the mean number of satel-
+lites, their altitudes, etc. Nevertheless, the binomial satellite
+point processes in [6]–[9] were not able to incorporate the
+fact that the satellites are located on approximately circular
+trajectories around the earth, namely their orbits. In this paper,
+we provide a tractable model that incorporates this fact in the
+multi-altitude LEO satellite case, by generalizing the work in
+Chang-Sik Choi is with Hongik University, South Korea. Franc¸ois
+Baccelli is with Inria Paris and Telecom Paris, France. (email: chang-
+[email protected], [email protected])
+[10] where all orbits are at the same altitude. Specifically,
+we present an analytical framework leveraging a Cox point
+process so that orbits are created first according to a Poisson
+point process on a cuboid and then satellites are distributed
+as Poisson point processes conditionally on these orbits. We
+derive key statistical properties of the proposed network model
+that are critical to obtain the performance of such satellite
+networks as functions of the altitude distribution, of the mean
+number of orbits, of the number of satellites, and of the
+Laplace transform of the random variable representing fading.
+B. Contributions
+Modeling of variable orbit LEO satellite constellations:
+This paper accounts for the geometric properties of practical
+LEO satellite systems that (i) satellites are always on orbits
+around the earth and (ii) such orbits are possibly at different
+altitudes. By developing a nonhomogeneous Poisson point
+process of mean λ in a cuboid, we creates a Poisson orbit
+process of orbits in the Euclidean space. Then, conditionally
+on the orbit process, satellites are distributed as linear Poisson
+point processes of mean µ on these orbits. Our motivation is
+to represent a general LEO satellite network where satellites
+are located at different altitude bands.
+Statistical properties of the proposed Cox point pro-
+cess: The proposed satellite Cox point process is built to be
+invariant by all rotations of the reference plane. This makes
+the statistical properties of the network to be the same for all
+perspectives seen from all points on earth. Leveraging this, we
+obtain the probability distribution function of the distance from
+the typical user to its nearest visible satellite and then derive
+the outage probability of the proposed network model. Using
+it, we derive the Laplace functional of the proposed satellite
+Cox point process and then give an integral expression for the
+Laplace transform of the total interference. These formulas
+are directly used to assess the network performance metrics
+such as the Signal-to-interference-plus-noise ratio (SINR) of
+the typical user.
+II. COX-MODELED SATELLITES
+A. Satellite Distribution
+The center of the earth is O = (0, 0, 0) and it is of radius
+re. The xy-plane is the reference plane and the x-axis is
+longitude reference direction. In this paper, we only focus on
+the snapshot of the network geometry and the movement of
+satellites is out of the scope.
+Consider a cuboid C = [ra, rb] × [0, π) × [0, π) where ra ≤
+rb the minimum and maximum altitudes and a Poisson point
+arXiv:2301.02469v1  [eess.SP]  6 Jan 2023
+
+2
+Reference: xy-plane
+x-axis
+A
+θ
+l(ρ,θ,φ)
+φ
+X: satellite 
+ω
+O
+~
+y-axis
+ρ
+z-axis
+Fig. 1.
+The orbital plane meets the reference plane at two points and the
+point with angle less than π is A. The angle θ is measured from the x-axis
+to the segment OA. The inclination ˜ϕ is measured from the reference plane
+to the orbital plane and the azimuth ϕ is given by π/2 − ˜ϕ. The angle ω for
+satellite X is measured from OA to OX over the orbital plane.
+process Ξ of intensity measure λν(dρ)/π2 in the cuboid C.
+We have
+� rb
+ra ν(dρ) = 1. Then, we build an orbit process by
+mapping each point of Ξ, say (ρ, θ, ϕ) into an orbit l(ρ, θ, ϕ)
+in the Euclidean space. Specifically, the first coordinate ρ is the
+orbit’s radius, θ is the orbit’s longitude, and ϕ is the orbit’s
+azimuth. See Fig. 1. For the Poisson point process on the
+cuboid, we write Ξ = �
+i Zi, where Zi is the point of Ξ.
+Since there are on average λ points of Ξ, there are on average
+λ orbits. The orbit process O in R3 is given by
+O =
+�
+Zi∈Ξ
+l(ρi, θi, ϕi).
+(1)
+Conditionally on Ξ, the locations of satellites on each orbit
+l(ρi, θi, ϕi) are modeled as a homogeneous Poisson point
+process ψi of intensity µ/(2πρi) on this orbit. Equivalently,
+the orbital angles of satellites on each orbit are modeled as
+a 1-dim homogeneous Poisson point process φi on segment
+[0, 2π) of intensity µ/(2π). Since the satellites are distributed
+conditionally on Ξ, the satellite point process Ψ is a Cox point
+process. The satellite Cox point process is
+Ψ =
+�
+i
+ψi.
+(2)
+Figs. 2 – 4 depict the proposed satellite Cox point process with
+λ, µ, ra and rb. In the figures, we use ν(dρ) =
+dρ
+rb−ra , i.e.,
+the radii of orbits are uniformly distributed on the interval
+[ra, rb]. The proposed model can be used to represent e.g.,
+multiple operators of LEO satellite networks where orbits are
+at different altitudes. The case of all satellites are located at
+the same altitude in [10] is a special case of the proposed
+model by taking ν(dρ) = δra(dρ), where ra is the radius of
+orbits.
+B. User Distribution
+Users are located on the surface of the earth {(x, y, z)|x2 +
+y2 +z2 = r2
+e} and the locations of network users are assumed
+to be independent of the locations of the LEO satellites.
+III. STATISTICAL RESULTS
+In this section, we derive/prove (i) the mean number of
+LEO satellites, (ii) the isotropy of Ψ, (iii) the distances from
+Fig. 2. The proposed Cox satellite model with ra = 7000 km, rb = 7100
+km. We use λ = 60, µ = 40, and ν(dρ) = dρ/(rb − ra).
+Fig. 3. The Cox-modeled satellite with ra = 7000 km and rb = 7500 km.
+We use λ = 30, µ = 60, and ν(dρ) = dρ/(rb − ra).
+Fig. 4. The Cox-modeled satellite with ra = 7000 km and rb = 8500 km.
+We use λ = 70, µ = 30, and ν(dρ) = dρ/(rb − ra).
+
+3
+the LEO satellites to an arbitrarily located user, (iv) the
+distribution of the distance to the nearest visible satellite, (v)
+the outage probability, (vi) the Laplace functional of Ψ, and
+(vii) the Laplace transform of the total interference under
+general fading. These statistical properties directly determine
+the performance of downlink LEO satellite communications
+in this context.
+Lemma 1. The average number of the proposed Cox satellite
+point process is λµ.
+Proof: The average number of satellites is given by
+E [Ψ(S)] = E
+�
+� �
+Zi∈Ξ
+E
+�
+� �
+Xj∈ψi
+1
+������
+Ξ
+�
+�
+�
+�
+= E
+� �
+Zi∈Ξ
+� 2π
+0
+µ
+2π dx
+����� Ξ
+�
+= µ
+�
+C
+λ
+π2 ν(dρ) dθ dϕ = λµ,
+where we use Campbell’s mean value theorem [11].
+Below we show that O is invariant w.r.t. rotations. This
+allows one to evaluate the performance of network seen by a
+typical user at the north pole.
+Lemma 2. The distribution of O and Ψ are invariant by all
+rotations of the reference space (O, x, y, z).
+Proof: The intensity measure of the proposed orbit pro-
+cess Ξ has the product form: ν(dρ) ×
+�
+λ/π2�
+dθ dϕ. This
+shows that the angles (θ, ϕ) form a homogeneous Poisson
+point process on the rectangle [0, π) × [0, π). [10] proved
+that the orbit process mapped by the very intensity measure
+�
+λ/π2�
+dθ dϕ is invariant by all rotations of the reference
+space. Hence, the law of O is also invariant by all rotations of
+the reference space (O, x, y, z). In the same vein, the law of
+Ψ is invariant by all rotations of the reference space as well.
+Lemma 3. Consider a satellite X of orbital angle ωj on the
+orbit l(ρi, θi, ϕi). The distance from (0, 0, re) to the satellite
+X(ρi, θi, ϕi, ωj) is given by
+�
+ρ2
+i − 2ρire sin(ωj) cos(ϕi) + r2e.
+(3)
+Proof: The coordinates (x, y, z) ∈ R3 of the satellite that
+has the orbital angle ωj on the orbit l(ρi, θi, ϕi) are given by
+x =
+�
+ρ2
+i cos2(ωj) + ρ2
+i sin2(ωj) cos2( ˜ϕi) cos
+�
+˜θ + θi
+�
+, (4)
+y =
+�
+ρ2
+i cos2(ωj) + ρ2
+i sin2(ωj) cos2( ˜ϕi) sin
+�
+˜θ + θi
+�
+, (5)
+z = ρi sin(ωj) sin( ˜ϕi),
+(6)
+˜θ = arctan (tan(ωj) cos( ˜ϕi)) ,
+(7)
+where ˜ϕ is the inclination: ˜ϕ = π/2 − ϕ.
+As a result, the distance from (0, 0, re) to the satellite is
+∥(x, y, z) − (0, 0, re)∥ =
+�
+ρ2
+i − 2ρire sin(ωj) cos(ϕi) + r2e.
+Note the distance is independent of the variable θ.
+U=(0,0,re)
+O
+A
+C
+E
+B
+F
+A`
+D
+ρ
+re
+Bottom of spherical cap
+Fig. 5. The arc of orbit l(ρ, θ, ϕ) in spherical cap C(ρ, d).
+A. The Lengths of Orbits’ Arcs
+Since (i) users are independent of Ψ and (ii) Ψ is invariant
+by rotations (Lemma 2 ), one can consider a typical user at
+(0, 0, re) and study the network performance it experiences,
+which will be typical.
+Let C(d) be the subset of S such that the distances from
+the typical observer u to the satellites on C(d) are less than
+a distance d. For any ra ≤ ρ ≤ rb, we define
+C(d) =
+�
+ra≤ρ≤rb
+C(ρ, d)
+=
+�
+ra≤ρ≤rb
+�
+(x, y, z) ∈ R3 |z ≥ re, x2 + y2 + z2 = ρ2,
+x2 + y2 + (z − re)2 ≤ d2�
+,
+where z ≥ re, since satellites with z-coordinates less than re
+are invisible to the user at (0, 0, re). C(ρ, d) is a spherical cap
+associated with the orbit of radius ρ. See Fig. 5.
+Lemma 4. The length of the arc given by the intersection of
+the spherical cap C(ρ, d) and the orbit l(ρ, θ, ϕ) is
+2ρ arcsin
+�
+�
+�
+1 −
+�ρ2 + r2e − d2
+2ρre cos(ϕ)
+�2
+�
+� ,
+(8)
+for ρ − re ≤ d ≤
+�
+ρ2 − r2e.
+Proof: Consider C(ρ, d). Let ξ be the angle ∠AOU in
+Fig. 5. Then, we use the law of Cosine to obtain cos(ξ) =
+(ρ2 + r2
+e − d2)/(2ρre).
+For the triangle △BCD, we have CD = ρ cos(ξ) tan(ϕ).
+Since the angle ∠BDC is π/2, we obtain
+BD =
+�
+ρ2 sin2(ξ) − ρ2 cos2(ξ) tan2(ϕ).
+For △BOD, OB = ρ and let κ′ = ∠BOD. Then we have
+sin(κ′) = BD/ρ =
+�
+sin2(ξ) − cos2(ξ) tan2(ϕ).
+Finally, the length of the arc >
+BF is given by
+ν(>
+BF) = 2ρ arcsin(
+�
+1 − cos2(ξ) sec2(ϕ)).
+where cos(ξ) = (ρ2 + r2
+e − d2)/(2ρre).
+In downlink LEO satellite communication networks, net-
+work users are meant to receive signals from their closest or
+
+4
+P(D > d) = exp
+�
+−2λ
+π
+� rb
+ra
+� ξ
+0
+�
+1 − e
+−µπ−1 arcsin
+�√
+1−cos2(ξ) sec2(ϕ)
+��
+dϕν(dρ)
+�
+,
+(9)
+P(D = ∞) = exp
+�
+−2λ
+π
+� rb
+ra
+� arccos(re/ρ)
+0
+�
+1 − e
+−µπ−1 arcsin
+�√
+1−r2e sec2(ϕ)/ρ2
+��
+dϕν(dρ)
+�
+,
+(10)
+L(f) = exp
+�
+− λ
+π2
+�
+C
+�
+1 − e− µ
+2π
+� 2π
+0
+1−exp (− ¯
+f(ρ,θ,ϕ,ω)) dω�
+ν(dρ) dθ dϕ
+�
+,
+(11)
+LΨ(f)f=sH∥X−U∥−α = exp
+�
+− λ
+π2
+�
+¯C
+�
+1 − e− µ
+2π
+�
+¯
+ω 1−LH(s(ρ2−2ρre sin(ω) cos(ϕ)+r2
+e)− α
+2 ) dω�
+ν(dρ) dθ dϕ
+�
+.
+(12)
+nearest satellites [9]. The distance D from a network user to
+its closest LEO satellite is a random variable. When there is
+no visible satellite, D
+def
+= ∞.
+Lemma 5. The cumulative distribution function of D is given
+by Eq. (9) where cos(ξ) = (ρ2 + r2
+e − d2)/(2ρre).
+Proof: For ra − re ≤ d ≤
+�
+r2
+b − r2e, we have
+P(D > d)
+(a)
+= P(∥X − u∥ > d, ∀X ∈ Ψ)
+(b)
+= P(∥Xj − u∥ > d, ∀Xj ∈ ψi, ∀Zi ∈ Ξ)
+= P
+�
+� �
+Zi∈Ξ
+P
+�
+� �
+Xj∈ψi
+∥Xj − u∥ > d
+������
+Ξ
+�
+�
+�
+� .
+To get (a), we use the fact that for R > r, all satellites should
+be at distances greater than r. We have (b) by using that the
+Cox satellite point process is comprised of the Poisson point
+processes conditionally on orbits. We have
+P
+�
+� �
+Xj∈ψi
+∥Xj − u∥ > r
+������
+Ξ
+�
+�
+= exp
+�
+−µπ−1 arcsin
+��
+1 − cos2(ξ)sec2(ϕi)
+��
+,
+where cos(ξ) = (ρ2
+i + r2
+e − d2)/(2ρire), as a function of the
+orbits’ radius. We use the facts that (i) in order to have no point
+at distance less than r, the arc created by the orbit l(ρi, ϕi, θi)
+and the set C(ρi, d) has to be empty of satellite points and (ii)
+the void probability of the Poisson point process of intensity
+µ on the arc is given by the negative exponential of µ times
+the arc length. Leveraging the facts that only the orbits with
+azimuth angles ϕ < ξ1, π − ξ1 < ϕ < π meet the spherical
+cap C(d), we have
+P(D > d)
+= P
+�ϕi<ξ1,π−ξ1<ϕi<π
+�
+Zi∈Ξ
+e
+−µπ−1 arcsin
+�√
+1−cos2(ξ) sec2(ϕi)
+��
+= exp
+�
+−2λ
+π
+� rb
+ra
+� ξ
+0
+�
+1 − e
+−µπ−1 arcsin
+�√
+1−cos2(ξ) sec2(ϕ)
+��
+dϕν(dρ)
+�
+,
+where cos(ξ) = (ρ2 + r2
+e − d2)/(2ρre). Above, we use the
+probability generating functional of the Poisson point process
+Ξ of intensity measure λν(dρ)/π2 in C .
+Definition 1. Outage occurs if the typical network user has
+no visible satellite. Equivalently, outage occurs if D = ∞.
+Lemma 6. The outage probability is given by Eq. (10).
+Proof: When there is no visible satellite, D = ∞. By
+using Lemma 5, the outage probability is given by
+P(D = ∞)
+= P(∥Xj − u∥ >
+�
+ρ2
+i − r2e, ∀Xj ∈ ψi, ∀Zi ∈ Ξ)
+= P
+�
+� �
+Zi∈Ξ
+P
+�
+� �
+Xj∈ψi
+∥Xj − u∥ >
+�
+ρ2
+i − r2e
+������
+Ξ
+�
+�
+�
+� ,
+where we have
+P
+�
+� �
+Xj∈ψi
+∥Xj − u∥ >
+�
+ρ2
+i − r2e
+������
+Ξ
+�
+�
+= exp
+�
+−µπ−1 arcsin
+��
+1 − r2e sec2(ϕi)/ρ2
+i
+��
+.
+We use that when d =
+�
+ρ2
+i − r2e, cos(ξ) = re/ρi. In other
+words, for a given ρ, the orbits with azimuth angles less than
+arccos(re/ρ) meet the spherical cap C(ρ,
+�
+ρ2 − r2e). The
+outage probability is then given by
+P
+� �
+Zi∈Ξ
+e−µπ−1 arcsin(√
+1−r2
+e sec2(ϕi)/ρ2
+i )
+�
+= exp
+�
+−2λ
+π
+� rb
+ra
+� arccos(re/ρ)
+0
+�
+1 − e
+−µπ−1 arcsin
+�√
+1−r2e sec2(ϕ)/ρ2
+��
+dϕν(dρ)
+�
+,
+where we use the probability generating functional of Ξ of
+intensity measure λν(dρ)/π2.
+Fig. 6 shows the outage probability obtained by Lemma 6.
+Lemma 7. Consider a function f(X) : R3 → R. The Laplace
+functional of the Cox point process is defined by LΨ(f) =
+EΨ
+�
+exp
+�
+− �
+Xi∈Ψ f(Xi)
+��
+. The Laplace functional is given
+by Eq. (11) where C = [ra, rb] × [0, π) × [0, π).
+
+5
+Fig. 6. The outage probability with ra = 7000 km and rb = 7500 km. We
+use λ = 72, µ = 22, and ν(dρ) = dρ/(rb − ra).
+Proof: The Laplace functional of the satellite Cox point
+process is given by
+LΨ(f)
+= E
+�
+e− �
+X∈Ψ f(X)�
+= EΞ
+�
+Eψ
+�
+e
+− �
+Zi∈Ξ
+�
+Xj ∈ψi f(X)��� Ξ
+��
+= EΞ
+� �
+Zi∈Ξ
+exp
+�
+− µ
+2π
+� 2π
+0
+1 − e− ¯
+f(ρi,θi,ϕi,ω) dω
+��
+= exp
+�
+− λ
+π2
+� rb
+ra
+� π
+0
+� π
+0
+�
+1 − e− µ
+2π
+� 2π
+0
+1−exp (− ¯
+f(ρ,θ,ϕ,ω)) dω�
+dϕ dθν(dρ)
+�
+,
+where we use the function ¯f(ρ, θ, ϕ, ω)=f(X) for any satellite
+X on the orbit l(ρ, θ, ϕ) with its orbital angle ω. Then, we
+use the probability generating functional of the Poisson point
+process of intensity measure λν(dρ)/π2 to get the result.
+Consider a random variable H modeling general fading. A
+received signal power of a user at u is then given by f(X) =
+H∥X −u∥−α where X is the location of the satellite and α is
+the path loss exponent. The total interference S is then given
+by the sum of the received signal powers from all satellites.
+S =
+�
+X∈ ¯Ψ
+H∥X − u∥−α, ¯Ψ = Ψ
+�
+�
+�
+ra<ρ≤rb
+C(ρ,
+�
+ρ2 − r2e)
+�
+� .
+Corollary 1. The Laplace transform of the total interference
+is given by Eq. (12) where ¯C = {(ρ, θ, ϕ) ∈ |l(ρ, θ, ϕ) ∩
+C(
+�
+r2
+b − r2a) ̸= ∅} and ¯ω = {ω ∈ [0, 2π]|X(ρ, θ, ϕ, ω) ∈
+C(
+�
+r2
+b − r2a), ∀(ρ, θ, ϕ) ∈ ¯C}.
+Proof: The Laplace transform in question is
+LΨ(f)f=sH∥X−U∥−α
+= EΞ
+�
+Eψ
+�
+e
+− �
+Zi∈Ξ
+�
+Xj ∈ψ sH∥Xj−u∥−α��� Ξ
+��
+= EΞ
+�
+� �
+Zi∈Ξ
+Eψ
+�
+� �
+Xj∈ψi
+LH(s∥Xj − u∥−α)
+�
+�
+�
+� ,
+where LH(κ) is the Laplace transform of the random variable
+H. Using a technique similar to Lemmas 3 and 7, we obtain
+the final result.
+IV. CONCLUSION
+This paper presents a stochastic geometry framework to
+model the locations of LEO satellites with multiple altitudes
+using a Cox point process. It provides analytical expressions
+for essential statistical properties such as the distribution of the
+distance from a typical user to the nearest satellite, the Laplace
+functional of the Cox point process, and the Laplace transform
+of the total interference, experienced by a typical user. These
+results can directly be used to determine the performance of
+multi-altitude LEO satellite communication networks. Future
+work will include (i) the analysis of the coverage probability of
+the typical user, (ii) the evaluation of the satellite coverage area
+underneath the Cox-modeled satellites, and (iii) an extension
+to a fixed-inclination orbit process.
+ACKNOWLEDGMENT
+The work of Chang-Sik Choi was supported by the NRF-
+2021R1F1A1059666. The work of Franc¸ois Baccelli was
+supported by the ERC NEMO grant 788851 to INRIA.
+REFERENCES
+[1] Y. Su, Y. Liu, Y. Zhou, J. Yuan, H. Cao, and J. Shi, “Broadband LEO
+satellite communications: Architectures and key technologies,” IEEE
+Wireless Commn., vol. 26, no. 2, pp. 55–61, 2019.
+[2] Z. Qu, G. Zhang, H. Cao, and J. Xie, “LEO satellite constellation for
+Internet of Things,” IEEE Access, vol. 5, pp. 18 391–18 401, 2017.
+[3] J. Khalife, M. Neinavaie, and Z. M. Kassas, “The first carrier phase
+tracking and positioning results with starlink LEO satellite signals,”
+IEEE Trans. Aerospace and Electronic Syst., vol. 58, no. 2, pp. 1487–
+1491, 2022.
+[4] A. Guidotti, A. Vanelli-Coralli, M. Conti, S. Andrenacci, S. Chatzinotas,
+N. Maturo, B. Evans, A. Awoseyila, A. Ugolini, T. Foggi, L. Gaudio,
+N. Alagha, and S. Cioni, “Architectures and key technical challenges for
+5G systems incorporating satellites,” IEEE Trans. Veh. Technol., vol. 68,
+no. 3, pp. 2624–2639, 2019.
+[5] 3GPP TR 38.821, “Solutions for NR to support non-terrestrial networks
+(NTN),” 3GPP TR 38.821.
+[6] N. Okati, T. Riihonen, D. Korpi, I. Angervuori, and R. Wichman,
+“Downlink coverage and rate analysis of low earth orbit satellite con-
+stellations using stochastic geometry,” IEEE Trans. Commun., vol. 68,
+no. 8, pp. 5120–5134, 2020.
+[7] A. Talgat, M. A. Kishk, and M.-S. Alouini, “Stochastic geometry-based
+analysis of LEO satellite communication systems,” IEEE Commun. Lett.,
+vol. 25, no. 8, pp. 2458–2462, 2021.
+[8] D.-H. Na, K.-H. Park, Y.-C. Ko, and M.-S. Alouini, “Performance
+analysis of satellite communication systems with randomly located
+ground users,” IEEE Trans. Wireless Commun., vol. 21, no. 1, pp. 621–
+634, 2022.
+[9] D.-H. Jung, J.-G. Ryu, W.-J. Byun, and J. Choi, “Performance analysis
+of satellite communication system under the shadowed-rician fading: A
+stochastic geometry approach,” IEEE Trans. Commun., vol. 70, no. 4,
+pp. 2707–2721, 2022.
+[10] C.-S. Choi and F. Baccelli, “An analytical framework for downlink LEO
+satellite communications based on Cox point processes,” arXiv preprint
+arXiv:2212.03549, 2022.
+[11] F. Baccelli and B. Błaszczyszyn, “Stochastic geometry and wireless
+networks: volume I theory,” Foundations and Trends in Networking,
+vol. 3, no. 3–4, pp. 249–449, 2010.
+
+10-2
+outage probability
+10-4
+10-6
+10
+20
+10
+20
+30
+30
+40
+50
+μ
+入

QNE0T4oBgHgl3EQfkQEN/content/tmp_files/load_file.txt

@@ -0,0 +1,307 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf,len=306
+page_content='1  Cox Point Processes for Multi-Altitude LEO  Satellite Networks  Chang-Sik Choi and Franc¸ois Baccelli  Abstract—We propose a simple analytical approach to describe  the locations of low earth orbit (LEO) satellites based on a  Cox point process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' We develop a variable-altitude Poisson orbit  process by accounting for the fact that satellites are always  located on circular orbits and these orbits may have different  altitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Then, the satellites on these orbits are modeled as  the Poisson point processes conditionally on the orbit process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  For this model, we derive the distribution of the distance to  the nearest visible satellite, the outage probability, the Laplace  functional of the proposed satellite Cox point process, and the  Laplace transform of the interference under a general fading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' The  derived statistics allow one to evaluate the performance of such  LEO satellite communication systems as functions of network  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  Index Terms—LEO satellite communications, Stochastic geom-  etry, Cox point process, Nearest distance, Total interference  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' INTRODUCTION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Motivation and Background  LEO satellites provide global connectivity to millions of  devices on earth [1]–[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' The applications of LEO satellite net-  works are numerous [1]: they provide Internet connections to  devices where ground infrastructure is unavailable [2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' local-  ization and emergency communications of aerial and ground  devices can be enabled by LEO satellites [3];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' LEO satellite  networks provide cheaper Internet connections to developing  countries [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' LEO satellite networks can even be integrated  with terrestrial networks to enable reliable connections to  devices in a small area [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' To support these applications, LEO  satellite networks will have a very large number of satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  The viability and performance of LEO satellite communi-  cations are significantly determined by the way satellites are  distributed in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Various evaluation methodologies have  been proposed to obtain the performance of LEO satellite  communication networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' For satellite layout, some studies  used probabilistic approaches including a binomial point pro-  cess [6]–[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' In contrast to the simulation-based approach,  the benefits of employing such analytical models lie in the  fact that they presents large-scale behaviors as functions of  network key parameters such as the mean number of satel-  lites, their altitudes, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Nevertheless, the binomial satellite  point processes in [6]–[9] were not able to incorporate the  fact that the satellites are located on approximately circular  trajectories around the earth, namely their orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' In this paper,  we provide a tractable model that incorporates this fact in the  multi-altitude LEO satellite case, by generalizing the work in  Chang-Sik Choi is with Hongik University, South Korea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Franc¸ois  Baccelli is with Inria Paris and Telecom Paris, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' (email: chang-  sik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='choi@hongik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='kr, francois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='baccelli@inria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='fr)  [10] where all orbits are at the same altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Specifically,  we present an analytical framework leveraging a Cox point  process so that orbits are created first according to a Poisson  point process on a cuboid and then satellites are distributed  as Poisson point processes conditionally on these orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' We  derive key statistical properties of the proposed network model  that are critical to obtain the performance of such satellite  networks as functions of the altitude distribution, of the mean  number of orbits, of the number of satellites, and of the  Laplace transform of the random variable representing fading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Contributions  Modeling of variable orbit LEO satellite constellations:  This paper accounts for the geometric properties of practical  LEO satellite systems that (i) satellites are always on orbits  around the earth and (ii) such orbits are possibly at different  altitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' By developing a nonhomogeneous Poisson point  process of mean λ in a cuboid, we creates a Poisson orbit  process of orbits in the Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Then, conditionally  on the orbit process, satellites are distributed as linear Poisson  point processes of mean µ on these orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Our motivation is  to represent a general LEO satellite network where satellites  are located at different altitude bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  Statistical properties of the proposed Cox point pro-  cess: The proposed satellite Cox point process is built to be  invariant by all rotations of the reference plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' This makes  the statistical properties of the network to be the same for all  perspectives seen from all points on earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Leveraging this, we  obtain the probability distribution function of the distance from  the typical user to its nearest visible satellite and then derive  the outage probability of the proposed network model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Using  it, we derive the Laplace functional of the proposed satellite  Cox point process and then give an integral expression for the  Laplace transform of the total interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' These formulas  are directly used to assess the network performance metrics  such as the Signal-to-interference-plus-noise ratio (SINR) of  the typical user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' COX-MODELED SATELLITES  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Satellite Distribution  The center of the earth is O = (0, 0, 0) and it is of radius  re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' The xy-plane is the reference plane and the x-axis is  longitude reference direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' In this paper, we only focus on  the snapshot of the network geometry and the movement of  satellites is out of the scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  Consider a cuboid C = [ra, rb] × [0, π) × [0, π) where ra ≤  rb the minimum and maximum altitudes and a Poisson point  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='02469v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='SP]  6 Jan 2023  2  Reference: xy-plane  x-axis  A  θ  l(ρ,θ,φ)  φ  X: satellite  ω  O  ~  y-axis  ρ  z-axis  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  The orbital plane meets the reference plane at two points and the  point with angle less than π is A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' The angle θ is measured from the x-axis  to the segment OA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' The inclination ˜ϕ is measured from the reference plane  to the orbital plane and the azimuth ϕ is given by π/2 − ˜ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' The angle ω for  satellite X is measured from OA to OX over the orbital plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  process Ξ of intensity measure λν(dρ)/π2 in the cuboid C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  We have  � rb  ra ν(dρ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Then, we build an orbit process by  mapping each point of Ξ, say (ρ, θ, ϕ) into an orbit l(ρ, θ, ϕ)  in the Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Specifically, the first coordinate ρ is the  orbit’s radius, θ is the orbit’s longitude, and ϕ is the orbit’s  azimuth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' For the Poisson point process on the  cuboid, we write Ξ = �  i Zi, where Zi is the point of Ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  Since there are on average λ points of Ξ, there are on average  λ orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' The orbit process O in R3 is given by  O =  �  Zi∈Ξ  l(ρi, θi, ϕi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  (1)  Conditionally on Ξ, the locations of satellites on each orbit  l(ρi, θi, ϕi) are modeled as a homogeneous Poisson point  process ψi of intensity µ/(2πρi) on this orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Equivalently,  the orbital angles of satellites on each orbit are modeled as  a 1-dim homogeneous Poisson point process φi on segment  [0, 2π) of intensity µ/(2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Since the satellites are distributed  conditionally on Ξ, the satellite point process Ψ is a Cox point  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' The satellite Cox point process is  Ψ =  �  i  ψi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  (2)  Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 2 – 4 depict the proposed satellite Cox point process with  λ, µ, ra and rb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' In the figures, we use ν(dρ) =  dρ  rb−ra , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=',  the radii of orbits are uniformly distributed on the interval  [ra, rb].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' The proposed model can be used to represent e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=',  multiple operators of LEO satellite networks where orbits are  at different altitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' The case of all satellites are located at  the same altitude in [10] is a special case of the proposed  model by taking ν(dρ) = δra(dρ), where ra is the radius of  orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' User Distribution  Users are located on the surface of the earth {(x, y, z)|x2 +  y2 +z2 = r2  e} and the locations of network users are assumed  to be independent of the locations of the LEO satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' STATISTICAL RESULTS  In this section, we derive/prove (i) the mean number of  LEO satellites, (ii) the isotropy of Ψ, (iii) the distances from  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' The proposed Cox satellite model with ra = 7000 km, rb = 7100  km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' We use λ = 60, µ = 40, and ν(dρ) = dρ/(rb − ra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' The Cox-modeled satellite with ra = 7000 km and rb = 7500 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  We use λ = 30, µ = 60, and ν(dρ) = dρ/(rb − ra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' The Cox-modeled satellite with ra = 7000 km and rb = 8500 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  We use λ = 70, µ = 30, and ν(dρ) = dρ/(rb − ra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  3  the LEO satellites to an arbitrarily located user, (iv) the  distribution of the distance to the nearest visible satellite, (v)  the outage probability, (vi) the Laplace functional of Ψ, and  (vii) the Laplace transform of the total interference under  general fading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' These statistical properties directly determine  the performance of downlink LEO satellite communications  in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' The average number of the proposed Cox satellite  point process is λµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  Proof: The average number of satellites is given by  E [Ψ(S)] = E  �  � �  Zi∈Ξ  E  �  � �  Xj∈ψi  1  ������  Ξ  �  �  �  �  = E  � �  Zi∈Ξ  � 2π  0  µ  2π dx  ����� Ξ  �  = µ  �  C  λ  π2 ν(dρ) dθ dϕ = λµ,  where we use Campbell’s mean value theorem [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  Below we show that O is invariant w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' This  allows one to evaluate the performance of network seen by a  typical user at the north pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' The distribution of O and Ψ are invariant by all  rotations of the reference space (O, x, y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  Proof: The intensity measure of the proposed orbit pro-  cess Ξ has the product form: ν(dρ) ×  �  λ/π2�  dθ dϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' This  shows that the angles (θ, ϕ) form a homogeneous Poisson  point process on the rectangle [0, π) × [0, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' [10] proved  that the orbit process mapped by the very intensity measure  �  λ/π2�  dθ dϕ is invariant by all rotations of the reference  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Hence, the law of O is also invariant by all rotations of  the reference space (O, x, y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' In the same vein, the law of  Ψ is invariant by all rotations of the reference space as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Consider a satellite X of orbital angle ωj on the  orbit l(ρi, θi, ϕi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' The distance from (0, 0, re) to the satellite  X(ρi, θi, ϕi, ωj) is given by  �  ρ2  i − 2ρire sin(ωj) cos(ϕi) + r2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  (3)  Proof: The coordinates (x, y, z) ∈ R3 of the satellite that  has the orbital angle ωj on the orbit l(ρi, θi, ϕi) are given by  x =  �  ρ2  i cos2(ωj) + ρ2  i sin2(ωj) cos2( ˜ϕi) cos  �  ˜θ + θi  �  , (4)  y =  �  ρ2  i cos2(ωj) + ρ2  i sin2(ωj) cos2( ˜ϕi) sin  �  ˜θ + θi  �  , (5)  z = ρi sin(ωj) sin( ˜ϕi),  (6)  ˜θ = arctan (tan(ωj) cos( ˜ϕi)) ,  (7)  where ˜ϕ is the inclination: ˜ϕ = π/2 − ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  As a result, the distance from (0, 0, re) to the satellite is  ∥(x, y, z) − (0, 0, re)∥ =  �  ρ2  i − 2ρire sin(ωj) cos(ϕi) + r2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  Note the distance is independent of the variable θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  U=(0,0,re)  O  A  C  E  B  F  A`  D  ρ  re  Bottom of spherical cap  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' The arc of orbit l(ρ, θ, ϕ) in spherical cap C(ρ, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' The Lengths of Orbits’ Arcs  Since (i) users are independent of Ψ and (ii) Ψ is invariant  by rotations (Lemma 2 ), one can consider a typical user at  (0, 0, re) and study the network performance it experiences,  which will be typical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  Let C(d) be the subset of S such that the distances from  the typical observer u to the satellites on C(d) are less than  a distance d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' For any ra ≤ ρ ≤ rb, we define  C(d) =  �  ra≤ρ≤rb  C(ρ, d)  =  �  ra≤ρ≤rb  �  (x, y, z) ∈ R3 |z ≥ re, x2 + y2 + z2 = ρ2,  x2 + y2 + (z − re)2 ≤ d2�  ,  where z ≥ re, since satellites with z-coordinates less than re  are invisible to the user at (0, 0, re).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' C(ρ, d) is a spherical cap  associated with the orbit of radius ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' The length of the arc given by the intersection of  the spherical cap C(ρ, d) and the orbit l(ρ, θ, ϕ) is  2ρ arcsin  �  �  �  1 −  �ρ2 + r2e − d2  2ρre cos(ϕ)  �2  �  � ,  (8)  for ρ − re ≤ d ≤  �  ρ2 − r2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  Proof: Consider C(ρ, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Let ξ be the angle ∠AOU in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Then, we use the law of Cosine to obtain cos(ξ) =  (ρ2 + r2  e − d2)/(2ρre).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  For the triangle △BCD, we have CD = ρ cos(ξ) tan(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  Since the angle ∠BDC is π/2, we obtain  BD =  �  ρ2 sin2(ξ) − ρ2 cos2(ξ) tan2(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  For △BOD, OB = ρ and let κ′ = ∠BOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Then we have  sin(κ′) = BD/ρ =  �  sin2(ξ) − cos2(ξ) tan2(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  Finally, the length of the arc >  BF is given by  ν(>  BF) = 2ρ arcsin(  �  1 − cos2(ξ) sec2(ϕ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  where cos(ξ) = (ρ2 + r2  e − d2)/(2ρre).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  In downlink LEO satellite communication networks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' net-  work users are meant to receive signals from their closest or  4  P(D > d) = exp  �  −2λ  π  � rb  ra  � ξ  0  �  1 − e  −µπ−1 arcsin  �√  1−cos2(ξ) sec2(ϕ)  ��  dϕν(dρ)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  (9)  P(D = ∞) = exp  �  −2λ  π  � rb  ra  � arccos(re/ρ)  0  �  1 − e  −µπ−1 arcsin  �√  1−r2e sec2(ϕ)/ρ2  ��  dϕν(dρ)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  (10)  L(f) = exp  �  − λ  π2  �  C  �  1 − e− µ  2π  � 2π  0  1−exp (− ¯  f(ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='ω)) dω�  ν(dρ) dθ dϕ  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  (11)  LΨ(f)f=sH∥X−U∥−α = exp  �  − λ  π2  �  ¯C  �  1 − e− µ  2π  �  ¯  ω 1−LH(s(ρ2−2ρre sin(ω) cos(ϕ)+r2  e)− α  2 ) dω�  ν(dρ) dθ dϕ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  (12)  nearest satellites [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' The distance D from a network user to  its closest LEO satellite is a random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' When there is  no visible satellite, D  def  = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' The cumulative distribution function of D is given  by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' (9) where cos(ξ) = (ρ2 + r2  e − d2)/(2ρre).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  Proof: For ra − re ≤ d ≤  �  r2  b − r2e, we have  P(D > d)  (a)  = P(∥X − u∥ > d, ∀X ∈ Ψ)  (b)  = P(∥Xj − u∥ > d, ∀Xj ∈ ψi, ∀Zi ∈ Ξ)  = P  �  � �  Zi∈Ξ  P  �  � �  Xj∈ψi  ∥Xj − u∥ > d  ������  Ξ  �  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  To get (a), we use the fact that for R > r, all satellites should  be at distances greater than r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' We have (b) by using that the  Cox satellite point process is comprised of the Poisson point  processes conditionally on orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' We have  P  �  � �  Xj∈ψi  ∥Xj − u∥ > r  ������  Ξ  �  �  = exp  �  −µπ−1 arcsin  ��  1 − cos2(ξ)sec2(ϕi)  ��  ,  where cos(ξ) = (ρ2  i + r2  e − d2)/(2ρire), as a function of the  orbits’ radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' We use the facts that (i) in order to have no point  at distance less than r, the arc created by the orbit l(ρi, ϕi, θi)  and the set C(ρi, d) has to be empty of satellite points and (ii)  the void probability of the Poisson point process of intensity  µ on the arc is given by the negative exponential of µ times  the arc length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Leveraging the facts that only the orbits with  azimuth angles ϕ < ξ1, π − ξ1 < ϕ < π meet the spherical  cap C(d), we have  P(D > d)  = P  �ϕi<ξ1,π−ξ1<ϕi<π  �  Zi∈Ξ  e  −µπ−1 arcsin  �√  1−cos2(ξ) sec2(ϕi)  ��  = exp  �  −2λ  π  � rb  ra  � ξ  0  �  1 − e  −µ��−1 arcsin  �√  1−cos2(ξ) sec2(ϕ)  ��  dϕν(dρ)  �  ,  where cos(ξ) = (ρ2 + r2  e − d2)/(2ρre).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Above, we use the  probability generating functional of the Poisson point process  Ξ of intensity measure λν(dρ)/π2 in C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Outage occurs if the typical network user has  no visible satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Equivalently, outage occurs if D = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' The outage probability is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  Proof: When there is no visible satellite, D = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' By  using Lemma 5, the outage probability is given by  P(D = ∞)  = P(∥Xj − u∥ >  �  ρ2  i − r2e, ∀Xj ∈ ψi, ∀Zi ∈ Ξ)  = P  �  � �  Zi∈Ξ  P  �  � �  Xj∈ψi  ∥Xj − u∥ >  �  ρ2  i − r2e  ������  Ξ  �  �  �  � ,  where we have  P  �  � �  Xj∈ψi  ∥Xj − u∥ >  �  ρ2  i − r2e  ������  Ξ  �  �  = exp  �  −µπ−1 arcsin  ��  1 − r2e sec2(ϕi)/ρ2  i  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  We use that when d =  �  ρ2  i − r2e, cos(ξ) = re/ρi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' In other  words, for a given ρ, the orbits with azimuth angles less than  arccos(re/ρ) meet the spherical cap C(ρ,  �  ρ2 − r2e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' The  outage probability is then given by  P  � �  Zi∈Ξ  e−µπ−1 arcsin(√  1−r2  e sec2(ϕi)/ρ2  i )  �  = exp  �  −2λ  π  � rb  ra  � arccos(re/ρ)  0  �  1 − e  −µπ−1 arcsin  �√  1−r2e sec2(ϕ)/ρ2  ��  dϕν(dρ)  �  ,  where we use the probability generating functional of Ξ of  intensity measure λν(dρ)/π2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 6 shows the outage probability obtained by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Consider a function f(X) : R3 → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' The Laplace  functional of the Cox point process is defined by LΨ(f) =  EΨ  �  exp  �  − �  Xi∈Ψ f(Xi)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' The Laplace functional is given  by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' (11) where C = [ra, rb] × [0, π) × [0, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  5  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' The outage probability with ra = 7000 km and rb = 7500 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' We  use λ = 72, µ = 22, and ν(dρ) = dρ/(rb − ra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  Proof: The Laplace functional of the satellite Cox point  process is given by  LΨ(f)  = E  �  e− �  X∈Ψ f(X)�  = EΞ  �  Eψ  �  e  − �  Zi∈Ξ  �  Xj ∈ψi f(X)��� Ξ  ��  = EΞ  � �  Zi∈Ξ  exp  �  − µ  2π  � 2π  0  1 − e− ¯  f(ρi,θi,ϕi,ω) dω  ��  = exp  �  − λ  π2  � rb  ra  � π  0  � π  0  �  1 − e− µ  2π  � 2π  0  1−exp (− ¯  f(ρ,θ,ϕ,ω)) dω�  dϕ dθν(dρ)  �  ,  where we use the function ¯f(ρ, θ, ϕ, ω)=f(X) for any satellite  X on the orbit l(ρ, θ, ϕ) with its orbital angle ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Then, we  use the probability generating functional of the Poisson point  process of intensity measure λν(dρ)/π2 to get the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  Consider a random variable H modeling general fading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' A  received signal power of a user at u is then given by f(X) =  H∥X −u∥−α where X is the location of the satellite and α is  the path loss exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' The total interference S is then given  by the sum of the received signal powers from all satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  S =  �  X∈ ¯Ψ  H∥X − u∥−α, ¯Ψ = Ψ  �  �  �  ra<ρ≤rb  C(ρ,  �  ρ2 − r2e)  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' The Laplace transform of the total interference  is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' (12) where ¯C = {(ρ, θ, ϕ) ∈ |l(ρ, θ, ϕ) ∩  C(  �  r2  b − r2a) ̸= ∅} and ¯ω = {ω ∈ [0, 2π]|X(ρ, θ, ϕ, ω) ∈  C(  �  r2  b − r2a), ∀(ρ, θ, ϕ) ∈ ¯C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  Proof: The Laplace transform in question is  LΨ(f)f=sH∥X−U∥−α  = EΞ  �  Eψ  �  e  − �  Zi∈Ξ  �  Xj ∈ψ sH∥Xj−u∥−α��� Ξ  ��  = EΞ  �  � �  Zi∈Ξ  Eψ  �  � �  Xj∈ψi  LH(s∥Xj − u∥−α)  �  �  �  � ,  where LH(κ) is the Laplace transform of the random variable  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Using a technique similar to Lemmas 3 and 7, we obtain  the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' CONCLUSION  This paper presents a stochastic geometry framework to  model the locations of LEO satellites with multiple altitudes  using a Cox point process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' It provides analytical expressions  for essential statistical properties such as the distribution of the  distance from a typical user to the nearest satellite, the Laplace  functional of the Cox point process, and the Laplace transform  of the total interference, experienced by a typical user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' These  results can directly be used to determine the performance of  multi-altitude LEO satellite communication networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Future  work will include (i) the analysis of the coverage probability of  the typical user, (ii) the evaluation of the satellite coverage area  underneath the Cox-modeled satellites, and (iii) an extension  to a fixed-inclination orbit process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  ACKNOWLEDGMENT  The work of Chang-Sik Choi was supported by the NRF-  2021R1F1A1059666.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' The work of Franc¸ois Baccelli was  supported by the ERC NEMO grant 788851 to INRIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  REFERENCES  [1] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Su, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Zhou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Yuan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Cao, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Shi, “Broadband LEO  satellite communications: Architectures and key technologies,” IEEE  Wireless Commn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 26, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 55–61, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  [2] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Qu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Cao, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Xie, “LEO satellite constellation for  Internet of Things,” IEEE Access, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 18 391–18 401, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  [3] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Khalife, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Neinavaie, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Kassas, “The first carrier phase  tracking and positioning results with starlink LEO satellite signals,”  IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Aerospace and Electronic Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 58, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 1487–  1491, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Guidotti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Vanelli-Coralli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Conti, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Andrenacci, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Chatzinotas,  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Maturo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Evans, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Awoseyila, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Ugolini, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Foggi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Gaudio,  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Alagha, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Cioni, “Architectures and key technical challenges for  5G systems incorporating satellites,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Veh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 68,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 2624–2639, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  [5] 3GPP TR 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='821, “Solutions for NR to support non-terrestrial networks  (NTN),” 3GPP TR 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='821.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  [6] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Okati, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Riihonen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Korpi, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Angervuori, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Wichman,  “Downlink coverage and rate analysis of low earth orbit satellite con-  stellations using stochastic geometry,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 68,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 5120–5134, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Talgat, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Kishk, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Alouini, “Stochastic geometry-based  analysis of LEO satellite communication systems,” IEEE Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=',  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 2458–2462, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  [8] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Na, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Park, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Ko, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Alouini, “Performance  analysis of satellite communication systems with randomly located  ground users,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 21, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 621–  634, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  [9] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Jung, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Ryu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Byun, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Choi, “Performance analysis  of satellite communication system under the shadowed-rician fading: A  stochastic geometry approach,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 70, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 4,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 2707–2721, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  [10] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Choi and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Baccelli, “An analytical framework for downlink LEO  satellite communications based on Cox point processes,” arXiv preprint  arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='03549, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  [11] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Baccelli and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' Błaszczyszyn, “Stochastic geometry and wireless  networks: volume I theory,” Foundations and Trends in Networking,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 3, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 3–4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content=' 249–449, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}
+page_content='  10-2  outage probability  10-4  10-6  10  20  10  20  30  30  40  50  μ  入' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'}

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@@ -0,0 +1,2302 @@
+An atomistic insight to moiré reconstruction in Twisted Bilayer Graphene beyond magic angle 
+An atomistic insight to moiré reconstruction in Twisted Bilayer Graphene 
+beyond magic angle 
+Aditya Dey∗,1, a) Shoieb Ahmed Chowdhury,1, a) Tara Peña,2 Sobhit Singh,1 Stephen M. Wu,2, b) and Hesam 
+Askari1 
+1) Department of Mechanical Engineering, University of Rochester, New York 
+2) Department of Electrical and Computer Engineering, University of Rochester, Rochester, 
+New York 
+(*Electronic mail: [email protected]) 
+Twisted bilayer graphene exhibits electronic properties that are highly correlated with the size and arrangement of 
+moiré patterns. While rigid rotation of two layers creates the topology of moiré patterns, local rearrangements of the 
+atoms due to interlayer van der Waals interactions result in atomic reconstruction within the moiré cells. The ability to 
+manipulate these patterns by controlling twist angle and/or externally applied strain provides a promising route to tune 
+their properties. While this phenomenon has been extensively studied for angles close to or smaller than the magic angle 
+(θm=1.1°), its extent for higher angles and how it evolves with strain is unknown and is believed to be mostly absent at 
+high angles. We use theoretical and numerical analyses to resolve reconstruction in angles above θm using interpretive 
+and fundamental physical measures. In addition, we propose a method to identify local regions within moiré cells and 
+track their evolution with strain for a range of representative high twist angles. Our results show that reconstruction is 
+actively present beyond the magic angle and its contribution to the evolution of the moiré cells is major. Our theoretical 
+method to correlate local and global phonon behavior provides further validation on the role of reconstruction at higher 
+angles. Our findings provide a better understanding of moiré reconstruction in large twist angles and the evolution of 
+moiré cells in the presence of strain, that might be very crucial for twistronics-based applications. 
+ 
+I. 
+Introduction 
+Engineering two-dimensional (2D) materials by control- 
+ling the stacking orientation of atomic layers have emerged 
+as a powerful technique to manipulate their mechanical and 
+opto-electronic properties. Bilayer graphene (BLG) is one of 
+the simplest van der Waals (vdW) structures that display di- 
+verse physical properties such as contrasting electronic struc- 
+tures depending on the stacking arrangement1–4. Introduc- 
+ing a relative rotation between the layers forms the Twisted 
+Bilayer Graphene (TBG) in which the atoms create a peri- 
+odic hexagonal superlattice called ‘moiré pattern’ (MP)5–7. 
+Emergence of this pattern is due to the atoms occupying dif- 
+ferent relative interlayer positions compared to BLG with a 
+global size that is inversely correlated with the twist angle 
+(θ ) as Lm = a/(2 sin(θ /2)) where a is the lattice constant of 
+graphene. Application of other mechanical stimuli such as in- 
+equivalent strain to the individual layers of TBG can further 
+manipulate the shape of the pattern. Thus, the combination of 
+hetero-straining process and twist provides a promising out- 
+look for creating unique shapes and geometries of MPs for 
+exciting opto-electronic applications8–10. 
+The atomic arrangements within MPs are influenced by 
+the interlayer vdW forces between the atoms that consider- 
+ably influence the atomic arrangement landscape. To manifest 
+this influence, we can consider a hypothetical intermediate 
+configuration where atoms are rigidly twisted in their plane 
+and consequently, the well-defined BLG stacking configura- 
+tions of AA, AB and SP types with their spatial variations 
+will emerge11,12. Upon allowing atomic reconfiguration, an 
+ 
+ 
+a)These authors contributed equally to this work 
+b)Department of Physics and Astronomy, University of Rochester, Rochester, 
+New York 
+atomic-scale reconstruction occurs and local stacked regions 
+evolve to their true minimum local energy configuration. This 
+process is known as moiré reconstruction13,14. Previous stud- 
+ies have reported this phenomenon for low angle TBGs, es- 
+pecially in the vicinity of or below the ’magic angle’ (θm = 
+1.1°)15,16. As the size of MP shrinks with an increase in θ and 
+leaves less space for reconfiguration of atoms, experimental 
+observation of moiré reconstruction becomes a challenge and 
+is generally assumed to be absent for θ > 2°15,17,18. Since the 
+large angle TBGs contain the same atomic registry but only 
+over a smaller region compared to the small twist angles, it is 
+unreasonable to expect moiré reconstruction should suddenly 
+become absent. The interplay between the in-plane elastic en- 
+ergy and interlayer vdW energy is still expected to contribute 
+to reconstruction at higher angles due to the same fundamen- 
+tal physics. Nevertheless, its extent remains unknown due to 
+the current limitations of experimental methods. 
+Recent experimental studies have demonstrated the abil- 
+ity to control TBGs with and without strain and characterize 
+moiré reconstruction for smaller θ systems8,14,19–23. Imag- 
+ing techniques such as STM and TEM become challenging 
+when feature size becomes comparable to their resolution. As 
+the size of MP decreases with increasing twist, imaging for 
+θ > 2° systems become unfeasible11,24. Therefore, the cur- 
+rent understanding of reconstruction through experimental vi- 
+sualization is limited to low angle twists and largely based 
+on image analysis techniques rather than physical measurable 
+quantities. Optical procedures such as Raman spectroscopy 
+offer an expedient method to characterize TBGs irrespective 
+of their size and twist angle25–28 but such methods predomi- 
+nantly extract the collective behavior of TBGs spanning nu- 
+merous MPs. Therefore the global vibrational behavior ob- 
+tained by Raman cannot be readily used to infer stacking and 
+the extent of reconstruction without an interrelation of phonon 
+
+− 
+i 
+An atomistic insight to moiré reconstruction in Twisted Bilayer Graphene beyond magic angle 
+2 
+ 
+FIG. 1: Atomistic model. Relaxed atomistic structures illustrate how periodic moiré superlattice is formed and how its shape 
+evolves with strain (a & b). Unlike BLG where a single interlayer distancing is expected, twist results in spatial variations of 
+interlayer distancing with as shown for (c) unstrained and (d) strained TBGs. Data presented for the twist angle of θ = 6° and 
+uniaxial strain of 1% . (e) Real space geometric analysis demonstrating the distortion of MPs with applied uniaxial tension to 
+the top layer. 
+ 
+ 
+behavior between local sub-domains and the bulk of TBG. 
+Atomistic analyses offer an alternative tool to study atomic ar- 
+rangements locally with a fine resolution and allow for track- 
+ing of atomistic evolution with varying twist angle11,20,29–31. 
+Current works are heavily concentrated on or below the magic 
+angle and do not explain the correlation of local and global be- 
+havior of TBGs and moreover, have not studied the evolution 
+of MPs with strain. As a result, there remains an outstanding 
+question about the viability and the role of reconstruction at 
+higher angles and how local and global vibrational properties 
+are correlated. 
+In this work, we utilized a combination of first-principles 
+and molecular statics atomistic simulations to examine the lo- 
+cal domains in TBGs and how global vibrational behavior is 
+tied to changes in local atomic registries. Based on physi- 
+cal parameters that include interlayer spacing and interlayer 
+energy, our method associates each atom to known stacking 
+types of the constituent bi-layer graphene and calculates their 
+resultant area fraction and traces the evolution of local sub- 
+domains, and demonstrates evidence of moiré reconstruction 
+for larger θ TBG systems. This paper presents an effective set 
+of criteria for the identification of local stacking and recon- 
+struction phenomena in TBGs that are valid with or without 
+the application of strain. In addition, we demonstrate the cor- 
+relation between local and global vibrational characteristics of 
+TBGs and how it validates our results on reconstructed struc- 
+tures, especially at higher angles. The methods presented in 
+this paper are devised for graphene but further adaptations are 
+ 
+possible for other 2D materials. 
+ 
+II. Methods 
+A.Atomistic modeling. 
+All the TBG structures are constructed by rotating the top 
+layer of Bernal stacked bilayer graphene with respect to its 
+bottom layer. The moiré lattice is created by identifying a 
+common periodic lattice for the two layers. Using the TBG 
+commensurability conditions, we have modeled their real and 
+reciprocal space lattice parameters32,33. The ⃗q vector or re- 
+ciprocal lattice parameter of TBG moirlattice is given as ⃗q = 
+⃗b′ ⃗b, where ⃗b and ⃗b′ denote the reciprocal lattice vectors 
+of the bottom layer and rotated top layer respectively. When 
+heterostrain is applied, the strained ⃗q vector is expressed as 
+q⃗ε = b⃗ε −⃗b, where b⃗ε denotes the strained top layer. The 
+mathematical expressions of b⃗ε are deduced in Supplementary 
+section II. All the atomistic models are relaxed using density 
+functional theory (DFT) simulations, except for θ = 1.08° sys- 
+tem. Because of a large moiré lattice for this structure (11164 
+atoms), DFT becomes forbiddingly inefficient and thus, we 
+use force-field potentials for relaxing this structure. 
+B.DFT calculations. 
+The real space lattices of TBG systems were constructed us- 
+ing ATOMISTIX TOOLKIT (QuantumATK) package34. All 
+the first-principles simulations were conducted with gener- 
+alized gradient approximation (GGA) assimilated in Quan- 
+tum Espresso open source package35,36. The Perdew-Burke- 
+
+3.55
+3.5
+鞋
+3.45
+鞋
+3.4
+3.35± 
+± 
+An atomistic insight to moiré reconstruction in Twisted Bilayer Graphene beyond magic angle 
+3 
+ 
+FIG. 2: Local stacking identification method. (a) Path PQ along the center of one moiré pattern to the other (θ = 6°) (b) 
+Illustration of interlayer energy (ILE) which is the energy contribution of vdW interactions. (c) ILE contour plot for unstrained 
+θ = 6° system. (d) Variation of interlayer spacing (ILS) with respect to moiré twist angles; Horizontal dotted line (magenta) 
+shows the minima of maximum ILS (dmax) obtained throughout a span of low and high angles TBGs. (e) Variation of ILE 
+difference for five representative θ (the dotted line shows the energy difference at soliton width boundary) (f) Contour plot 
+demonstrating individual stacking type locally, obtained after implementing classification method. 
+ 
+ 
+Ernzerhof (PBE) form along with GGA has been used as the 
+exchange-correlation functional37. Ion-electron interactions 
+for carbon atoms in TBGs have been described by ultrasoft 
+pseudopotentials38. All technical details about DFT parame- 
+ters are given in Supplementary information-Section I. 
+ 
+C.MS simulations. 
+Molecular statics simulations were done using LAMMPS 
+open source software39,40. The unstrained, DFT relaxed TBG 
+moiré lattice was transformed into an orthogonal cell for per- 
+forming MS simulations. The simulation box is considered 
+with free surface boundary conditions allowing us to account 
+for the aperiodic crystal geometry (or moiré lattice mismatch) 
+due to strain applied to one of the layers. The uniaxial strain 
+was incremented by  0.1% up to the final strain magnitude of 
+1%. The snapshots of the structure at different strain mag- 
+nitudes were taken in Ovito open visualization tool. Further 
+computational details are mentioned in Supplementary section 
+I. 
+ 
+III. Results and Discussions 
+A. Global structural analysis of pristine and strained TBGs. 
+We have studied a number of TBG systems between θ = 
+1.08° and 13.2° to perform our analysis on MPs close to θm 
+as well as outside the limit of small angles. For simplic- 
+ity, most of the presented data include three representative 
+TBG systems θ = 1.08°, 6° and 13.2°. The MP geometries 
+ 
+are modeled using the well-defined commensurability con- 
+ditions of TBG systems and relaxed using first-principles or 
+force field optimization techniques (see Methods) (Fig. 1(a)). 
+Since the local domains in TBG evolve through high symme- 
+try BLG stacking, we can observe topographical variation in 
+the structure41,42 represented by interlayer spacing (ILS) con- 
+tour plot (Fig. 1(c)). The centers of hexagonal MPs have re- 
+gions of atoms where AA stacking exists13,43. These central 
+regions are surrounded by two domains, AB and BA stack- 
+ing, which are energetically degenerate but topologically in- 
+equivalent. Since both of these stacking represent the Bernal 
+graphene, they can be categorized as one44,45. The boundaries 
+of these AB/BA regions are separated by segments referred 
+as strain solitons. The shear strain which generates due to 
+two inequivalent stacking domains facing each other is con- 
+fined within those segments with characteristic width referred 
+to as the soliton width43. The atomic structure in the soliton 
+regions corresponds to SP stacking which is an intermediate 
+configuration between AB (or BA) and AA. A TBG system 
+displays an out-of-plane corrugation in its structure caused by 
+local ILS variation with AA regions having the highest spac- 
+ing followed by SP and AB regions11,12,30. 
+On employing heterostrain, we observed a similar topo- 
+graphical feature with distorted MPs due to the inequiva- 
+lence of strain in each layer that resulted in an oblique moiré 
+arrangement8 (Fig. 1(b), (d) for tension and Fig. S1 for com- 
+
+(LE (meV/atom)
+28
+17
+14
+3.6
+dmax (AA stacking)
+12
+10
+3.5
+dmin (AB stacking)
+8
+6
+1.10
+3.4
+3.48°
+4
+4.410
+d max
+60
+d min
+7.34°
+3.3
+4
+8
+12
+16
+20
+24
+28
+0
+0.2
+0.4
+0.6
+0.8
+Twist angle (00)
+Normalized distance| 
+| 
+| 
+| 
+| 
+| ̸ 
+| 
+| 
+An atomistic insight to moiré reconstruction in Twisted Bilayer Graphene beyond magic angle 
+4 
+ 
+pression). A geometric analysis is represented to explicate 
+the angular change due to distortion and rigid rotation (Fig. 
+1(e)) by deducing the expressions of their reciprocal lattice 
+(⃗q) vectors (see Supplementary). The change in ⃗q vector with 
+uniaxial strain triggers the distortion in MPs27,46. As shown 
+in Fig. 1(e), the boundaries of MPs resemble a hexagon. On 
+connecting the centers of adjacent MPs, we can draw a tri- 
+angle (∆ABC) with A⃗B and B⃗C as the moiré lattice vectors 
+and α being the angle between them. In unstrained condition, 
+the magnitude of vectors A⃗B = B⃗C = Lm (Lm = Length of 
+MP) and the angles are α = 60°, φ = 120°. As the ⃗q vector 
+changes with uniaxial heterostrain, ∆ABC transforms to ∆A′B 
+′C′ such that A⃗′B′ = B⃗′C′ . The deformed moiré lattice can 
+be quantified with a change in α with the applied strain (Fig. 
+S2). The expressions of moiré reciprocal lattice vectors, show 
+the geometrical changes enforced upon hetero-straining these 
+systems (Supplementary section II). 
+ 
+B. Classification method to identify local domains. 
+The deformation of MP with strain gives rise to changes in 
+their local sub-domains and it is important to examine them 
+for quantifying their contribution to global physical behavior. 
+Traversing along a diagonal of MP (path PQ in Fig. 2(a)), 
+i.e., from the center of one moiré pattern to the center of its 
+second nearest neighbor, we expect to cross all the locally 
+stacked regions: AA, AB, SP, BA, and AA11,43,45. Since we 
+aim to develop a criteria to classify each atom into one of these 
+stacking, we first examined the atoms along the path PQ. To 
+perform the stacking identification, we initially used the ILS 
+parameter d because the local domains in TBGs vary in in- 
+terlayer distancing. Since pristine BLG stacking follows an 
+increasing ILS trend from AB to SP and finally the AA re- 
+gion, dmax (maximum ILS) and dmin (minimum ILS) in TBGs 
+can be respectively understood as the ILS of AA and AB re- 
+gions. By examining the range of ILS (dmax and dmin) over 
+different possible twist angles (Fig. 2(d)) we identify the min- 
+imum value of dmax (3.475Å) and classify atoms above this 
+ILS threshold as AA. It should be noted that this does not mis- 
+classify AB and SP because this threshold is quite above the 
+ILS of pristine AB (3.33Å) and SP (3.38Å). Due to the small 
+ILS difference between AB and SP, the same ILS parameter 
+cannot be used to identify the rest of the stackings. 
+We introduced another parameter called ′interlayer energy′ 
+(ILE) to distinguish between AB and SP according to their 
+energy, rather than ILS. The ILE is a physical measure of 
+vdW interaction between atoms in two different layers, as il- 
+lustrated by the schematic in Fig. 2(b). It is obtained by com- 
+puting the vdW part of the total potential energy between C- 
+atoms in different layers Fig. 2(c). Since these local domains 
+have indistinguishable and strong in-plane covalent bonds, 
+their total potential energy is predominantly sourced from the 
+in-plane interactions, which show little variance risen from 
+their interlayer configuration. Moreover, with applied strain, 
+the changes in total potential energy due to stretching and 
+compressing of the in-plane bonds are orders of magnitude 
+higher than their interlayer vdw counterparts. This motivates 
+the use of vdW interaction energy and its variations for iden- 
+tification purposes. However, being a per-atom quantity there 
+ 
+are a lot of fluctuations in ILE magnitudes, most prominently 
+observed in AB regions (2(c)). Moreover, if the average ILE 
+magnitude is used with respect to their bonded neighbors, it 
+will result in an insignificant difference between AB and SP 
+sub-domains. Hence to account for this, we calculated the av- 
+erage ILE difference (∆EILE ) of each atom with its bonded 
+neighbors. Although it can be difficult to separate AA and 
+SP regions since they have minimal fluctuations in ILE, this 
+parameter easily allows to classify AB stacked atoms as they 
+have the highest variations in energy with neighbors. Based 
+on the ∆EILE analysis for five representative TBGs (Fig. 2e), 
+we have identified the ∆EILE threshold at the soliton bound- 
+ary (SP width) and classified atoms above that threshold as 
+AB. The infinitesimal difference in these thresholds allowed 
+us to define a θ -independent ∆EILE value for identifying the 
+two stackings (see Supplementary for details). It is important 
+to note that the same approach can be used for classification 
+in the presence of strain because the physical parameters used 
+do not depend on strain. Although the magnitude of inter- 
+layer energy can be expected to vary, we observed a negligible 
+change in ∆EILE threshold with strain (see Supplementary). 
+Thus using these criterion based on ILS and ILE, we could 
+classify atoms into their local stacking as shown in Fig. 2(f), 
+which applies to TBGs with any twist angle and strain (Fig. 
+3(a)). 
+ 
+ 
+On implementing the classification method, we obtained 
+area fractions (AF) of each sub-domains present in a TBG 
+structure. Using this measure to monitor the evolution of lo- 
+cal domains in the presence of strain, we observed that the 
+sub-domains’ AF remain almost unchanged (Fig. 3(b), Fig. 
+S4 for tension and Fig S5 for compression). It demonstrates 
+a characteristic tendency of these local regions to retain their 
+registry with an external strain applied globally. The varia- 
+tion of AF as a function of twist angle (Fig. 3(c)) shows that 
+area fraction of AB (AFAB) and SP (AFSP) increases whereas 
+that of AA (AFAA) decreases with decreasing θ . This can 
+be attributed to the potential energy of soliton (SP) regions 
+contributing to in-plane forces, that displace atoms to max- 
+imize the area of AB/BA (most stable BLG-stacking) local 
+domains30. Such observations are well-interpreted in exper- 
+iments, particularly for systems close to θm (1.08°). Hence 
+we compared our theoretically estimated AF for θ = 1.08° 
+(and additional θ = 1.21°, 1.37°) systems with experimen- 
+tally interpreted area fractions from graphical analysis of STM 
+images19, as marked in Fig. 3(c). The close similitude be- 
+tween these sets of area fraction values provides a valida- 
+tion of our stacking classification method. We believe our 
+approach interprets the physical behavior of sub-domains at 
+atomic-level and with high accuracy. Besides, as our method 
+is based on physical parameters such as energy, it directly en- 
+capsulates the underlying physics while in contrast, the previ- 
+ously reported data rely on a graphical interpretation of gradi- 
+ent in image intensity and contrast from experiments. Hence, 
+our methodology is more accurate and able to resolve atom- 
+istic insights even at a higher twist angle where the moiré cell 
+size shrinks drastically. 
+
+An atomistic insight to moiré reconstruction in Twisted Bilayer Graphene beyond magic angle 
+5 
+ 
+FIG. 3: Evolution of local regions with twist angle and strain. (a) Contour plot demonstrating local stacking type for 
+heterostrained θ = 6° system (1% tension). Area fractions of individual stacking domain with respect to (b) strain (tension) and 
+(c) twist angle. The red markings in (c) are extracted from reported work by Kazmierczak et. al.19 to compare our results with 
+data obtained by analyzing experimental measurements . 
+ 
+ 
+C. Detecting moiré reconstruction in high twist angle TBGs. 
+We further utilized this method to study the extent of atomic 
+reconstruction in TBG systems. Moiré reconstruction can be 
+studied by examining local regions in rigidly twisted (R-TBG) 
+structure and comparing with their relaxed geometry13–16. 
+The rigidly twisted TBG refers to its unrelaxed geometry, con- 
+sidered in a conceptual intermediate configuration, in which 
+the layers of BLG are twisted by a certain angle but the 
+atoms are not allowed to reconfigure to form their true equi- 
+librium structure. During reconstruction, local sites in the 
+structure prefer to diverge from energetically unfavorable AA 
+stacking by atomic displacements. This is achieved by rear- 
+rangement of the atoms to minimize vdW energy and obtain- 
+ing the nearly commensurate Bernal-stacked (AB/BA) BLG 
+structure partitioned by the SP segments after reconstruction. 
+The emergence of soliton (SP) domains is one of the pre- 
+dominant features of reconstruction phenomena in 2D mate- 
+rials. Previous studies have attributed the minor atomic dis- 
+placements of large θ relaxed TBGs to insignificant change 
+in atomic registry of local domains indicating the absence 
+of reconstruction15–17,47. However, examining TBG systems 
+with an atomistic insight and employing our sub-domain iden- 
+ 
+tification method, we show considerable changes in the local 
+registries for larger θ TBGs. We utilized the area fraction 
+measure to capture the structural changes in local domains of 
+relaxed and unrelaxed geometries. The stacking identification 
+assessment of R-TBG is conducted similarly to the relaxed 
+TBG (see Supplementary). For θ = 6° structure (4(a)-(c)), the 
+AA regions shrink upon relaxation and conversely, the AB/BA 
+regions expand to approximate triangular domains. Undoubt- 
+edly, this structural change was expected and prominently ob- 
+served for θ = 1.08° system (4(d)-(f)). But we encountered 
+a similar observation for a large θ structure. Hence, contrary 
+to the general idea that reconstruction diminishes at higher 
+angles, we show clear evidence demonstrating moiré recon- 
+struction in higher θ (>2°) TBG systems. This observation 
+indicates that irrespective of how small the atomic displace- 
+ments are, the change in AF of local domains for higher θ 
+TBGs show pronounced variation in atomic registries upon 
+relaxation. 
+
+0.5
+AA
+SP
+0.4
+AB
+fraction
+0.3
+0.2
+0.1
+0%strain
+0.5%strain
+1%strain
+0.5
+0.4
+Area fraction
+0.3
+0.2
+-→AB
+-CAA
+--SP
+0.1
+AB(Kazmierczaket.al.)
+AA(Kazmierczaket.al.)
+SP(Kazmierczaket.al.)
+0
+0
+2
+4
+6
+8
+10
+12
+Twist Angle (0)AFrigid 
+An atomistic insight to moiré reconstruction in Twisted Bilayer Graphene beyond magic angle 
+6 
+ 
+ 
+FIG. 4: Demonstration of moiré reconstruction. Stacking contour plot for rigid (a) θ = 6°, (d) θ = 1.08° and relaxed (b) θ = 
+6°, (e) θ = 1.08° TBG systems. (c), (f) Comparison of area fractions for each stacking , showing the change in local atomic 
+registries before and after relaxation that signifies the extent of reconstruction. 
+ 
+ 
+D. Analyzing extent of reconstruction in strained and 
+unstrained TBGs. 
+Using this approach, we have also studied the extent of 
+moiré reconstruction in high angle TBGs in the presence 
+of heterostrain. Lattice deformation due to heterostrain in- 
+duces distortion in MPs, which is minimized by sustaining the 
+formed domain-wall-like boundary lines (SP regions) due to 
+superlattice reconstruction15,23,48. Similar to the unstrained 
+case, we have compared the local AF of rigid and relaxed 
+systems under heterostrain (Fig. 5). The rigid system for 
+strained TBGs refers to its unrelaxed structure obtained af- 
+ter employing strain to the relaxed geometry of pristine TBG 
+structure (see Supplementary). We observed that our assess- 
+ment could capture the variations in local atomic registry of 
+strained TBGs (Fig. 5(a)-(c)). The substantial change in AF 
+of AA and AB regions and perpetual of SP domains, signifies 
+the tendency of preserving the SP boundaries with change in 
+local atomic registry of AA and AB domains, thus indicating 
+the presence of atomic reconstruction in large θ strained TBG 
+systems. To assess the extent of change in local registries, we 
+have calculated the percentage change in local AF upon re- 
+laxing the structures, i.e., ∆AF(%) = ( 
+AFrelaxed−AFrigid ) × 100. 
+On examining the variation of ∆AF over unstrained (Fig 5(c)) 
+and strained (tensile Fig. 5(e) and compressive Fig. 5(f)) 
+TBGs spanning a wide range of twist angles, it is observed 
+that ∆AF for all local stackings monotonically decreases with 
+increasing θ . Although this implies that, as expected, the ef- 
+ 
+fect of reconstruction reduces with increasing twist angle, AFs 
+data shows that it can not be disregarded. It is noticed that 
+for both pristine and strained cases, the AB stacked domains 
+show ample variation in rigid and relaxed configurations even 
+for higher angles. This variation rapidly decreases for AA and 
+SP regions, especially at very high twist angles. Nonetheless, 
+this analysis reveals the existence of local atomic reconstruc- 
+tion for both unstrained and strained large θ TBG systems. 
+It has been previously argued that for a large twist angle, 
+the gaining vdW energy cannot compensate for the losing in- 
+tralayer elastic energy15,17,23. This results in no change of 
+vdW stacking energy between rigid and relaxed structures, ul- 
+timately indicating the absence of reconstruction. However, 
+our analysis of ILE over different θ (Fig. S6) clearly shows 
+a small but relatively significant difference between the rigid 
+and relaxed structures of higher θ TBGs. Although we ob- 
+served a quick increase and gradual decrease in energies of re- 
+laxed and R-TBG respectively with increasing θ , the relaxed 
+(or reconstructed) system has the lower energy throughout. 
+Thus, even for large twist angles the reconstructed structure 
+formed as a consequence of local atomic changes is their en- 
+ergetically favorable configuration, which directly establishes 
+the presence of reconstruction. It is not surprising that such 
+minor changes in atomic registries for large twist angles are 
+challenging to capture in experiments given the length scale 
+limitations. But based on our results, reconstruction should 
+not be neglected for higher angles and motivate the study of 
+the implications of reconstruction for large θ TBGs. 
+
+0.5
+RigidTBLG
+RelaxedTBLG
+Area Fraction
+0.4
+0.3
+0.2
+0.1
+AA
+AB
+SP
+Stackings
+RigidTBLG
+0.5
+RelaxedTBLG
+0.4
+0.1
+AA
+AB
+SP
+Stackings− 
+An atomistic insight to moiré reconstruction in Twisted Bilayer Graphene beyond magic angle 
+7 
+ 
+FIG. 5: Moiré reconstruction in hetero-strained TBGs. Stacking contour plot of (a) rigid and (b) relaxed θ = 6° structure in 
+the presence of 0.5% uniaxial tension. (c) Change in local stacking area fractions before and after relaxation for the strained 
+structure. Percentage change in local AF of rigid and relaxed θ = 6° structures (∆AF) with respect to twist angle for (d) pristine 
+(unstrained), (e) 1% strained (uniaxial tension) and (f) -1% strained (uniaxial compression) TBGs. Positive and negative values 
+of ∆AF (%) respectively indicates increase and decrease in respective local AFs 
+ 
+E. Mapping local and global physical property (phonon 
+behavior) to changes in local atomic registry. 
+Further validation on the presence of reconstruction at high 
+angles lies within an interrelation of local stacking domains 
+and global vibrational properties. To accomplish this, we 
+have studied their phonon behavior that can be directly trans- 
+lated to Raman scattering frequencies, which is an efficient 
+experimental technique for examining these systems, espe- 
+cially under strain49–52. We have examined the phonon dis- 
+persion spectra of TBGs and their local domains with ab-inito 
+simulations. Initially, we obtained the phonon spectra of un- 
+strained TBG systems using DFT (See Methods and Supple- 
+mentary). As compared to phonon spectrum of BLG, the 
+difference in phonon modes for TBG is quite small due to 
+weaker interlayer interaction (Fig. S7). Although we noticed 
+some differences in low-frequency acoustic phonons, the ef- 
+fect is substantially feeble for optical modes that correspond 
+to the experimentally observed Raman peaks53,54. Pertaining 
+to our goal of probing Raman spectra of TBGs, we analyzed 
+the high frequency optical (Longitudinal (LO) and Transverse 
+(TO)) branches of its phonon spectra55. We have indepen- 
+dently computed the phonon behavior of each sub-domain for 
+comparing them to the global optical vibrational behavior (see 
+Supplementary) as shown in Fig. 6(a). To analyze the minute 
+difference between phonon frequencies of all the structures, 
+we have plotted the optical phonon frequency difference (∆ω) 
+of each stacking with respect to the whole TBG structure, 
+∆ω = ωTBG ωstacking (Fig. 6(b) shows ∆ω for LO). We ob- 
+served that the phonon frequency magnitude of AA and AB 
+regions are smaller than TBG, whereas larger for SP region. 
+A similar trend is observed while comparing the TO phonon 
+modes (Fig. S8). The optical phonon behavior of AB stacking 
+is the closest to that of TBG which indicates that AB-stacked 
+domains predominantly control the overall phonon behavior 
+in TBGs. This is because unfolded phonon branches of TBG 
+exhibit an infinitesimal difference when compared to that of 
+Bernal stacked BLG49,54. The correlation of AF measure with 
+local and global phonon behavior is discussed in the following 
+sub-sections. 
+ 
+1. Correlating local area fraction measure and phonon 
+behavior using Bond-Order-Length-Strength theory 
+To further establish a connection between the optical 
+phonons modes of TBG and phonon frequencies of its sub- 
+domains with individual stacking AF, we utilized the Bond 
+Order Length Strength (BOLS) theory56. BOLS can correlate 
+Raman peaks and their shifts in terms of constitutive struc- 
+tural parameters such as bond length and bond energy56–58. 
+It explains that the intrinsic association of bonds with their 
+physical properties can describe the extrinsic process of opti- 
+cal electron scattering captured by their phonon spectra. This 
+theory provides an independent method of calculating phonon 
+
+0.5
+Rigid
+0.4
+Relaxed
+0.1
+AE
+AA
+AB
+SP
+100
+100
+100
+&XX
+0%
+&XX
+-0.5%
+50
+50
+50
+AAF (%)
+AAF (%)
+△AF (%)
+0
+0
+-50
+.·AA
+-50
+AA
+-50
+-AA
+.--AB
+.--AB
+.--AB
+.-SP
++SP
+-100
+1
+-1005
+-100
+0
+2
+4
+6
+8
+10
+12
+14
+0
+2
+4
+6
+8
+10
+12
+14
+0
+2
+4
+6
+8
+10
+12
+14
+Twist angle (00)
+Twist angle (00)
+Twist angle (00)dz 
+ε=0 
+ε=  1% 
+2g 
+√ 
+− 
+An atomistic insight to moiré reconstruction in Twisted Bilayer Graphene beyond magic angle 
+8 
+frequencies of TBG based on the AFs of each sub-domain. 
+TABLE I: List of β (eV 1/2Å−1) pre-factor values 
+Therefore, the comparison of the results from BOLS theory 
+ 
+ 
+and ab-initio phonon frequencies of TBG can further validate 
+the accuracy of our sub-domain categorizations. The details of 
+BOLS formulation and the parameters involved are explained 
+in Supplementary section I. To obtain the vibrational prop- 
+erties of various structures using BOLS correlation, we can 
+deduce the phonon frequency shift based on bond length (dz), 
+bond energy (Ez), reduced mass (µ) and atomic coordination 
+number (z) using the following relation: 
+∆ω ∝  z
+  
+Ez 
+(1) 
+analyzed both the βTBG values using a times improvement ba- 
+sis (mi). Using this we compared the weighted βTBG, first by 
+dz 
+µ 
+taking our calculated local reconstructed AF as the weights 
+and second by randomly assigning equal AF (33.33% weight 
+for three regions) to each individual stacking. We calculated 
+∆ω = k
+  z  
+Ez
+ 
+(2) 
+∆ω = ωstructure −ωbulk = k (β ) 
+(3) 
+where, k is the proportionality constant in Eq. 1 (µ is con- 
+stant because we have only carbon-based systems). ∆ω is the 
+difference of the optical phonon frequency of a system and a 
+reference material considered in bulk form (see Supplemen- 
+tary). Hence ∆ω = kβ , where β is the pre-factor containing 
+the variable parameters, such that β =  Ez(z/dz). The mag- 
+nitude of this pre-factor directly relates to the optical phonon 
+frequency of a structure ωstructure, and thus can help in calcu- 
+lating its phonon behavior in terms of the associated physical 
+parameters (i.e., z, dz and Ez). Hence, we have utilized this 
+BOLS theory based pre-factor β to study the phonon behav- 
+ior of TBGs and their local domains, including their strained 
+configurations. 
+The calculated β magnitudes for global TBG structure 
+(βTBG) and its sub-domains are listed in Table I and values 
+of all the parameters such as, d, z and E are listed in Table 
+SI. Although the β magnitudes are numerically close, they 
+follow a trend as βSP > βTBG > βAB > βAA, on careful in- 
+spection. This trend also aligns with the observation made 
+while comparing the optical frequencies of these structures 
+(6(b)). Interestingly, this shows how effectively the BOLS 
+theory could endorse the characteristic trend in their phonon 
+behavior. Furthermore, we employed the local stacking AF 
+values of reconstructed structures in BOLS expression to in- 
+still an alternate estimation of phonon frequencies in an at- 
+tempt to authenticate our classification method, as explained 
+hereon. We analyzed the phonon behavior of global TBG 
+their error % with actual βTBG and obtained the relative er- 
+ror comparison or times improvement with respect to actual 
+βTBG values. The mi values in Table II show significant times 
+improvement on considering our estimated AF values of re- 
+constructed structures. The similitude between global βTBG 
+and weighted βTBG using local AFs signify that the physical 
+attributes of local regions in a TBG structure directly correlate 
+with the global vibrational comportment. Besides, this analy- 
+sis shows another evidence that our stacking classification is 
+an effective method for wide-ranging θ and strain magnitudes, 
+which is shown to detect reconstruction in these structures. 
+ 
+2. Comparison of BOLS-estimated phonon frequencies with 
+experimental Raman to validate sub-domain area fraction 
+measure 
+To authenticate our reconstructed AF measures with DFT- 
+based phonon calculations and AFs driven BOLS theory, we 
+first calculated the phonon spectra of strained TBGs using 
+DFT simulations followed by calculating Raman frequencies 
+using BOLS (see Supplementary). Figure 6(c) shows the op- 
+tical phonon branches of TBG (θ = 6°) including tensile and 
+compressive uniaxial heterostrain. We have considered Ra- 
+man G band frequency in this study, which can be obtained 
+at Γ point in high symmetry Brillouin Zone path55,59. We ob- 
+served strain-induced phonon band splitting due to inequiv- 
+alent strain present in both the layers59–62 (supplementary 
+section IV). This phenomenon is observed in Raman spec- 
+troscopy as represented by the schematic of G-band Raman 
+peaks in hetero-strained TBGs (Fig. 6(d)). Due to weak inter- 
+layer vdW interaction in TBGs, their interlayer shear strength 
+is negligible which results in slippage between the layers. 
+Hence, the bottom layer remains mostly unstrained when 
+structure based on two approaches, the first being βTBG cal- 
+straining the top layer62,63. The Raman spectra of heteros- 
+trained TBG show significant individual peaks of unstrained 
+culated directly from BOLS expression. For the other ap- 
+proach, we have taken a weighted average of β values of in- 
+bottom layer (p1 
+) and strained top layer (p2′ 
+± ). The 
+dividual stacking with their reconstructed AF as the weights, 
+i.e., βTBG = AFAAβAA + AFABβAB + AFSPβSP. On comparing 
+the actual and weighted βTBG, i.e., eactual = (βTBG(weighted) 
+βTBG(actual))/βTBG(actual), we observed that they align very 
+peak of strained layer redshifts or blueshifts depending on the 
+nature of strain. Also, for the case of graphene, an increase in 
+the magnitude of strain further splits the G-band peaks cor- 
+responding to the doubly degenerate E+ and E2
+−
+g phonons 
+well with a small error %, including for strained systems (Ta- 
+ble II). However, given the seemingly small difference in β 
+2′′ 
+ε=±1% in Fig 6(d)-(f))8,64. 
+AF values of reconstructed systems 
+values of the structures, it may be argued that these small er- 
+rors are not much intriguing. Therefore, we have additionally 
+We then used the local 
+in BOLS expression to estimate Raman G-band frequencies 
+for comparison with experiments. and establish a connec- 
+(p 
+Stacking 
+θ = 1.08° 
+θ = 6° 
+θ = 13.2° 
+AA 
+3.084 
+3.198 
+3.418 
+AB 
+3.126 
+3.294 
+3.450 
+SP 
+3.180 
+3.376 
+3.491 
+TBG(βBOLS) 
+3.135 
+3.306 
+3.474 
+TBG(βweighted ) 
+3.141 
+3.292 
+3.466 
+ 
+
+An atomistic insight to moiré reconstruction in Twisted Bilayer Graphene beyond magic angle 
+9 
+ 
+TABLE II: Error table for BOLS-estimated β pre-factors based on actual and weighted βTBG, for systems with and without 
+strain. 
+ 
+ 
+Strain (%) 
+ 
+eactual 
+θ = 1.08° 
+ 
+mi 
+ 
+eactual 
+θ = 6° 
+ 
+mi 
+ 
+eactual 
+θ = 13.2° 
+ 
+mi 
+0 
+0.38  
+6 
+0.27 
+ 
+5 
+0.22  
+5 
+0.2% 
+- 
+ 
+- 
+0.42 
+ 
+5 
+0.29  
+4 
+0.5% 
+- 
+ 
+- 
+0.60 
+ 
+5 
+0.35  
+4 
+0.7% 
+- 
+ 
+- 
+0.51 
+ 
+5 
+0.49  
+4 
+1% 
+- 
+ 
+- 
+0.69  
+5 
+0.45  
+3 
+ 
+FIG. 6: Phonon behavior of TBGs with respect to its local domains. (a) Optical phonon modes of TBLG (θ = 6°) and its 
+individual counterparts. (b) Longitudinal optical (LO) phonon frequency difference with respect to TBG system, (c) Phonon 
+band splitting with heterostrain (tension and compression) (d) Schematic of a typical Raman G-peak splitting with inequivalent 
+strain employed in a bilayer system. Comparison of G-band frequencies for (e) θ = 6° with uniaxial compression and (f) θ = 
+13.2° with uniaxial tension. Solid lines in (e), (f) denote the Raman G-peak data obtained from DFT-based phonon calculations. 
+Heterostrain-assisted peak splitting of top and bottom layer (as shown in the schematic) is also denoted. Sub-figures(e)-(f) also 
+shows the close alignment of Bond Order Length Strength (BOLS)-estimated data using reconstructed AFs with 
+DFT-calculated and experimental data (reported by Peña et. al.65) as compared to that using rigid TBG AFs. 
+ 
+ 
+tion between global and local vibrational behavior. We first 
+extracted the G-band frequency (ωG) from DFT-simulated 
+phonon spectra for both unstrained and strained structures. 
+Figure 6(e) and 6(f) respectively shows the variation of ωG 
+for 6° and 13.2° with strain. To demonstrate both directions 
+of uniaxial strain, we showed the case of compression for 6° 
+and tension for 13.2°. In both cases, we observed that ωG at 
+zero strain is 1588 cm−1, which changes negligibly for the 
+unstrained bottom layer. In Fig. 6(e) due to compression, we 
+observed blueshift in ωG and redshift for tensile strain in Fig. 
+6(f) (see Supplementary section V). On comparing our results 
+for 6° and 13.2° systems with the experimental data reported 
+by Pena et. al.65 and Gao et. al.8 respectively, we found a 
+good agreement between them (magenta data points in Fig. 
+ 
+6(e) and (f)). Finally, to achieve an experimental validation of 
+our stacking identification method as well as to highlight that 
+the global behavior such as Raman scattering is tied to local 
+structural configurations, we used our calculated AFs of re- 
+constructed TBGs in BOLS to predict the Raman G-band fre- 
+quencies of heterostrained systems (see Supplementary sec- 
+tion I for details). 
+We found a qualitative agreement between BOLS estimated 
+and DFT calculated ωG Raman peaks shown in Fig. 6(e) and 
+Fig. 6(f) (green dots). It must be noted that since BOLS ap- 
+proach encompasses mathematical interpolation for project- 
+ing the phonon frequencies, it can not resolve the further band 
+splitting of the strained top layer. We have also used the rigid 
+TBG AFs to check how it compares with the estimated G- 
+
+20
+1650
+LO
+SP stacking
+1650
+(cm
+ABstacking
+(cm
+10
+AAstacking
+Phonon frequency
+TO
+Phonon frequency
+1500
+1500
+AOLO
+1350
+1350
+TBG(0=6°)
+AB stacking
+-10
+Bottom layer (g=0%)
+AAstacking
+Top layer (=+1%)
+SP stacking
+1200
+1200
+- Top layer (c=-1%)
+K
+M
+-20
+K
+M
+K
+M
+BOLSo.(reconstructedAF)
+1600
+Tension
+Compression
+BOLS . (rigidAF)
+Bottomlayerp
+1650
+Experimental o(Pena et. al.)
+Top layer p
+1575
+Top layer p
+Intensity
+(cm
+Top layer p
+1550
+p=0
+1600
+(Bottomlayer)
+BOLSo.(reconstructedAF)
+=-1%
+Bottomlayerp
+Experimental o. (Gao et. al.)
+=1%
+8-1%
+p--1%p
+1525
+(Top layer)(Top layer)
+(Toplayer)(Toplayer)
+1575
+Ramano
+peak frequencies
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.2
+0.4
+0.6
+0.8
+Uniaxial compressive strain (%)
+Uniaxial tensile strain (%)An atomistic insight to moiré reconstruction in Twisted Bilayer Graphene beyond magic angle 
+10 
+ 
+band frequencies. We observed a distinct misalignment of 
+BOLS-estimated Raman data using rigid AFs with that us- 
+ing reconstructed AFs and experimentally obtained data as 
+well. Hence, our analysis clearly demonstrates the difference 
+in vibrational behavior of reconstructed and rigid structures 
+and also shows that the reconstructed systems align closely 
+with the experimentally obtained measurements. This cer- 
+tainly implies that the physical behavior of TBGs such as 
+their vibrational properties is governed by their reconstructed 
+phases even for a large θ system and hence establishes an 
+additional validation on the presence of moiré reconstruc- 
+tion in their structures. Moreover, an agreement between the 
+AF utilized BOLS-estimated Raman data and DFT-calculated 
+phonon shows a theoretical approach to calculate Raman fre- 
+quencies at a comparatively lower computational cost. We 
+have calculated the G-band data for the heterostrained 1.08° 
+system using BOLS formulation (Fig. S9). As a whole uti- 
+lizing our stacking classification method and analyzing their 
+Raman signature using BOLS, we established a precise au- 
+thentication about reconstruction in high twist angles and also 
+demonstrated a connection of the global phonon shift of a 
+TBG system with changes in its local atomic registries. 
+ 
+IV. Conclusion 
+Using atomistic simulations, we studied the characteristics 
+of locally stacked domains in TBG moiré patterns and demon- 
+strated a comprehensive approach to study atomic reconstruc- 
+tion phenomena in these structures, including the presence of 
+heterostrain. We proposed a way to classify TBGs into their 
+stacking types (AA, AB, and SP) and calculated their area 
+fractions to track structural evolution as a function of θ and 
+strain. Our classification scheme allowed us to exhibit the 
+existence of moiré reconstruction even for larger twist angle 
+(>2°) TBG systems, which is difficult to detect experimen- 
+tally. We showed how the moiré patterns of these large-angle 
+TBGs can be distorted by applying strain. Besides, the atomic 
+reconstruction in the presence of strain (in terms of area frac- 
+tion change of commensurate domain) can be manipulated by 
+an amount between 55% to 73% (for θ = 6°) with an applied 
+strain of only 0.5%, opening up a massive opportunity for 
+large angle TBGs to be used in strain engineering applica- 
+tions. 
+We studied the extent of reconstruction over a wide range 
+of θ and realized how it evolves in the presence of strain. 
+To further analyze this finding and validate the AF measure, 
+we utilized DFT-based phonon calculations and a theoretical 
+approach (BOLS theory) to deduce Raman frequencies and 
+compare them with experimental data. Using BOLS theory, 
+we discovered that global phonon behavior is directly related 
+to the physical features of local regions. Further, we real- 
+ized that the Raman data using reconstructed AFs in BOLS 
+aligns closely with DFT-calculated as well as experimental 
+data. Moreover, on comparing the Raman data with rigid AFs, 
+our results show a clear difference with that using the recon- 
+structed sub-domains and hence imply that the latter governs 
+the physical behavior in TBGs even for higher angles. Hence, 
+our study shows a self-consistent approach to characterize lo- 
+cal regions in TBGs and utilize them to examine as well as 
+ 
+validate moiré reconstruction phenomena, based on physical 
+measures. Our findings on the presence of reconstruction in 
+large θ TBGs might open up an interesting research outlook 
+in twistronics. Moreover, our methodologies can be utilized to 
+identify stacking types and perform similar analyses in other 
+twisted vdW systems, especially in the presence of strain. 
+ 
+Acknowledgments 
+We wish to acknowledge the support from the National 
+Science Foundation (OMA-1936250) and National Science 
+Foundation Graduate Research Fellowship Program (DGE- 
+1939268). 
+ 
+Data Availability Statement 
+The data that support the findings of this study are available 
+from the corresponding author upon reasonable request. 
+ 
+References 
+ 
+1X.-J. Zhao, H. Hou, X.-T. Fan, Y. Wang, Y.-M. Liu, C. Tang, S.-H. Liu, P.- 
+P. Ding, J. Cheng, D.-H. Lin, et al., “Molecular bilayer graphene,” Nature 
+communications 10, 1–7 (2019). 
+2K. Lee, B. Fallahazad, J. Xue, D. C. Dillen, K. Kim, T. Taniguchi, 
+K. Watanabe, and E. Tutuc, “Chemical potential and quantum hall fer- 
+romagnetism in bilayer graphene,” Science 345, 58–61 (2014). 
+3T. Ohta, A. Bostwick, T. Seyller, K. Horn, and E. Rotenberg, “Controlling 
+the electronic structure of bilayer graphene,” Science 313, 951–954 (2006). 
+4E. McCann and M. Koshino, “The electronic properties of bilayer 
+graphene,” Reports on Progress in physics 76, 056503 (2013). 
+5A. Luican, G. Li, A. Reina, J. Kong, R. Nair, K. S. Novoselov, A. K. 
+Geim, and E. Andrei, “Single-layer behavior and its breakdown in twisted 
+graphene layers,” Physical review letters 106, 126802 (2011). 
+6A. Luican, G. Li, A. Reina, J. Kong, R. R. Nair, K. S. Novoselov, A. K. 
+Geim, and E. Y. Andrei, “Single-layer behavior and its breakdown in 
+twisted graphene layers,” Phys. Rev. Lett. 106, 126802 (2011). 
+7Y. Yu, K. Zhang, H. Parks, M. Babar, S. Carr, I. M. Craig, M. Van Winkle, 
+A. Lyssenko, T. Taniguchi, K. Watanabe, et al., “Tunable angle-dependent 
+electrochemistry at twisted bilayer graphene with moiré flat bands,” Nature 
+Chemistry 14, 267–273 (2022). 
+8X. Gao, H. Sun, D.-H. Kang, C. Wang, Q. J. Wang,  and D. Nam, 
+“Heterostrain-enabled dynamically tunable moiré superlattice in twisted bi- 
+layer graphene,” Scientific reports 11, 1–8 (2021). 
+9L. Huder, A. Artaud, T. Le Quang, G. T. De Laissardiere, A. G. Jansen, 
+G. Lapertot, C. Chapelier, and V. T. Renard, “Electronic spectrum of 
+twisted graphene layers under heterostrain,” Physical review letters 120, 
+156405 (2018). 
+10J.-B. Qiao, L.-J. Yin, and L. He, “Twisted graphene bilayer around the first 
+magic angle engineered by heterostrain,” Physical Review B 98, 235402 
+(2018). 
+11F. Gargiulo and O. V. Yazyev, “Structural and electronic transformation in 
+low-angle twisted bilayer graphene,” 2D Materials 5, 015019 (2017). 
+12N. N. Nam and M. Koshino, “Lattice relaxation and energy band modula- 
+tion in twisted bilayer graphene,” Physical Review B 96, 075311 (2017). 
+13K. Zhang and E. B. Tadmor, “Structural and electron diffraction scaling of 
+twisted graphene bilayers,” Journal of the Mechanics and Physics of Solids 
+112, 225–238 (2018). 
+14Y.-W. Liu, Y. Su, X.-F. Zhou, L.-J. Yin, C. Yan, S.-Y. Li, W. Yan, S. Han, 
+Z.-Q. Fu, Y. Zhang, Q. Yang, Y.-N. Ren, and L. He, “Tunable lattice recon- 
+struction, triangular network of chiral one-dimensional states, and band- 
+width of flat bands in magic angle twisted bilayer graphene,” Phys. Rev. 
+Lett. 125, 236102 (2020). 
+15H. Yoo, R. Engelke, S. Carr, S. Fang, K. Zhang, P. Cazeaux, S. H. Sung, 
+R. Hovden, A. W. Tsen, T. Taniguchi, et al., “Atomic and electronic recon- 
+struction at the van der waals interface in twisted bilayer graphene,” Nature 
+materials 18, 448–453 (2019). 
+
+An atomistic insight to moiré reconstruction in Twisted Bilayer Graphene beyond magic angle 
+11 
+ 
+16Y. Choi, J. Kemmer, Y. Peng, A. Thomson, H. Arora, R. Polski, Y. Zhang, 
+H. Ren, J. Alicea, G. Refael, et al., “Electronic correlations in twisted 
+bilayer graphene near the magic angle,” Nature Physics 15, 1174–1180 
+(2019). 
+17T. A. de Jong, T. Benschop, X. Chen, E. E. Krasovskii, M. J. de Dood, R. M. 
+Tromp, M. P. Allan, and S. J. Van der Molen, “Imaging moiré deformation 
+and dynamics in twisted bilayer graphene,” Nature Communications 13, 1– 
+8 (2022). 
+18A. C. Gadelha, D. A. Ohlberg, C. Rabelo, E. G. Neto, T. L. Vasconcelos, 
+J. L. Campos, J. S. Lemos, V. Ornelas, D. Miranda, R. Nadas, et al., “Local- 
+ization of lattice dynamics in low-angle twisted bilayer graphene,” Nature 
+590, 405–409 (2021). 
+19N. P. Kazmierczak, M. Van Winkle, C. Ophus, K. C. Bustillo, S. Carr, H. G. 
+Brown, J. Ciston, T. Taniguchi, K. Watanabe, and D. K. Bediako, “Strain 
+fields in twisted bilayer graphene,” Nature materials 20, 956–963 (2021). 
+20T. C. Barbosa, A. C. Gadelha, D. A. Ohlberg, K. Watanabe, T. Taniguchi, 
+G. Medeiros-Ribeiro, A. Jorio, and L. C. Campos, “Raman spectra of 
+twisted bilayer graphene close to the magic angle,” 2D Materials 9, 025007 
+(2022). 
+21Z.-B. Dai, Y. He, and Z. Li, “Effects of heterostrain and lattice relaxation 
+on the optical conductivity of twisted bilayer graphene,” Phys. Rev. B 104, 
+045403 (2021). 
+22L. Zhang, Y. Wang, R. Hu, P. Wan, O. Zheliuk, M. Liang, X. Peng, Y.-J. 
+Zeng, and J. Ye, “Correlated states in strained twisted bilayer graphenes 
+away from the magic angle,” Nano letters 22, 3204–3211 (2022). 
+23A. Kerelsky, L. J. McGilly, D. M. Kennes, L. Xian, M. Yankowitz, S. Chen, 
+K. Watanabe, T. Taniguchi, J. Hone, C. Dean, et al., “Maximized electron 
+interactions at the magic angle in twisted bilayer graphene,” Nature 572, 
+95–100 (2019). 
+24J. Campos-Delgado, G. Algara-Siller, C. Santos, U. Kaiser,  and J.-P. 
+Raskin, “Twisted bi-layer graphene: Microscopic rainbows,” Small 9, 
+3247–3251 (2013). 
+25Y. Wang, Z. Su, W. Wu, S. Nie, N. Xie, H. Gong, Y. Guo, J. Hwan Lee, 
+S. Xing, X. Lu, et al., “Resonance raman spectroscopy of g-line and folded 
+phonons in twisted bilayer graphene with large rotation angles,” Applied 
+Physics Letters 103, 123101 (2013). 
+26A. Jorio and L. G. Cançado, “Raman spectroscopy of twisted bilayer 
+graphene,” Solid State Communications 175, 3–12 (2013). 
+27J. Campos-Delgado, L. G. Cançado, C. A. Achete, A. Jorio, and J.-P. 
+Raskin, “Raman scattering study of the phonon dispersion in twisted bi- 
+layer graphene,” Nano Research 6, 269–274 (2013). 
+28R. He, T.-F. Chung, C. Delaney, C. Keiser, L. A. Jauregui, P. M. Shand, 
+C. Chancey, Y. Wang, J. Bao, and Y. P. Chen, “Observation of low en- 
+ergy raman modes in twisted bilayer graphene,” Nano letters 13, 3594–3601 
+(2013). 
+29M. Koshino and Y.-W. Son, “Moiré phonons in twisted bilayer graphene,” 
+Physical Review B 100, 075416 (2019). 
+30S. Dai, Y. Xiang, and D. J. Srolovitz, “Twisted bilayer graphene: Moiré 
+with a twist,” Nano letters 16, 5923–5927 (2016). 
+31M. Koshino, “Electronic transmission through a b-b a domain boundary in 
+bilayer graphene,” Physical Review B 88, 115409 (2013). 
+32M. Van Wijk, A. Schuring, M. Katsnelson, and A. Fasolino, “Relaxation 
+of moiré patterns for slightly misaligned identical lattices: graphene on 
+graphite,” 2D Materials 2, 034010 (2015). 
+33V. Carozo, C. M. Almeida, E. H. Ferreira, L. G. Cancado, C. A. Achete, 
+and A. Jorio, “Raman signature of graphene superlattices,” Nano letters 11, 
+4527–4534 (2011). 
+34W. Hou, S. A. Chowdhury, A. Dey, C. Watson, T. Peña, A. Azizimanesh, 
+H. Askari, and S. M. Wu, “Nonvolatile ferroelastic strain from flexoelectric 
+internal bias engineering,” Physical Review Applied 17, 024013 (2022). 
+35P. Kumar, A. Dey, J. Roques, L. Assaud, S. Franger, P. Parida, and V. Biju, 
+“Photoexfoliation synthesis of 2d materials,” ACS Materials Letters 4, 263– 
+270 (2022). 
+36V. Kumar, A. Dey, S. Thomas, M. A. Zaeem, and D. R. Roy, “Hydrogen- 
+induced tunable electronic and optical properties of a two-dimensional 
+penta-pt 2 n 4 monolayer,” Physical Chemistry Chemical Physics 23, 
+10409–10417 (2021). 
+37A. Dey, R. Sharma, and S. A. Dar, “An extensive investigation of structural, 
+electronic, thermoelectric and optical properties of bi-based half-huesler 
+alloys by first principles calculations,” Materials Today Communications 
+ 
+25, 101647 (2020). 
+38A. Dey, B. A. Baraiya, S. Adhikary, and P. K. Jha, “First-principles calcu- 
+lations of the effects of edge functionalization and size on the band gap of 
+be3n2 nanoribbons: Implications for nanoelectronic devices,” ACS Applied 
+Nano Materials 4, 493–502 (2020). 
+39A. P. Thompson, H. M. Aktulga, R. Berger, D. S. Bolintineanu, W. M. 
+Brown, P. S. Crozier, P. J. in’t Veld, A. Kohlmeyer, S. G. Moore, T. D. 
+Nguyen, et al., “Lammps-a flexible simulation tool for particle-based ma- 
+terials modeling at the atomic, meso, and continuum scales,” Computer 
+Physics Communications 271, 108171 (2022). 
+40S. A. Chowdhury, K. Inzani, T. Peña, A. Dey, S. M. Wu, S. M. Griffin, and 
+H. Askari, “Mechanical properties and strain transfer behavior of molybde- 
+num ditelluride (mote2) thin films,” Journal of Engineering Materials and 
+Technology 144 (2022). 
+41L. J. McGilly, A. Kerelsky, N. R. Finney, K. Shapovalov, E.-M. Shih, 
+A. Ghiotto, Y. Zeng, S. L. Moore, W. Wu, Y. Bai, et al., “Visualization 
+of moiré superlattices,” Nature Nanotechnology 15, 580–584 (2020). 
+42J. Lin, W. Fang, W. Zhou, A. R. Lupini, J. C. Idrobo, J. Kong, S. J. 
+Pennycook, and S. T. Pantelides, “Ac/ab stacking boundaries in bilayer 
+graphene,” Nano letters 13, 3262–3268 (2013). 
+43X. Lin, D. Liu, and D. Tománek, “Shear instability in twisted bilayer 
+graphene,” Physical Review B 98, 195432 (2018). 
+44L. Gong, R. J. Young, I. A. Kinloch, S. J. Haigh, J. H. Warner, J. A. Hinks, 
+Z. Xu, L. Li, F. Ding, I. Riaz, et al., “Reversible loss of bernal stacking dur- 
+ing the deformation of few-layer graphene in nanocomposites,” Acs Nano 
+7, 7287–7294 (2013). 
+45R. Miwa, P. Venezuela, and E. S. Morell, “Periodic arrays of intercalated 
+atoms in twisted bilayer graphene: An ab initio investigation,” Physical 
+Review B 92, 115419 (2015). 
+46Y. Hou, S. Zhang, Q. Li, L. Liu, X. Wu, and Z. Zhang, “Evaluation local 
+strain of twisted bilayer graphene via moiré pattern,” Optics and Lasers in 
+Engineering 152, 106946 (2022). 
+47M. Koshino and N. N. Nam, “Effective continuum model for relaxed 
+twisted bilayer graphene and moiré electron-phonon interaction,” Physical 
+Review B 101, 195425 (2020). 
+48L. Zhang, Y. Wang, R. Hu, P. Wan, O. Zheliuk, M. Liang, X. Peng, Y.-J. 
+Zeng, and J. Ye, “Correlated states in strained twisted bilayer graphenes 
+away from the magic angle,” Nano Letters 22, 3204–3211 (2022), pMID: 
+35385281, https://doi.org/10.1021/acs.nanolett.1c04400. 
+49F. Wu, A. H. MacDonald, and I. Martin, “Theory of phonon-mediated su- 
+perconductivity in twisted bilayer graphene,” Physical review letters 121, 
+257001 (2018). 
+50R. W. Havener, H. Zhuang, L. Brown, R. G. Hennig, and J. Park, “Angle- 
+resolved raman imaging of interlayer rotations and interactions in twisted 
+bilayer graphene,” Nano letters 12, 3162–3167 (2012). 
+51A. C. Gadelha, D. A. Ohlberg, F. C. Santana, G. S. Eliel, J. S. Lemos, 
+V. Ornelas, D. Miranda, R. B. Nadas, K. Watanabe, T. Taniguchi, et al., 
+“Twisted bilayer graphene: a versatile fabrication method and the detection 
+of variable nanometric strain caused by twist-angle disorder,” ACS Applied 
+Nano Materials 4, 1858–1866 (2021). 
+52M. Huang, H. Yan, T. F. Heinz, and J. Hone, “Probing strain-induced elec- 
+tronic structure change in graphene by raman spectroscopy,” Nano letters 
+10, 4074–4079 (2010). 
+53A. I. Cocemasov, D. L. Nika, and A. A. Balandin, “Phonons in twisted 
+bilayer graphene,” Physical Review B 88, 035428 (2013). 
+54M. Lamparski, B. Van Troeye, and V. Meunier, “Soliton signature in the 
+phonon spectrum of twisted bilayer graphene,” 2D Materials 7, 025050 
+(2020). 
+55H. Wang, Y. Wang, X. Cao, M. Feng, and G. Lan, “Vibrational properties of 
+graphene and graphene layers,” Journal of Raman Spectroscopy: An Inter- 
+national Journal for Original Work in all Aspects of Raman Spectroscopy, 
+Including Higher Order Processes, and also Brillouin and Rayleigh Scatter- 
+ing 40, 1791–1796 (2009). 
+56X. Yang, Y. Wang, J. Li, W. Liao, Y. Liu, and C. Q. Sun, “Graphene phonon 
+softening and splitting by directional straining,” Applied Physics Letters 
+107, 203105 (2015). 
+57C. Q. Sun, “Size dependence of nanostructures: Impact of bond order defi- 
+ciency,” Progress in solid state chemistry 35, 1–159 (2007). 
+58C. Q. Sun, “Relaxation of the chemical bond,” Springer Ser. Chem. Phys 
+108, 807 (2014). 
+
+An atomistic insight to moiré reconstruction in Twisted Bilayer Graphene beyond magic angle 
+12 
+ 
+59D. Yoon, Y.-W. Son, and H. Cheong, “Strain-dependent splitting of the 
+double-resonance raman scattering band in graphene,” Physical review let- 
+ters 106, 155502 (2011). 
+60O. Frank, M. Bousa, I. Riaz, R. Jalil, K. S. Novoselov, G. Tsoukleri, 
+J. Parthenios, L. Kavan, K. Papagelis, and C. Galiotis, “Phonon and struc- 
+tural changes in deformed bernal stacked bilayer graphene,” Nano Letters 
+12, 687–693 (2012). 
+61T. Sohier, M. Gibertini, M. Calandra, F. Mauri, and N. Marzari, “Break- 
+down of optical phonons’ splitting in two-dimensional materials,” Nano let- 
+ters 17, 3758–3763 (2017). 
+62C. Androulidakis, E. N. Koukaras, G. Paterakis, G. Trakakis, and C. Galio- 
+tis, “Tunable macroscale structural superlubricity in two-layer graphene via 
+strain engineering,” Nature communications 11, 1–11 (2020). 
+63K. Wang, C. Qu, J. Wang, W. Ouyang, M. Ma, and Q. Zheng, “Strain engi- 
+neering modulates graphene interlayer friction by moiré pattern evolution,” 
+ 
+ACS applied materials & interfaces 11, 36169–36176 (2019). 
+64Y. Lu, L. Yan, S. Hussain, M. Sun, Z. Zhang, and H. Zheng, “Interlayer 
+coulomb interaction in twisted bilayer graphene nanofragments character- 
+ized by the vibrational mode of gr+ band,” Applied Physics Letters 120, 
+083103 (2022). 
+65T. Peña, A. Dey, S. A. Chowdhury, A. Azizimanesh, W. Hou, A. Sewaket, 
+C. L. Watson, H. Askari, and S. M. Wu, “Moiré engineering in 2d het- 
+erostructures with process-induced strain,” arXiv (2022). 
+66Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, 
+and P. Jarillo-Herrero, “Unconventional superconductivity in magic-angle 
+graphene superlattices,” Nature 556, 43–50 (2018). 
+67A. Nimbalkar and H. Kim, “Opportunities and challenges in twisted bilayer 
+graphene: a review,” Nano-Micro Letters 12, 1–20 (2020). 
+
+1 
+ 
+An atomistic insight to moir´e reconstruction in Twisted Bilayer Graphene beyond 
+magic angle 
+Aditya Dey,1, a) Shoieb Ahmed Chowdhury,1, a) Tara Pen˜a,2 Sobhit Singh,1 Stephen M. 
+Wu,2, b) and Hesam Askari1 
+1)Department of Mechanical Engineering, University of Rochester, 
+New York 
+2)Department of Electrical and Computer Engineering, University of Rochester, 
+Rochester, New York 
+ 
+Supplementary information 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+a)These authors contributed equally to this work 
+b)Department of Physics and Astronomy, University of Rochester, Rochester, New York 
+
+2 
+ 
+I. 
+COMPUTATIONAL AND THEORETICAL METHODS 
+ 
+A. 
+DFT calculations 
+ 
+The real space lattices of TBG systems were constructed using ATOMISTIX TOOLKIT 
+(QuantumATK) commercial package. All the first principles simulations were conducted 
+with generalized gradient approximation (GGA)1,2 assimilated in Quantum Espresso open 
+source package. The Perdew-Burke-Ernzerhof (PBE) form along with GGA has been used 
+as the exchange-correlation functional3. Ion-electron interactions for carbon atoms in TBGs 
+have been described by ultrasoft pseudopotentials. The vdW interaction has been incor- 
+porated as well using the semi-empirical Grimme functional4. Wavefunctions are expanded 
+using a plane wave basis set with an energy cutoff and charge density of 55 Ry and 450 Ry 
+respectively. We used 14 × 14 × 1 k-point grid within Monkhorst-Pack5,6 scheme to sample 
+the reciprocal space Brillouin zone. The structures were optimized until all the atomic forces 
+were less than 0.01 eV/˚A. The in-plane lattice constants were relaxed including the non- 
+periodic out-of plane lattice (25 ˚A space) to elude interactions in that direction. Phonon 
+dispersion spectra of all TBG structures were simulated using self-consistent density func- 
+tional perturbation theory (DFPT)7,8. The dynamical matrices were first computed on an 
+adequate q-point grid. The inter-atomic constants used in computing the phonon dispersion 
+were obtained from the Fourier interpolation of these dynamical matrices. 
+ 
+ 
+B. MS simulations 
+ 
+Molecular statics simulations were done using LAMMPS open source software. The 
+unstrained, DFT-relaxed TBG moir´e lattice was transformed into an orthogonal cell with 
+approximate dimensions of 32 nm × 20 nm for all the TBG structures. The number of MPs 
+generated in each structure is dependent on the twist angle, for example θ = 6◦has 72 MPs, 
+and θ = 13.2◦has 288 MPs respectively. A vacuum space of 50 ˚A is inserted along the out- 
+plane-direction to avoid interactions with the periodic images. Hydrogen passivation was 
+done along the free surfaces to obtain the most stable structure. The TBG structures were 
+minimized using a conjugate gradient energy minimization method to have minimum energy 
+configurations. A reactive empirical bond order (REBO) potential was used for the intra- 
+lyer covalent bonds9 and for the interlayer van der Waals interaction a registry-dependent 
+
+3 
+ 
+8z 
+C 
+= 
+Kolmogorov-Crespi (KC) potential10 was selected. As TBG contains different local stacking 
+configurations, an interatomic potential that considers registry different than equilibrium 
+minimum energy stacking is needed11,12. Subsequently, we loaded the structure with con- 
+stant incremental strain to the top layer. We limit the magnitude of applied strain to 1% 
+for impeding our analysis within the contended boundaries of the experimental capability of 
+straining such systems13,14. Between each loading step, the atoms of the top layer were kept 
+stationary at the applied strain level and energy minimization was performed. The snap- 
+shots of the structure at different strain magnitudes were taken in Ovito open visualization 
+tool15. 
+ 
+C. 
+BOLS formulation 
+ 
+The BOLS notion explains the bond contraction and bond strengthening phenomena 
+using the following expressions16: 
+dz 
+= Cz 
+b 
+2 
+= 1 + exp[ 12−z ] 
+ 
+(1) 
+ 
+ 
+Eb 
+z 
+m 
+(2) 
+z 
+Here, the subscripts z and b respectively represent the coordination number (CN) of 
+a particular atomic structure and its bulk counterpart as a standard. The terms d and 
+E denote bond length and bond energy respectively. Cz represents the bond contraction 
+coefficient that varies with atomic structures having different z. The bond nature index 
+is denoted by m which is 2.56 for carbon bonds17. Since we are dealing with graphitic 
+structures in this study, we consider the bulk counterpart as diamond. Using the bond 
+length of the diamond (db = 1.54˚A) and bond lengths dz for each stacking configuration, we 
+can calculate Cz and z for each configuration using equations (1) and (2). Again using the 
+relation given in equation (2), we can calculate the bond energy for each individual stacking. 
+For diamond, the single C-C bond energy can be obtained from its total cohesive energy, 
+which is known to us, i.e., Eb = 0.614 eV17. Having known z, dz and Ez, we calculate the 
+β pre-factor values for each stacking using equation (2). The relation stated in equation 1 
+in the main text can be derived by equating the vibrational energy of a harmonic system to 
+the first-order approximated Taylor series of its interatomic potential as16: 
+d 
+E 
+
+4 
+ 
+TBG 
+G 
+ref 
+ref 
+G 
+ref 
+TBG 
+− ω 
+β ϵ 
+TBG 
+i 
+| | 
+i 
+dz 
+µ 
+= 
+i 
+1µ(∆ω)2x2 ∼= 1 δu(r) x2 ∝ 1 Ez x2 
+(3) 
+2 
+2 δr2 
+2 dz2 
+⇒ ∆ω ∝ z /
+Ez
+ 
+ 
+The BOLS correlation is also used to estimate the phonon frequencies pertaining to 
+Raman G-band peaks. To achieve this, we perform some steps of mathematical interpolation 
+for equation (3). We can write the equation as ∆ωG = ωG 
+− ωref = k (β), where ωG is 
+the G band frequency of any reference material. Now we can calculate ωG 
+for each TBG 
+system with respect to their bulk counterpart (diamond) by comparing respective β pre- 
+factors as, 
+G 
+TBG − ωref 
+= βTBG . After obtaining ωG , we can exercise ωG,ϵ=0 (ωG 
+ωdiamond − ωG 
+βdiamond 
+ref 
+TBG 
+TBG 
+at zero strain) and β pre-factors of strained and unstrained TBG systems to estimate their 
+ωG,ϵ=0 − ωG 
+βϵ=0 
+G-band frequency in strained configuration (ωG,ϵ ), as 
+TBG 
+G,ϵ 
+TBG 
+ref 
+G 
+ref 
+=  T BG . Operating 
+TBG 
+this individually for top and bottom layers, we can obtain their G-peak frequencies for both 
+directions and various magnitudes of applied strain. The βϵ 
+values for the strained top 
+layer are listed in Table SIV. Since the bottom layer remains unstrained, we observe negligible 
+differences between their β pre-factor values for strained and unstrained configurations. 
+ 
+II. 
+GEOMETRIC ANALYSIS OF STRAINED MSCS 
+ 
+We deduce the expressions of their reciprocal lattice (⃗q ) vectors to quantify the structural 
+changes in strained MSCs18,19. The reciprocal lattice vectors of TBG moir´e lattices20 (⃗q) is 
+given as ⃗q = b⃗′  − ⃗b, where b⃗′  and ⃗b denote the reciprocal lattice vectors of the rotated top 
+layer and bottom layer in a TBG structure respectively. The length of moir´e pattern (MP), 
+4π 
+Lm can be derived using the magnitude of ⃗q vector as Lm = √3 ⃗q . When strain is applied 
+to the top layer, the mathematical expression of its reciprocal lattice vector19 (b⃗ε) can be 
+written as b⃗ε = (I⃗ + S⃗ )
+−1
+b⃗′  , where I⃗ is the identity matrix and S⃗ denotes the strain tensor 
+i 
+i 
+which can be written as the following for the case of uniaxial tension, 
+S⃗ = 
+ε 
+0 
+ 
+0 −νε  
+Here, ε is the nominal strain applied and ν denotes the Poisson’s ratio. So, the reciprocal 
+lattice vector of TBG with heterostrain can be expressed as ⃗ε 
+b⃗ε − b⃗i. As shown in 
+ω 
+ω 
+q 
+
+5 
+ 
+Fig 1(e) in the main text, the boundaries of MPs resemble a hexagon and we can draw a 
+triangle (∆ABC) with A⃗B and B⃗C as the MP lattice vectors and α being the angle between 
+them (α = 60°, ϕ = 120°). The variation of α and ϕ with the applied strain is shown in 
+Fig. S2. With uniaxial tension, we see a monotonic decrease in these angles and vice-versa 
+for uniaxial compression. The changes in expressions of ⃗q vectors are associated with the 
+geometrical changes enforced upon hetero-straining these systems. 
+ 
+ 
+III. 
+EXPLANATION OF STACKING IDENTIFICATION METHOD (FOR 
+UNSTRAINED AND STRAINED SYSTEMS) 
+Firstly, we performed the identification of atoms that should be classified as ’AA’ type 
+using ILS. As observed in main text Fig. 1(c) and (d), the spacing between two layers of 
+TBG varies due to out-of-plane displacements of atoms. The ILS of equilibrium structures 
+follows this trend: AA > SP > AB. Hence, in a TBG system, the maximum ILS (dmax) 
+corresponds to AA region and the minimum distance (dmin) represents AB region. It is 
+observed that dmax and dmin vary with increasing twist angle up to 21°, after which we 
+noticed a plateaued regime21. This results due to the depletion of perfectly stacked AA and 
+AB configurations, as the length of the MPs, reduces with increasing θ. We obtained the 
+maximum and minimum magnitudes of dmax (3.589˚A and 3.475˚A) and dmin (3.456˚A and 
+3.338˚A). Using the lower bound of dmax for all the twist angles, i.e., 3.475˚A, we classified 
+the atoms with local ILS greater than 3.475˚A as ’AA’ stacking type. On the other hand, 
+considering the upper bound of dmin and identifying the regions with ILS below that value 
+as AB stacking can lead to the misclassification of AB and SP types. For the wide range of 
+twist angle considered in this study, the ILS alone cannot provide a margin of separation for 
+classifying AB and SP stacked atoms. To address this issue, we considered interlayer energy 
+or ILE (per atom) in the structure. Perusing the ILE contour plot, we observed that the 
+center of MPs has the highest energy followed by the SP segments. The AB (or BA) has 
+the lowest energy corresponding to the ground state configuration of BLG. But, being a per 
+atom quantity, the C atoms in AB stacking that are present directly on top of a C atom on 
+the other layer show the highest ILE value as shown in main text Fig. 2(c). 
+To obtain the same measure of energy for AB stacked atoms whether they are located 
+at the center of a lattice hexagon or at the corner, we calculated the difference of interlayer 
+
+6 
+ 
+energy of each atom with its three bonded neighbors and consider their average. The 
+interlayer energy difference with neighboring atoms allows us to easily classify AB stacked 
+atoms as they have the highest fluctuation of energy with neighbors compared to AA or SP 
+stacked regions where the quantity is quite uniform. To obtain a classification threshold of 
+interlayer energy difference for AB stacking, we first calculated the soliton width of different 
+TBG systems, i.e., the width of SP regions similarly as explained by Gargiulo et al21. On 
+analyzing the path from the center of AB domain to the center of another AB (or BA) 
+region, we traverse across the SP segment. Calculating the ILS and plotting it along the 
+centers of triangular (AB) regions, we observed a small peak (Fig S3). This peak corresponds 
+to the SP region and its full width at half maxima (FWHM) gives us the soliton width21. 
+Considering this soliton width (varies with twist angle), we obtained the interlayer energy 
+difference value at the boundary of SP domains. This process is repeated for different twist 
+angles to establish a unique threshold that can be applied to any TBG system. The energy 
+difference threshold lies in a diminutive range, 8.22-8.31 meV for the angles considered (Fig 
+2(e) in main text). On averaging these magnitudes, we defined a ∆EILE threshold of 8.24 
+meV/atom, above which an atom is classified as AB stacking type. The contour plot of TBG 
+(θ = 6◦) system in main Fig 2(f) shows the outcome of applying the method where each atom 
+has been classified as belonging to either AA or AB or SP stacked. We utilized the same 
+approach for classifying the local domains in strained systems. Since the ILS parameter 
+defines the out-of-plane distancing of pristine structures, it is not affected by an in-plane 
+applied strain. However, the interlayer energy of the structure is expected to change because 
+an externally applied strain disturbs the interlayer interactions. But since the mechanical 
+deformation is applied globally, the local regions will experience a similar change in ILE 
+with respect to their nearest neighbors and hence ∆EILE remains approximately unchanged 
+(see Table S1). 
+ 
+ 
+IV. 
+STACKING IDENTIFICATION OF RIGID STRUCTURES 
+ 
+We followed the same approach used for reconstructed or relaxed systems to classify local 
+regions in rigid structures. The atomistic structure of rigid TBGs (R-TBGs) is different 
+from reconstructed systems. Since they are created by simply employing a rigid twist to a 
+Bernal stacked bilayer graphene, they do not have a variation of interlayer spacing, which is 
+
+7 
+ 
+present in reconstructed TBGs pertaining to the formation of local stackings in the structure. 
+When a R-TBG is modeled from Bernal stacked (or AB) graphene, it has an ILS equal to 
+that of AB stacked graphene throughout its structure. Hence to account for this we defined 
+their uniform ILS, which is different from their initial geometry. We first considered their 
+relaxed structure and obtained an average ILS value considering all the interlayer distances 
+throughout the structure. Then, we re-modeled the rigid TBG structure by adjusting the 
+layers with respect to the average ILS value. Since different structures have varying fractions 
+of local interlayer regions, this average ILS changes for systems with certain twist angles. 
+It must be noted that we have not utilized this average ILS to define any threshold to 
+classify local atoms, rather it is used only to define the respective rigid structures. Further, 
+following the same method as relaxed systems we obtained their interlayer energy followed by 
+calculating the ILE difference (∆EILE) per atom. Now to classify the individual stackings, 
+we referred back to the ILS and ∆EILE thresholds obtained for relaxed systems. Having 
+known the ILS threshold for AA region (3.475 ˚A ) ,  we then identified the ∆EILE value at the 
+location corresponding to that ILS value by traversing along path PQ (Fig. 2(a) main text). 
+Then, we employed this value in ∆EILE calculation for R-TBG and specified atoms above 
+that threshold (6.88 meV/atom) as AA. For identifying AB type, we have considered the 
+∆EILE threshold (8.24 meV/atom) corresponding to its location on the path PQ. Similarly, 
+we then used that location to detect ∆EILE threshold for AB type in R-TBG structure 
+(5.92 meV/atom, so it lies between 5.92 and 6.88 meV/atom). After classifying AB and 
+AA, we have assigned the remaining atoms as SP. Further, we have used this same method 
+to identify the local stackings in rigid structures of strained configurations. To model rigid 
+systems of strained TBGs in a way that physically makes sense, we first considered the 
+relaxed or reconstructed structure of pristine TBG. Now the top layer is stretched such 
+that an unrelaxed hetero-strained TBG system is generated, which is referred to as the 
+rigid structure in the presence of strain. Relaxing this strained structure results in a fully 
+optimized system, pertaining to the reconstructed TBG configuration with strain. 
+
+8 
+ 
+V. 
+PHONON DISPERSION SPECTRA OF TBG AND ITS LOCAL 
+DOMAINS 
+The simulations for phonon dispersion spectra were performed for θ = 6° and 13.2° 
+sys- tems. Due to the computational cost of DFT-based phonon simulations for large 
+MPs, we computed phonon spectra only for θ > 4.41° systems. We discussed an approach 
+using BOLS correlation to predict the Raman peaks pertaining to optical phonon modes 
+for larger TBG systems. As described by Cocemasov et al, TBGs contain hybrid folded 
+phonon branches that require to be unfolded onto the single layer first BZ22. Using the 
+PhononUnfolding package23, we simplified the phonon spectra of TBGs along Γ-K-M-Γ 
+high symmetry path (Fig S7 shows unfolded spectra of θ = 6°). To obtain the phonon 
+spectra of local sub- domains, we first identified the atomic positions of each local 
+stacking as defined by our identification method and extract the data from the main 
+structure. Then, we calculated the average bond length lavg of each configuration and 
+deduce their respective lattice con- 
+stant as a  
+stacking = √3l  
+avg . With the calculated unit cell parameters, we have computed 
+their phonon spectrum. 
+ 
+ 
+VI. 
+PHONON BAND SPLITTING WITH HETEROSTRAIN 
+ 
+A combination of Molecular statics and first principles simulations has been used to 
+compute phonon dispersion spectra of TBGs with heterostrain. By freezing the obtained 
+configuration from LAMMPS, we have extracted the atomic data of strained periodic moir´e 
+lattice and further minimized the supercell in DFT to obtain first-principles-level fidelity, 
+followed by phonon spectra calculations. We observed strain-induced phonon band splitting 
+due to inequivalent strain present in both layers. With tension, the atomic bonds in a crystal 
+are stretched relative to their unstrained condition. When the bond length is increased, 
+and the force constant remains unchanged, as a result, the vibrational frequency decreases. 
+Conversely for compression, the bond length reduces which leads to an increase in vibrational 
+frequency. That is why we observe redshift and blueshift in phonon frequencies for tensile 
+and compressive strain respectively24. The redshift and blueshift of Raman G-band for θ = 
+1.08°, shown in Fig. S9 is a good demonstration of this phenomenon. 
+
+9 
+ 
+ 
+ 
+ 
+FIG. 1: (a) Relaxed atomistic structure and (b) interlayer spacing contour plot of θ= 6° 
+TBG system under 1% uniaxial compressive strain. 
+ 
+ 
+ 
+ 
+TABLE I: Average ∆EILE threshold value considering five representative TBG systems (θ 
+= 1.1°, 3.48°, 4.41°, 6° and 7.34°) in the presence of strain. 
+ 
+Strain (%) ∆EILE  (meV/atom) 
+0 
+8.24 
++0.5 
+8.223 
+-0.5 
+8.21 
++1 
+8.23 
+-1 
+8.207 
+
+3.6
+Spacing (A)
+鞋
+鞋
+3.5
+cocal Interlayer
+鞋
+3.45
+3.4
+.3510 
+ 
+ 
+ 
+ 
+FIG. 2: Variation of angles α and ϕ with strain demonstrating the deformation of moir´e 
+patterns (for TBG system θ= 6°) 
+ 
+ 
+ 
+ 
+TABLE II: Evolution of area fractions f of local stacking domains with uniaxial tension 
+and compression applied to the top layer 
+ 
+ 
+ 
+ 
+Strain (%) 
+ 
+θ = 1.1°  
+ 
+θ = 6°  
+ 
+θ = 13.2°  
+fAA 
+fAB 
+fSP 
+fAA 
+fAB 
+fSP 
+fAA 
+fAB 
+fSP 
+0 
+0.135 
+0.474 
+0.391 
+0.25 
+0.39 
+0.36 
+0.272 
+0.379 
+0.349 
++0.2 
+- 
+- 
+- 
+0.261 
+0.376 
+0.363 
+0.293 
+0.338 
+0.369 
+-0.2 
+- 
+- 
+- 
+0.239 
+0.407 
+0.354 
+0.257 
+0.399 
+0.344 
++0.5 
+- 
+- 
+- 
+0.274 
+0.356 
+0.37 
+0.309 
+0.317 
+0.374 
+-0.5 
+- 
+- 
+- 
+0.218 
+0.432 
+0.35 
+0.239 
+0.419 
+0.342 
++0.7 
+- 
+- 
+- 
+0.289 
+0.339 
+0.372 
+0.322 
+0.301 
+0.377 
+-0.7 
+- 
+- 
+- 
+0.2 
+0.455 
+0.345 
+0.218 
+0.443 
+0.339 
++1 
+- 
+- 
+- 
+0.302 
+0.319 
+0.379 
+0.330 
+0.291 
+0.379 
+-1 
+- 
+- 
+- 
+0.188 
+0.471 
+0.341 
+0.2 
+0.462 
+0.338 
+
+EUniaxial tension
+EUniaxialtension
+64
+124
+G  Uniaxial compression
+G  Uniaxial compression
+62
+122
+(c)0 
+(o)
+Angle 
+60
+e
+120
+58
+118
+56
+116
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+Strain(%)
+Strain(%)11 
+ 
+θ = 1.1° 
+θ = 6° 
+θ = 13.2° 
+ 
+ 
+ 
+FIG. 3: Normalized spatial interlayer spacing difference (∆d) profiles traversing between 
+centers of moir´e pattern, i.e., path PQ in Fig. 2(a) (for TBG system θ= 6°) 
+ 
+ 
+ 
+ 
+TABLE III: Parameters for calculating βBOLS pre-factors for TBGs and their respective 
+sub-domains. 
+ 
+ 
+ 
+Parameters TBG 
+AA 
+AB 
+SP 
+TBG 
+AA 
+AB 
+SP 
+TBG 
+AA 
+AB 
+SP 
+dz (˚A )  
+1.406 1.40 1.405 1.411 1.424 1.417 1.423 1.43 
+1.438 1.431 1.437 1.441 
+z 
+5.008 4.88 4.987 5.12 
+5.43 5.185 5.409 5.612 5.851 5.67 5.792 5.911 
+Cz 
+0.913 0.909 0.912 0.916 0.926 0.918 0.924 0.929 0.933 0.929 0.933 0.935 
+Ez (eV) 
+0.775 0.783 0.776 0.768 0.752 0.764 0.751 0.741 
+0.73 0.743 0.733 0.727 
+
+AA
+AA
+3.6
+3.55
+3.5
+3.45
+SP
+3.4
+AB
+AB
+3.35
+P
+Q12 
+ 
+ 
+ 
+FIG. 4: Variation of area fractions of individual stacking domain with respect to 
+heterostrain (tension) for different twist angles 
+
+0.5
+0.5
+0 = 3.48°
+AA
+0 = 4.410
+AA
+■SP
+SP
+AB
+AB
+0.4
+0.4
+0.3
+Area
+0.2
+0.1
+0.1
+0% strain
+0.5% strain
+1%strain
+0% strain
+0.5% strain
+1% strain
+(a)
+(b)
+0.5
+0.5
+0 = 5.08°
+AA
+0 = 7.34°
+AA
+■SP
+SP
+AB
+AB
+0.4
+0.4
+uo
+0.3
+0.1
+0.1
+0% strain
+0.5% strain
+1% strain
+0%strain
+0.5% strain
+1%strain
+(c)
+(d)
+0.5
+0.5
+0 = 9.349
+AA
+0 = 13.10
+AA
+-SP
+■SP
+AB
+AB
+0.4
+0.4
+ion
+on
+cti
+0.3
+fra
+0.1
+0.1
+0% strain
+0.5%strain
+1%strain
+0%strain
+0.5%strain
+1%strain
+(e)
+(f)13 
+ 
+ 
+ 
+ 
+FIG. 5: Variation of area fractions of individual stacking domain with respect to 
+heterostrain (compression) for θ = 3.48°, 6° and 13.2° 
+ 
+ 
+ 
+ 
+ 
+ 
+FIG. 6: Interlayer energy or vdW stacking energy for rigid and relaxed TBG systems. The 
+ILE of relaxed TBG system is always lower than rigid TBG even for larger twist angles. 
+
+0.5
+0.5
+0.5
+0=3.48°
+AA
+0 = 60
+AA
+0 = 13.29
+AA
+■SP
+SP
+SP
+0.4
+AB
+0.4
+AB
+0.4
+AB
+fract
+0.1
+0.1
+0% strain
+-0.5% strain
+-1% strain
+0% strain
+0.5% strain
+-1% strain
+0% strain
+-0.5% strain
+-1% strain-10
+ILE (meV/atom)
+-15
+Interlayer energy, 1
+20
+25
+Rigid TBG
+Relaxed TBG
+-30
+0
+2
+4
+6
+8
+10
+12
+14
+Twist angle (0o)14 
+ 
+ 
+ 
+ 
+ 
+FIG. 7: Unfolded phonon spectra of TBG system θ = 6° along high symmetry points of its 
+Brillouin zone. Phonon dispersion spectra of Bernal stacked BLG is also shown for 
+comparison. 
+
+1500
+(cm 
+1200
+Phonon frequency (
+900
+600
+300
+Bernal stackedpristine BLG
+TBG (0 = 6)
+K
+M15 
+ 
+ 
+ 
+ 
+FIG. 8: Transverse optical (TO) phonon frequency difference with respect to TBG system 
+θ = 6° 
+
+20
+10
+-10
+AA stacking
+AB stacking
+SP stacking
+-20
+K
+M16 
+ 
+ 
+ 
+FIG. 9: BOLS predicted Raman G band frequencies of θ = 1.1° TBG system as a function 
+of applied heterostrain (tension and compression) 
+
+1610
+Bottom layer (unstrained)
+ Top layer (tension)
+1600
+Top layer (compression)
+1590
+(cm
+1580
+1570
+1560
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+Strain(%)17 
+ 
+TBG 
+TABLE IV: Calculated βϵ 
+pre-factor values of strained top layer using BOLS 
+parameters with respect to strain 
+ 
+ 
+Strain (%) 
+θ = 1.1° 
+θ = 6° 
+θ = 13.2° 
+0 
+3.135 
+3.306 
+ 
+3.466 
++0.2 
+3.207 
+3.355 
+ 
+3.527 
+-0.2 
+3.078 
+3.214 
+ 
+3.421 
++0.5 
+3.311 
+3.451 
+ 
+3.619 
+-0.5 
+2.988 
+3.064 
+ 
+3.356 
++0.7 
+3.398 
+3.506 
+ 
+3.674 
+-0.7 
+2.732 
+2.961 
+ 
+3.312 
++1 
+3.475 
+3.592 
+ 
+3.773 
+-1 
+2.602 
+2.795 
+ 
+3.248 
+ 
+
+18 
+ 
+REFERENCES 
+ 
+1A. Dey, R. Sharma, S. A. Dar, and H. H. Raza, “A computational investigation on struc- 
+tural, mechanical, electronic, magnetic, thermoelectric, and optical properties of crxpb 
+(x= sc, ti) half-heusler alloys,” Journal of Superconductivity and Novel Magnetism 34, 
+781–796 (2021). 
+2A. Dey, B. A. Baraiya, S. Adhikary, and P. K. Jha, “First-principles calculations of the ef- 
+fects of edge functionalization and size on the band gap of be3n2 nanoribbons: Implications 
+for nanoelectronic devices,” ACS Applied Nano Materials 4, 493–502 (2020). 
+3R. Sharma, A. Dey, S. A. Dar, and V. Srivastava, “A dft investigation of csmgx3 (x= cl, 
+br) halide perovskites: Electronic, thermoelectric and optical properties,” Computational 
+and Theoretical Chemistry 1204, 113415 (2021). 
+4 E´ . D. Murray, K. Lee, and D. C. Langreth, “Investigation of exchange energy density func- 
+tional accuracy for interacting molecules,” Journal of Chemical Theory and Computation 
+5, 2754–2762 (2009). 
+5A. Dey, R. Sharma, S. A. Dar, and I. H. Wani, “Cubic pbgeo3 perovskite oxide: A 
+compound with striking electronic, thermoelectric and optical properties, explored using 
+dft studies,” Computational Condensed Matter 26, e00532 (2021). 
+6A. Dey and D. Chakraborty, “Engineering the band structures of zigzag blue phosphorene 
+and arsenene nanoribbons by incorporating edge corrugations: A first principles explo- 
+ration,” Journal of Nanoscience and Nanotechnology 21, 5929–5936 (2021). 
+7V. Kumar, A. Dey, S. Thomas, M. A. Zaeem, and D. R. Roy, “Hydrogen-induced tunable 
+electronic and optical properties of a two-dimensional penta-pt 2 n 4 monolayer,” Physical 
+Chemistry Chemical Physics 23, 10409–10417 (2021). 
+8J. A. Abraham, R. Sharma, S. Ahmad, and A. Dey, “Dft investigation on the electronic, 
+optical and thermoelectric properties of novel half-heusler compounds scaux (x= si, ge, sn, 
+pb) for energy harvesting technologies,” The European Physical Journal Plus 136, 1091 
+(2021). 
+9D. W. Brenner, O. A. Shenderova, J. A. Harrison, S. J. Stuart, B. Ni, and S. B. Sinnott, 
+“A second-generation reactive empirical bond order (rebo) potential energy expression for 
+hydrocarbons,” Journal of Physics: Condensed Matter 14, 783 (2002). 
+10A. N. Kolmogorov and V. H. Crespi, “Registry-dependent interlayer potential for graphitic 
+
+19 
+ 
+systems,” Physical Review B 71, 235415 (2005). 
+11K. Zhang and E. B. Tadmor, “Energy and moir´e patterns in 2d bilayers in translation and 
+rotation: A study using an efficient discrete–continuum interlayer potential,” Extreme 
+Mechanics Letters 14, 16–22 (2017). 
+12S. A. Chowdhury, K. Inzani, T. Pen˜a, A. Dey, S. M. Wu, S. M. Griffin, and H. Askari, 
+“Mechanical properties and strain transfer behavior of molybdenum ditelluride (mote2) 
+thin films,” Journal of Engineering Materials and Technology 144 (2022). 
+13X. Gao, H. Sun, D.-H. Kang, C. Wang, Q. J. Wang, and D. Nam, “Heterostrain-enabled 
+dynamically tunable moir´e superlattice in twisted bilayer graphene,” Scientific reports 11, 
+1–8 (2021). 
+14C. Androulidakis, E. N. Koukaras, G. Paterakis, G. Trakakis, and C. Galiotis, “Tunable 
+macroscale structural superlubricity in two-layer graphene via strain engineering,” Nature 
+communications 11, 1–11 (2020). 
+15A. Stukowski, “Visualization and analysis of atomistic simulation data with ovito–the open 
+visualization tool,” Model. Simul. Mater. Sci. Eng 18, 015012 (2009). 
+16W. Zheng and C. Sun, “Energy environ. sci. 4, 627 (2011),”. 
+17X. Yang, Y. Wang, J. Li, W. Liao, Y. Liu, and C. Q. Sun, “Graphene phonon softening 
+and splitting by directional straining,” Applied Physics Letters 107, 203105 (2015). 
+18J. Campos-Delgado, L. G. Can¸cado, C. A. Achete, A. Jorio, and J.-P. Raskin, “Raman 
+scattering study of the phonon dispersion in twisted bilayer graphene,” Nano Research 6, 
+269–274 (2013). 
+19Y. Hou, S. Zhang, Q. Li, L. Liu, X. Wu, and Z. Zhang, “Evaluation local strain of twisted 
+bilayer graphene via moir´e pattern,” Optics and Lasers in Engineering 152, 106946 (2022). 
+20V. Carozo, C. M. Almeida, E. H. Ferreira, L. G. Cancado, C. A. Achete, and A. Jorio, 
+“Raman signature of graphene superlattices,” Nano letters 11, 4527–4534 (2011). 
+21F. Gargiulo and O. V. Yazyev, “Structural and electronic transformation in low-angle 
+twisted bilayer graphene,” 2D Materials 5, 015019 (2017). 
+22A. I. Cocemasov, D. L. Nika, and A. A. Balandin, “Phonons in twisted bilayer graphene,” 
+Physical Review B 88, 035428 (2013). 
+23F. Zheng and P. Zhang, “Phonon unfolding: A program for unfolding phonon dispersions 
+of materials,” Computer Physics Communications 210, 139–144 (2017). 
+24D. Yoon, Y.-W. Son, and H. Cheong, “Strain-dependent splitting of the double-resonance 
+
+20 
+ 
+raman scattering band in graphene,” Physical review letters 106, 155502 (2011). 
+25A. Dey, R. Sharma, and S. A. Dar, “An extensive investigation of structural, electronic, 
+thermoelectric and optical properties of bi-based half-huesler alloys by first principles cal- 
+culations,” Materials Today Communications 25, 101647 (2020). 
+

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+3D photophoretic aircraft made from ultralight porous 
+materials can carry kg-scale payloads in the mesosphere 
+Thomas Celenza, Andy Eskenazi and Igor Bargatin 
+ 
+ 
+We show that photophoretic aircraft would greatly benefit from a three-dimensional (3D) hollow geometry 
+that pumps ambient air through sidewalls to create a high-speed jet.  To identify optimal geometries, we 
+developed a theoretical expression for the lift force based on both Stokes (low-Re) and momentum (high-
+Re) theory and validated it using finite-element fluid-dynamics simulations. We then systematically varied 
+geometric parameters, including Knudsen pump porosity, to minimize the operating altitude or maximize 
+the payload. Assuming that the large vehicles can be made from previously demonstrated nanocardboard 
+material, the minimum altitude is 55 km while the payload can reach 1 kilogram for 3D structures with 10-
+meter diameter at 80 km altitude. In all cases, the maximum areal density of the sidewalls cannot exceed a 
+few grams per square meter, demonstrating the need for ultralight porous materials. 
+ 
+ 
+For centuries, humans have been exploring Earth’s atmosphere and outer space, a quest that has 
+led to discoveries in fields ranging from aerodynamics to astronomy and climate modeling [1-3]. However, 
+the study of certain regions of the atmosphere is hindered by available propulsion technologies. For instance, 
+in Earth’s mesosphere, anthropogenic emissions of carbon dioxide are counterintuitively producing rapid 
+cooling [4]. The shrinking of the atmosphere resulting from this cooling [5] can be problematic, given that 
+a contracting mesosphere can result in reduced satellite drag, which could translate into a greater 
+accumulation of space debris [6]. Unfortunately, uncertainties in calculations of these effects are currently 
+large because experimental observations within the mesosphere are challenging [7], given that this region, 
+extending from fifty to eighty kilometers above the surface of Earth, has air pressures too low to sustain 
+planes or balloons and too high for orbiting satellites.  
+ 
+Another region of significant interest is the Martian atmosphere, where most recently the 
+helicopter Ingenuity achieved near-surface flight [8]. Even with this milestone, sustained flight at high 
+altitudes in Mars, e.g., from Olympus Mons, is not yet possible due to decreasing atmospheric density 
+[9,10]. Like the study of Earth’s mesosphere, the exploration of Mars’ atmosphere at high altitudes is 
+limited by the lack of long-duration methods of flight and propulsion at ambient pressures below ~1 mbar 
+(100 Pa). As a result, developing an airborne platform that can operate in a very thin atmosphere, both on 
+Mars and on Earth, would be extremely useful in helping collect valuable and atmospheric data related to 
+wind patterns, temperature and pressure variations, as well as the concentrations of atmospheric gases. 
+ 
+One promising concept, based on the lightweight light-powered centimeter-scale microflyers 
+developed by Cortes et al. [11], can potentially overcome the issues faced by the current propulsion 
+mechanisms and achieve sustained flight in Earth’s mesosphere and the Martian atmosphere. These devices, 
+composed of porous plates, can levitate due to photophoresis, a light-driven propulsion mechanism where 
+a jet is created using Knudsen pumping of ambient gas [12]. Knudsen pumps have no moving parts and 
+instead exploit temperature gradients to induce gas flows through these plates. Known as “nanocardboard”, 
+these ultralight porous plates are composed of nanometer-thick (25–400 nm) aluminum oxide face sheets 
+that are connected by channels with micrometer-scale width and height. They offer an areal density of only 
+~1 g/m2 and a bending stiffness orders of magnitude higher relative to solid plates of the same mass [13].  
+ 
+Photophoretic levitation is typically enabled by a difference in physical properties between the top 
+and bottom of the plate. For instance, in the study performed by Cortes et al. [12], the bottom side of the 
+nanocardboard was coated with carbon nanotubes (CNTs), which absorbed the incident light and 
+subsequently increased in temperature relative to the top side. This difference in temperatures caused the 
+Knudsen pumping, which pushed air down through the channels of nanocardboard from the cold to the hot 
+side and thus creating a downward jet below the nanocardboard that levitated plates with centimeter-scale 
+sizes [11]. This mechanism works best in low pressure environments (1-100 Pa) [14], such as in Earth’s 
+mesosphere or near the top of Olympus Mons on Mars [15]. If the lift forces are large enough to carry tiny 
+“smart dust” sensor payloads [16], many such microflyers can be deployed on Earth or on Mars to record 
+data in these regions of the atmosphere.  
+ 
+In this work, we propose much larger photophoretic vehicles, which are many meters in diameter, 
+three-dimensional rather than planar, and use porous sidewalls that push air into an inner chamber and out 
+
+of a small nozzle (Fig. 1). Using the nozzle increases the speed of the air jet, and such 3D photophoretic 
+vehicles can not only increase the resulting lift force but also widen the range of operating pressures. 
+Combining design concepts from the previously demonstrated photophoretic levitation of planar 
+nanocardboard [11] and analytical tools we used for solid mylar-CNT composite disks [17], we analyzed 
+3D geometries with porous alumina nanocardboard walls and CNTs deposited on their inner side. Because 
+alumina is transparent, CNTs on the inside of the structure would absorb the incident light, inducing the 
+Knudsen pumping of air from the outside into the interior chamber through the pores and then out of the 
+chamber through the exit nozzle, producing a jet as illustrated in Fig. 1.  
+ 
+ 
+Figure 1: A hollow sphere with porous alumina-CNT composite walls flying in Earth’s mesosphere (a) and over the 
+top of Olympus Mons in Mars (b). The cross-sectional view (c) of the sphere shows the air flow in, with velocity 𝑣𝑓𝑡, 
+due to Knudsen pumping (across the nanocardboard walls, as seen on the zoomed-in view) and out as a jet through 
+the exit nozzle, with velocity 𝑣𝑗𝑒𝑡. As depicted in (c), A is the nanocardboard channel width, L the nanocardboard 
+channel height, and r the structure’s outlet radius, while D the structure’s overall size dimension.  Background Earth 
+and Mars Image Credits: NASA. 
+ 
+To identify the optimal 3D geometry that maximized payload, we considered three representative 
+geometries (a sphere, a cone, and a rocket), and performed a series of simulations to determine the 
+parameters that would yield the greatest lift forces. However, first, it was necessary to develop an analytical 
+expression that predicted the lift forces produced by such structures across a wide range of Reynold 
+numbers. To determine this expression, we modeled these 3D structures with outlet jet velocities as small 
+as 10-6 m/s to as large as ~100 m/s and at various atmospheric altitudes up to 80 km using computational 
+fluid dynamics simulations in ANSYS Fluent, as detailed in the supplementary information. For each fluid-
+flow simulation, we found the reaction forces induced from the air flow (equal and opposite to the lift force), 
+and then fitted the collected data using the equation  
+ 
+ 
+𝐹 = 𝐶18𝜇𝐷𝑣𝑓𝑡 + 𝐶2𝜌𝐴𝑣jet
+2 . 
+(1) 
+ 
+
+a
+Exterior
+Vft
+Vft
+Interior
+Vft
+Cross-
+sectional
+viewHere, 𝜇 corresponded to the fluid viscosity, 𝜌 to the density, 𝐴 = 𝜋𝑟2 is the area of a nozzle with radius r, 
+D is the geometry’s characteristic (i.e., largest) dimension, while 𝑣𝑓𝑡 is the flow-through velocity of the 
+fluid flow through the porous material and 𝑣𝑗𝑒𝑡 is the velocity of the fluid exiting the structure through the 
+small nozzle. As outlined in the supplementary information, 𝑣𝑓𝑡 depends on the light intensity, I, the 
+altitude dependent air pressure, P, and the geometric parameters of the nanocardboard. The upper limit of 
+the flow-through velocity typically scales as 𝑣𝑓𝑡 ≈ 0.03 𝐼/𝑃 (see supplementary information), resulting in 
+velocities of less than 1 mm/s under natural sunlight (~1000 W/m2) and standard atmospheric pressure (105 
+Pa) but increasing by many orders of magnitude as the pressure drops at higher altitudes. 
+ 
+In Eqn. (1), the first term is based on Stokes’ drag on a disk, obtained from a linearization of the 
+steady-state Navier-Stokes equations in the case of dominating viscous forces, i.e., in the low-Re limit. 
+Cortes et al. previously showed that at vanishingly low air flow speeds, the lift of a stationary 
+nanocardboard plate with air flowing through it was equal to the Stokes drag for a solid disk [11]. In contrast, 
+at high jet speeds, the inertial terms dominate, and the lift is mostly dependent on the velocity of the jet 
+exiting the nozzle. The helicopter-momentum theory equation, which can be derived from a simple 
+application of Reynolds Transport Theorem and represents the second term in Eqn. (1), can model the lift 
+in this high-Re limit. Summing both terms results in a simple interpolation between the two operating 
+regimes that gives an estimate for the lift force at all pressures and velocities (and, therefore, all values of 
+Re). Table 1 summarizes the average fitted C1 and C2 parameters, both on the order of 1, obtained from 
+fitting the results for 27 ANSYS Fluent simulations using 3 different altitudes (0 km, 40 km and 70 km), 3 
+geometry types (sphere, cone, and rocket), and 3 different structure sizes (1cm, 5cm and 10cm).  
+ 
+Fitting Parameters for Each Geometry 
+Geometry 
+Cone 
+Sphere 
+Rocket 
+C1 
+1.2 
+1.3 
+1.4 
+C2 
+0.9 
+0.9 
+0.4 
+ 
+Table 1: Fitting parameters for the three geometries in addition to key dimensions. Notice that these ANSYS 
+simulations were performed assuming a 100% porosity along each one of these structures’ walls.  
+ 
+After determining the coefficients C1 and C2, we proceeded to numerically optimize the various 
+parameters controlling the overall 3D shape and nanocardboard porous microstructure to maximize the 
+payload capabilities. The developed MATLAB code [18] was based on the photophoretic levitation theory 
+for nanocardboard [11] adapted to axisymmetric 3D structures, as detailed in the supplementary 
+information. The code also took into account how temperature and pressure depend on the altitude in the 
+atmosphere, employing empirical relations developed from standard atmospheric data [19]. Our 
+optimization sought the combination of A (nanocardboard channel width), L (nanocardboard channel 
+height), and r (the structure’s outlet/nozzle radius) that resulted in the highest payload or achieved flight at 
+the lowest altitude as a function of the overall aircraft size D (diameter for sphere and cone, and length for 
+the rocket). All these geometric parameters are illustrated in Fig. 1c. 
+ 
+Our numerical optimizations revealed that the optimal nanocardboard porosity parameters A and 
+L were of the same order of magnitude across all geometries and dimensions D. When optimized for 
+achieving flight at the minimum altitude (55 km with zero-payload), A and L were ≈ 0.20 mm and ≈ 0.21 
+mm, respectively. When optimized for maximum payload (achieved at 80 km altitude), A and L were 0.90 
+mm and 0.91 mm, or about a factor of 4 greater. Because these parameters are of the same order of 
+magnitude despite the approximately 40-fold change in ambient pressure at the minimum possible altitude 
+of 55 km and the max payload altitude of 80 km, we can make structures that simultaneously achieve 
+levitation at low altitudes while carrying significant payload at higher altitudes. 
+ 
+The maximum areal densities, i.e., the maximum lift force divided by specific gravity g and the 
+area of nanocardboard, were also comparable for all structures. Table 2 shows that the typical value of 
+maximum areal density was ≈ 7.1 g/m2 (grams per square meter) for small aircraft (D = 10 cm) compared 
+to ≈ 5.5 g/m2 for large aircraft (D = 10 m). Both these densities are in the same order of magnitude as the 
+theoretical upper limit derived for the high-Re case in the supplementary information, of 0.016 𝐼/
+(𝑣𝑎𝑣𝑔𝑔) ≈  0.004  kg/m2 = 4 g/m2. Here, 𝑣𝑎𝑣𝑔 = √8𝑅𝑎𝑖𝑟𝑇/𝜋  ≈ 400  m/s is average speed of air 
+molecules at 55-80 km altitudes, while 𝑅𝑎𝑖𝑟 = 𝑅𝑢/𝑀𝑎𝑖𝑟 = 287 𝐽/(𝑘𝑔 ∙ 𝐾) is the gas-specific ideal constant 
+of air, equal to the universal gas constant 𝑅𝑢 divided by the average molar mass of air 𝑀𝑎𝑖𝑟. Fig 2a shows 
+how the maximum areal densities varies with aircraft size D and, therefore, the airflow’s Reynolds number. 
+
+The permissible areal densities of each structure decrease with increasing size and Re and stabilize at ~5.5 
+g/m2 for larger aircraft that carry payloads of 1 gram or more. 
+ 
+Areal Densities and Areas Ratio 
+Geometry 
+Cone 
+Sphere 
+Rocket 
+D = 10 cm 
+D = 10 m 
+D = 10 cm 
+D = 10 m 
+D = 10 cm 
+D = 10 m 
+Max Areal 
+Density 
+For Max. 
+Payload 
+6.6 g/m2 
+5.4 g/m2 
+7.8 g/m2 
+5.5 g/m2 
+6.9 g/m2 
+5.7 g/m2 
+Area 
+Ratios 
+For Min. 
+Altitude 
+18 
+26 
+26 
+27 
+23 
+25 
+For Max. 
+Payload 
+5 
+5 
+5 
+6 
+6 
+6 
+ 
+Table 2: Summary of the parametric studies results for the Cone, Sphere and Rocket, for values of D = 10 cm and D 
+= 10 m (full data for all the probed values of D can be found in the supplementary information section). Here, the area 
+ratio refers to the 𝐴𝑡𝑜𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 ratio, of the structure’s total surface area to its outlet area.  
+ 
+Figure 2: Areal Density versus Characteristic Size (a) and Maximum Payload versus Surface Area (b) for the three 
+considered 3D geometries at 80-km altitude. Each data point corresponds to the optimized geometry at each of the 
+probed values of the parameter D. The overlap between the curves, in particular starting at surface areas larger than 
+0.01 m2, suggests that the geometries have similar areal densities and maximum payload capabilities. 
+  
+Plotting maximum payloads against the structure surface area in Fig. 2b revealed that, for a given 
+surface area, the maximum payload was very similar across all three geometries. While the sphere 
+outperformed at smallest sizes, all three shapes (cone, sphere, and rocket) offered essentially the same 
+performance at the largest sizes, i.e., for sizes that maximize the payload and are most promising for 
+practical applications. Fig. 3 below illustrates optimized shapes for the 10-meter cone (a), sphere (b) and 
+rocket (c), which could carry 780, 540, and 1020 grams of payload, respectively. This is sufficient capacity 
+to carry modern communication devices [20] and similar to the payload of a typical CubeSat [21].  
+a 
+b 
+𝑅𝑒 = 𝜌𝑣𝑓𝑡𝐷
+𝜇
+ 
+
+D=10m
+a
+b
+c
+D= 10m
+D=
+10m
+r=4.97m
+r=3.67m
+r=4.97m
+Payload: 780 g
+Payload: 540 g
+Payload: 1020 gMax.Payload againstGeometrySurfaceArea
+100
+Max. Payload (kg)
+0
+10°
+Sphere
+Cone
+Rocket
+10-4
+10~2
+100
+102
+Surface Area (m3)Max.Areal Density against characteristic D
+ReynoldsNumber
+100
+10l
+102
+103
+25
+Sphere
+Cone
+20
+Rocket
+15
+10
+10-2
+10~1
+100
+10l
+D (m) 
+Figure 3: Geometrically optimized cone (a), sphere (b) and rocket (c) for maximum payload capabilities with a fixed 
+characteristic dimension of D = 10 meters. D represents the cone and sphere diameter, and the rocket length. Achieving 
+a payload of 1kg required a D of 11.5 and 14 m for the cone and sphere, respectively.  
+ 
+Finally, as demonstrated in Table 2 and the supplementary information section, we noticed that 
+the 𝐴𝑡𝑜𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 ratio, of the total surface area to the outlet area, was approximately constant for the optimal 
+geometries. For the minimum altitude case, this ratio ranged from 17 to 42, averaging ≈ 23 across the three 
+geometries and sizes. For the maximum payload case, the typical value of this ratio was approximately 6, 
+resulting in relative nozzle sizes shown in Fig. 3. Due to mass conservation, the outlet jet speed needs to 
+be larger than the flow-through velocity by the same factor as precisely the 𝐴𝑖𝑛/𝐴𝑜𝑢𝑡 area ratio. Therefore, 
+recalling the 𝑣𝑓𝑡 ≈ 0.03 𝐼/𝑃 relationship, at the maximum payload altitude of 80 km, we can approximate 
+𝑣𝑗𝑒𝑡 = 𝑣𝑓𝑡𝐴𝑡𝑜𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 ≈ 0.18 𝐼/𝑃 ≈ 0.18 × 1300 𝑊 𝑚−2/1 𝑃𝑎 = 234 𝑚/𝑠 , while at the minimum 
+altitude of 55 km (i.e., for zero payload),  𝑣𝑗𝑒𝑡 = 𝑣𝑓𝑡𝐴𝑡𝑜𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 ≈ 0.70 𝐼/𝑃 ≈ 0.70 × 1200 𝑊 𝑚−2/
+10 𝑃𝑎 =  84 𝑚/𝑠. Notice that for the payload altitude of 80 km, the jet speed approaches but remains below 
+the speed of sound, given by 𝑣𝑠𝑜𝑢𝑛𝑑 = √𝛾𝑅𝑎𝑖𝑟𝑇80𝑘𝑚 ≈ √1.4 × 287 𝐽/(𝑘𝑔 𝐾) × 200 𝐾 ≈ 280  m/s, 
+where 𝛾 is the adiabatic constant of air, while 𝑇80𝑘𝑚 ≈ 200 𝐾 is the air temperature at 80 km altitude. 
+Achieving kg-scale payloads in the mesosphere will therefore require building 10m-scale photophoretic 
+aircraft out of ultralight materials that simultaneously possess low areal densities (≈ 1 g/m2) and sufficient 
+structural integrity. However, these aircraft do not necessarily have to be rigid; instead, it is possible to 
+make use of flexible parachute or balloon-like structures with overall dimensions similar to those shown in 
+Fig. 3.  
+ 
+In the calculations above, we assumed that all surfaces are illuminated with 1000 W/m2 light 
+intensity, which is not always realistic. The direct sunlight intensity in the mesosphere is similar to that in 
+outer space, ~1360 W/m2. Additional ~500 W/m2 of sunlight will be reflected from the clouds and Earth 
+below the aircraft due to Earth’s planetary albedo of approximately 0.3. Depending on the elevation of the 
+Sun in the sky and the orientation of the surface, it may be exposed to anywhere between essentially zero 
+and almost 2000 W/m2 of combined direct and reflected sunlight. If the aircraft ends up rotating as balloons 
+often do, all walls will experience an average flux on the order of 1000 W/ m2 or slightly less. For reference, 
+we also performed simulations at a reduced intensity of 500 W/m2, which results in payloads ~4 times lower 
+than those shown above. One last important aspect to note about these photophoretic aircraft is that they 
+only create lift when exposed to light (i.e., during the day), limiting the steady operation to ~12 hours at 
+most latitudes, after which the aircraft will start to descend to the ground. However, near the poles, the 
+polar day can last many months and extended operations of up to several months may be possible. 
+ 
+To conclude, we show that 3D photophoretic aircraft with porous walls made of ultralight, 
+ultrathin materials are capable of carrying kg-scale payloads, comparable to those of typical CubeSats. The 
+results presented above can be easily generalized for high-altitude operation on Mars using a Martian 
+atmospheric model [22]. This work opens the way to creating persistent, low-cost, sensor-carrying aircraft 
+in the previously inaccessible atmospheric regions at 55-80 km altitudes on Earth and 20-40 km altitudes 
+on Mars, enabling a greater understanding of our planet and the worlds beyond. 
+ 
+ 
+References 
+ 
+[1] Baum, S. D. (2009). Cost–benefit analysis of space exploration: Some ethical considerations. Space 
+Policy, 25(2), 75-80. 8 
+ 
+[2] Bainbridge, W. S. (2009). Motivations for space exploration. Futures, 41(8), 514-522. 
+ 
+[3] Lindgren, E. A., Sheshadri, A., Podglajen, A., & Carver, R. W. (2020). Seasonal and latitudinal 
+variability of the gravity wave spectrum in the lower stratosphere. Journal of Geophysical Research: 
+Atmospheres, 125(18), e2020JD032850. 
+ 
+[4] Laštovička, J. (2017). A review of recent progress in trends in the upper atmosphere. Journal of 
+Atmospheric and Solar-Terrestrial Physics, 163, 2-13. 
+ 
+
+[5] Bailey, S. M., Thurairajah, B., Hervig, M. E., Siskind, D. E., Russell III, J. M., & Gordley, L. L. (2021). 
+Trends in the polar summer mesosphere temperature and pressure altitude from satellite observations. 
+Journal of Atmospheric and Solar-Terrestrial Physics, 220, 105650. 
+ 
+[6] Tran, L. “NASA Satellites See Upper Atmosphere Cooling and Contracting.” NASA. Goddard Space 
+Flight Center, June 28, 2021. https://www.nasa.gov/feature/goddard/2021/nasa-satellites-see-upper-
+atmosphere-cooling-contracting-climate-change. 
+ 
+[7] Goessling, H. F., & Bathiany, S. (2016). Why CO 2 cools the middle atmosphere–a consolidating model 
+perspective. Earth System Dynamics, 7(3), 697-715. 
+ 
+[8] Gohd, C. “Mars Helicopter Ingenuity: First Aircraft to Fly on Red Planet.” Space.com. Space, May 22, 
+2021. https://www.space.com/ingenuity-mars-helicopter-perseverance-rover. 
+ 
+[9] Squyres, S. W., Arvidson, R. E., Baumgartner, E. T., Bell III, J. F., Christensen, P. R., Gorevan, S., ... 
+& Romero, R. A. (2003). Athena Mars rover science investigation. Journal of Geophysical Research: 
+Planets, 108(E12). 
+ 
+[10] Young, L. A., Delaune, J., Johnson, W., Withrow, S., Cummings, H., Sklyanskiy, E., ... & Bhagwat, 
+R. (2020). Design Considerations for a Mars Highland Helicopter. In ASCEND 2020 (p. 4027). 
+ 
+[11] Cortes, J., Stanczak, C., Azadi, M., Narula, M., Nicaise, S. M., Hu, H., & Bargatin, I. (2020). 
+Photophoretic levitation of macroscopic nanocardboard plates. Advanced Materials, 32(16), 1906878. 
+ 
+[12] Pharas, K., & McNamara, S. (2010). Knudsen pump driven by a thermoelectric material. Journal of 
+Micromechanics and Microengineering, 20(12), 125032. 
+ 
+[13] Lin, C., Nicaise, S. M., Lilley, D. E., Cortes, J., Jiao, P., Singh, J., ... & Bargatin, I. (2018). 
+Nanocardboard as a nanoscale analog of hollow sandwich plates. Nature communications, 9(1), 1-8. 
+ 
+[14] Horvath, H. (2014). Photophoresis–a forgotten force??? KONA Powder and Particle Journal, 31, 181-
+199. 
+ 
+[15] 
+European 
+Space 
+Agency. 
+“Facts 
+about 
+Mars.” 
+https://www.esa.int/Science_Exploration/Space_Science/Mars_Express/Facts_about_Mars. 
+ 
+[16] Niccolai, L., Bassetto, M., Quarta, A. A., & Mengali, G. (2019). A review of Smart Dust architecture, 
+dynamics, and mission applications. Progress in Aerospace Sciences, 106, 1-14. 
+ 
+[17] Azadi, M., Popov, G. A., Lu, Z., Eskenazi, A. G., Bang, A. J. W., Campbell, M. F., ... & Bargatin, I. 
+(2021). Controlled levitation of nanostructured thin films for sun-powered near-space flight. Science 
+Advances, 7(7), eabe1127. 
+ 
+[18] Eskenazi, A., Celenza, T., & Bargatin, I. (2022). MATLAB-fluid-flow-parametric-studies. 
+https://github.com/andyeske/MATLAB-fluidflow-parametric-studies 
+ 
+[19] 
+Engineering 
+ToolBox. 
+(2003). U.S. 
+Standard 
+Atmosphere 
+vs. 
+Altitude. 
+https://www.engineeringtoolbox.com/standard-atmosphere-d_604.html 
+ 
+[20] Saeed, N., Elzanaty, A., Almorad, H., Dahrouj, H., Al-Naffouri, T. Y., & Alouini, M. S. (2020). 
+Cubesat communications: Recent advances and future challenges. IEEE Communications Surveys & 
+Tutorials, 22(3), 1839-1862. 
+ 
+[21] NASA. (2017). CubeSat 101: Basic Concepts and Processes for First-Time CubeSat Developers. 
+https://www.nasa.gov/sites/default/files/atoms/files/nasa_csli_cubesat_101_508.pdf 
+ 
+[22] Justh, H. L., Cianciolo, A. D., & Hoffman, J. (2021). Mars Global Reference Atmospheric Model 
+(Mars-GRAM): User Guide (No. NASA/TM-20210023957). 
+ 
+ 
+
+Page 
+1 
+3D photophoretic aircraft made from ultralight porous 
+materials can carry kg-scale payloads in the mesosphere 
+Supplementary Information 
+Thomas Celenza, Andy Eskenazi and Igor Bargatin 
+ 
+In this document, we present and expand on the computational and theoretical framework behind our work. 
+The first section is devoted to the ANSYS Fluent simulations, covering the solver set-up and the theory 
+behind the force calculations. The second section of this document focuses on the MATLAB code, 
+specifically the derivation of the equations used in the optimization of the geometrical and channel 
+parameters of the 3D geometries, including the rocket, cone and sphere.  
+ 
+1. ANSYS Fluent Simulations 
+ 
+The goal of the ANSYS Fluent simulations was to determine an analytical expression to estimate the lift 
+forces produced by various types of 3D structures. Because we sought geometries that operated across a 
+wide range of velocities and altitudes (and thus air pressures, densities, temperatures and viscosities), the 
+expression for the lift force needed to be valid across a wide range of Reynolds (Re) numbers as well. In 
+particular, this equation needed to reasonably accurately model the transition between the low-Re (Stokes) 
+regime to the high-Re regime. As the main paper argues, an appropriate expression is 
+ 
+ 
+𝐹 = 𝐶18𝜇𝐷𝑣𝑓𝑡 + 𝐶2𝜌𝐴𝑣jet
+2 . 
+(S1) 
+ 
+Here, 𝜇 corresponds to the fluid viscosity, 𝜌 to the density, 𝐴 = 𝜋𝑟2 is the area of a nozzle with radius r, D 
+is the geometry’s characteristic (usually largest) dimension, while 𝑣𝑓𝑡 is the flow-through velocity of the 
+fluid through the porous material and 𝑣𝑗𝑒𝑡 is the velocity of the fluid exiting the structure through the small 
+nozzle. The fitting parameters 𝐶1 and 𝐶2 depended on the geometry and were determined using ANSYS 
+simulations. In this work, we considered three geometries, a cone, sphere, and rocket, shown in Figure S1. 
+Figure S1: Main geometric parameters for the cone (a), rocket (b) and sphere (c). Notice that here, the variable D 
+serves as an overall indicator of the size of the geometry, while the variable r controls the outlet radii of the nozzle. 
+
+Isometric
+View
+Isometric
+View
+Side
+View
+121
+h
+a
+Side
+View
+Isometric
+View
+Side
+View
+2r
+bPage 
+2 
+Through the ANSYS Simulations, we determined the average 𝐶1 and 𝐶2 coefficients for each structure and 
+examined how these would evolve with overall size of the structure or the altitude. We performed 9 sets of 
+simulations for each geometry, where we varied three different inlet/outlet area ratios at three different 
+altitudes, resulting in flow-through velocities as small as 10-6 m/s or as large as 1 m/s.  
+ 
+Figure S2 shows boundary conditions 
+employed in our simulations using a sphere 
+as an example. To make our simulations 
+computationally more efficient, we took 
+advantage of the axial symmetry of our 
+three geometries and thus constructed our 
+models 
+in 
+a 
+2D, 
+axisymmetric 
+environment, which allowed us to only 
+simulate fluid flow on the top half of each 
+structure. We formed these geometries 
+using ANSYS’ “Design Modeler” module, 
+and they were essentially composed of 
+three spaces: an outer air box, and inner air 
+box, and the nanocardboard geometry itself 
+(whose interior was “subtracted” from the 
+inner air box, as seen in Figure S2).   
+ 
+The next step was to specify mesh 
+elements, shown in Figure S3. Plot (a) 
+shows the larger, outer air box with coarser 
+mesh elements, while plot (b) is a zoomed-
+in view into the smaller, inner air box, 
+containing smaller mesh elements. By 
+dividing the air box into these two regions, we optimized the overall number of mesh elements in the 
+simulation by providing a higher resolution just in the area close to the geometry. We created the mesh by 
+selecting edges and dividing them into a discrete number of points; to enforce a uniform grid pattern, we 
+used the quadrilaterals face meshing command. For the sphere, this resulted in 184,180 elements (185,408 
+nodes); for the cone, 194,322 elements (195,865 nodes); for the rocket, 293,053 elements (294,616 nodes). 
+These were the final numbers of mesh elements obtained as a result of performing a convergence analysis 
+until observing negligible changes in the computed lift forces.  
+ 
+ 
+The final step was to establish Fluent’s “set-up” module parameters. For the model, we chose the viscous 
+k-omega, with the low-Re (viscous) corrections feature enabled. Next, we fixed the boundary conditions as 
+described by Figure S2, and manually modified operating conditions (environment pressure, fluid density 
+and fluid viscosity) matching the chosen altitude. Since our fluid was air, we extracted its properties as 
+tabulated in altitude-dependent standard atmospheric tables, summarized in Table 1 below for 0 km, 40 km 
+and 70 km (our probed altitudes). Last, we specified the inlet velocity as a variable parameter, since that 
+Figure S2: ANSYS Simulations boundary conditions. As the 
+illustration shows, the inner wall of the geometry (red) was chosen 
+as the flow-velocity inlet (inducing the air to flow from the into the 
+structure), while the outer wall (violet) was selected as the outlet 
+(mass outflow in Fluent, inducing the air to pass through the 
+structure’s walls). For the purposes of these simulations, we are 
+assuming we have 100% porous walls through which the air flows 
+at velocity 𝑣𝑓𝑡  (an idealization of the actual nanocardboard 
+geometry). 
+b 
+a 
+Figure S3: Sample meshing of the axisymmetric sphere simulation in ANSYS Fluent. Here, plot (a) provides an 
+overall picture of the air box (which is more than ten times larger than the geometry in question in each dimension), 
+while plot (b) shows a zoomed-in image of the area immediately surrounding the sphere. The size of the outer air 
+box was not arbitrary, but rather resulted from a series of simulations that gradually increased its dimensions until 
+force values converged.  
+  
+
+OuterAir BoX
+(coarsemeshelements
+-InerAirBox
+Cfimrmcsh clements
+-AirOut
+Airiln
+-Axis of SymmetryPage 
+3 
+allowed us to sweep through values ranging from 10-6 m/s to 1 m/s in 7 logarithmically equally spaced 
+points.  
+ 
+Summary of Altitude-Dependent Atmospheric Properties 
+Altitude 
+0 km 
+40 km 
+70 km 
+Atmospheric Pressure (Pa) 
+101300 
+275.47 
+4.66 
+Atmospheric Temperature (K) 
+288 
+251 
+220 
+Air Density (kg/m3) 
+1.23 
+3.83*10-3 
+7.38*10-5 
+Air Viscosity (Pa * s) 
+1.796*10-5 
+1.610*10-5 
+1.447*10-5 
+ 
+Table 1: Tabulated altitude-dependent atmospheric conditions for 0 km, 40 km and 70 km. These values were manually 
+inputted for each simulation set into the Fluent solver. 
+ 
+We repeated this process 36 times, to construct 18 simulations for the cone, 9 for the sphere and 9 for the 
+rocket, using operating conditions corresponding to 3 different altitudes (0 km, 40 km and 70 km) and 3 
+different geometry sizes. In each case, we computed the reaction force in the axisymmetric direction using 
+a line integral along the walls of the outer air box, resulting in the force values shown in Figures S4–S7. 
+This computation made use of the fact that under steady-state operation, the reaction force is equal to the 
+lift force. The 𝐶1 and 𝐶2 coefficients were then determined by performing a non-linear fitting in MATLAB 
+to equation (S1), resulting in the values that are shown in the same figures and tabulated in Tables 2-5. In 
+general, most curves of Figures S4–S7 (in the logarithmic scale) show a transition from the viscous, low-
+Re regime to the high-Re regime that is manifested through a change in the slopes of the force curves. 
+However, at 70 km in altitude, the lift force stayed in the Stokes (low-Re) regime and the high-Re 𝐶2 
+coefficients remained uncertain at this particular altitude. Thus, when computing the overall average 𝐶1 and 
+𝐶2, we did not incorporate the 𝐶2 corresponding to the 70 km altitude.  
+ 
+ 
+Fitting Parameters for the Rocket, Dia. = 2 cm 
+Altitude 
+Length = 1 cm 
+Length = 5 cm 
+Length = 10 cm 
+C1 
+C2 
+C1 
+C2 
+C1 
+C2 
+0 km 
+2.0 
+(1.6–2.4) 
+1.1 
+(0.9–1.3) 
+1.0 
+(0.8–1.2) 
+1.1 
+(0.9–1.2) 
+0.9 
+(0.7–1.1) 
+1.1 
+(0.9–1.2) 
+40 km 
+2.24 
+(2.12–2.38) 
+0.73 
+(0.62–0.85) 
+1.1 
+(1.0–1.3) 
+0.8 
+(0.6–1.0) 
+1.0 
+(0.9–1.2) 
+0.8 
+(0.6–1.0) 
+70 km 
+2.361 
+(2.353–2.368) 
+ 
+1.20 
+(1.20–1.20) 
+ 
+1.08 
+(1.08–1.10) 
+ 
+Average 
+2.22 
+0.91 
+1.12 
+0.92 
+1.00 
+0.95 
+ 
+Table 2: 𝐶1 and 𝐶2 coefficients computed for the rocket geometry of different lengths (1 cm, 5 cm and 10 cm), alongside 
+the 66% confidence intervals for each fitting parameter (tabulated below each coefficient entry).  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+a 
+b 
+c 
+Figure S4: Results from the altitude-dependent rocket simulations in ANSYS Fluent; each data point corresponds 
+to a different flow-through velocity, ranging from 10-6 m/s to 1 m/s, while plots (a), (b) and (c) correspond to 
+different rocket lengths. 
+
+Reactionforcesforvariousflow-throughvelocities
+RocketGeometry:Dia.=2cm,Len.=1cm
+102
+ANSYS Force (Altitude: 0 km)
+Fit:C1-2.00,C2=1.10
+104
+ANSYSForce (Altitude: 40km)
+Fit:C1=2.24,C2=0.73
+ANSYSForce (Altitude:70km)
+Force (N)
+Fit:C1=2.36,C2=0.06
+10-6
+10-8
+10-10
+10-12
+10~6
+104
+102
+100Reactionforcesforvariousflow-throughvelocities
+100
+RocketGeometry:Dia.=2cm,Len.5cm
+ANSYSForce (Altitude:0km)
+Fit:C1-1.04,C2=1.10
+ANSYSForce (Altitude:40km)
+Fit:C1=1.14,C2=0.77
+ANSYSForce(Altitude:70km)
+Force (N)
+Fit:C1=1.20,C2=0.17
+10-5
+10-10
+10-6
+104
+102
+100
+V, (m/s)Reactionforcesforvariousflow-throughvelocities
+100
+RocketGeometry:Dia=2cm,Len.=10cm
+ANSYSForce (Altitude:0km)
+Fit:C1-0.90,C2=1.08
+ANSYSForce(Altitude:40km)
+Fit:C1=1.02,C2=0.82
+ANSYSForce(Altitude:70km)
+Force (N)
+Fit:C1=1.08,C2=0.19
+105
+10-10
+10-6
+104
+102
+V, (m/s)
+100Page 
+4 
+ 
+Fitting Parameters for the Sphere, Dia. = 2 cm 
+Altitude 
+rout = 0.1 cm 
+rout = 0.5 cm 
+rout = 1 cm 
+C1 
+C2 
+C1 
+C2 
+C1 
+C2 
+0 km 
+1.4 
+(0.7–2.0) 
+0.29 
+(0.21–0.37) 
+1.5 
+(1.3–1.7) 
+1.06 
+(0.95–1.18) 
+0.9 
+(0.8–1.0) 
+1.5 
+(1.4–1.7) 
+40 km 
+1.4 
+(1.0–1.9) 
+0.6 
+(0.4–0.8) 
+1.5 
+(1.3–1.6) 
+0.9 
+(0.7–1.0) 
+0.91 
+(0.89–0.93) 
+0.99 
+(0.91–1.08) 
+70 km 
+1.65 
+(1.63–1.67) 
+ 
+1.58 
+(1.52–1.64) 
+ 
+0.95 
+(0.94–0.96) 
+ 
+Average 
+1.48 
+0.45 
+1.50 
+0.97 
+0.91 
+1.26 
+ 
+Table 3: 𝐶1 and 𝐶2 coefficients computed for the sphere geometry of different outlet radii (0.1 cm, 0.5 cm and 1 cm), 
+alongside the 66% confidence intervals for each fitting parameter (tabulated below each coefficient entry).  
+ 
+ 
+ 
+ 
+ 
+Fitting Parameters for the Cone, Dia. = 2 cm 
+Altitude 
+Length = 2 cm 
+Length = 5 cm 
+Length = 10 cm 
+C1 
+C2 
+C1 
+C2 
+C1 
+C2 
+0 km 
+0.7 
+(0.5–1.0) 
+0.9 
+(0.7–1.1) 
+0.7 
+(0.5–0.9) 
+0.9 
+(0.8–1.1) 
+0.7 
+(0.4–1.0) 
+0.9 
+(0.7–1.1) 
+40 km 
+1.0 
+(0.8–1.2) 
+0.6 
+(0.3–0.8) 
+0.9 
+(0.7–1.1) 
+0.6 
+(0.4–0.8) 
+0.8 
+(0.7–1.0) 
+0.7 
+(0.6–0.9) 
+70 km 
+1.07 
+(0.98–1.16) 
+ 
+1.01 
+(0.94–1.07) 
+ 
+0.98 
+(0.95–1.02) 
+ 
+Average 
+0.94 
+0.72 
+0.88 
+0.76 
+0.84 
+0.82 
+ 
+Table 4: 𝐶1 and 𝐶2 coefficients computed for the cone geometry (2 cm diameter) of different lengths (2 cm, 5 cm and 
+10 cm), alongside the 66% confidence intervals for each fitting parameter (tabulated below each coefficient entry).  
+ 
+a 
+b 
+c 
+Figure S5: Results from the altitude-dependent sphere simulations in ANSYS Fluent; each data point corresponds 
+to a different flow-through velocity, ranging from 10-6 m/s to 1 m/s, while plots (a), (b) and (c) correspond to 
+different sphere outlet radii. 
+a 
+b 
+c 
+Figure S6: Results from the altitude-dependent cone (2 cm diameter) simulations in ANSYS Fluent; each data point 
+corresponds to a different flow-through velocity, ranging from 10-6 m/s to 1 m/s, while plots (a), (b) and (c) 
+correspond to different cone lengths. 
+
+Reactionforcesfor various flow-through velocities
+Sphere Geometry:Dia.=2 cm,r
+out
+=0.1 cm
+100
+ANSYSForce (Altitude:0km)
+Fit:C1-1.35,C2=0.29
+ANSYSForce (Altitude; 40km)
+Fit:CI-1.43,C2=0.62
+ANSYS Force (Altitude: 70 km)
+Fit:C1 1.65,C20.08
+10-5
+Force
+1o-to
+10-6
+10-4
+102
+100Reactionforcesforvariousflow-throughvelocities
+SphereGeometry:Dia.=2cm,r
+=0.5 cm
+100
+ou
+ANSYSForce (Altitude:0km)
+Fit:C1=1.46,C2-1.06
+ANSYSForce (Altitude:40km)
+Fit:C1=1.47,C2=0.87
+ANSYS Force (Altitude: 70km)
+Force (N)
+Fit:C1 1.58,C2.0.41
+10-s
+10-10
+10~6
+104
+102
+100
+Va(m/s)Reactionforcesforvariousflow-throughvelocities
+SphereGeometry:Dia,=2 cm,r
+=1 cm
+102
+out
+ANSYSForce (Altitude:0km)
+Fit:C1=0.88,C2=1.54
+104
+ANSYSForce (Altitude:40km)
+Fit:C1=0.91,C2=0.99
+"
+ANSYSForce (Altitude:70km)
+Force (N)
+10*6
+Fit:CI=0.95,C2=0.47
+10-8
+10-10
+10~/2
+10-6
+104
+10-2
+100Reaction forcesfor various flow-through velocities
+100
+Cone Geometry:Dia.=2 cm, Len.=2cm
+ANSYSForce(Altitude:0km)
+Fit: C1 =0.74, C2=0.87
+ANSYSForce(Altitude:40km)
+Fit:C11.00,C2-0.56
+ANSYSForce(Altitude:70km)
+Fit:C1-1.07.C2-0.01
+Force(
+10-5
+10-10
+10-6
+104
+10-2
+100Reactionforcesfor various flow-through velocities
+100
+Cone Geometry:Dia.=2 cm, Len.=5cm
+ANSYSForce (Altitude:0km)
+Fit:C1=0.71,C2=0.92
+ANSYSForce(Altitude:40km)
+Fit:C10.93,C2-0.60
+ANSYSForce(Altitude:70km)
+Fit:C1-1.01.C2-0.01
+Force(
+10-5
+10-10
+10-6
+104
+10-2
+100
+Va (m/s)Reactionforcesfor various flow-through velocities
+ConeGeometry:Dia.=2cm,l/日@Q价
+100
+ANSYSForce(Altitude:0km)
+FitCI=0.69,C2=0.90
+ANSYSForce (Altitude:40km)
+Fit:C10.84,C2-0.74
+ANSYSForce(Altitude:70km)
+Fit:C10.98,C2-0.10
+Force(
+10-5
+10-lo
+106
+104
+102
+100Page 
+5 
+ 
+Fitting Parameters for the Cone, Dia. = 4 cm 
+Altitude 
+Length = 2 cm 
+Length = 5 cm 
+Length = 10 cm 
+C1 
+C2 
+C1 
+C2 
+C1 
+C2 
+0 km 
+0.9 
+(0.7–1.1) 
+1.0 
+(0.8–1.2) 
+1.0 
+(0.8–1.2) 
+1.0 
+(0.8–1.1) 
+1.0 
+(0.7–1.3) 
+1.0 
+(0.8–1.1) 
+40 km 
+1.4 
+(1.1–1.7) 
+0.6 
+(0.4–0.9) 
+1.2 
+(1.0–1.3) 
+0.7 
+(0.6–0.9) 
+1.1 
+(1.0–1.2) 
+0.8 
+(0.7–1.0) 
+70 km 
+1.5 
+(1.3–1.6) 
+ 
+1.24 
+(1.22–1.25) 
+ 
+1.19 
+(1.18–1.20) 
+ 
+Average 
+1.27 
+0.82 
+1.13 
+0.86 
+1.09 
+0.89 
+ 
+Table 5: 𝐶1 and 𝐶2 coefficients computed for the cone geometry (4 cm diameter) of different lengths (2 cm, 5 cm and 
+10 cm), alongside the 66% confidence intervals for each fitting parameter (tabulated below each coefficient entry).  
+ 
+ 
+As we increased in altitude, the value of the 𝐶1 parameter increased while that of 𝐶2 decreased. All in all, 
+Table 6 below summarizes the average 𝐶1 and 𝐶2 coefficients obtained for each geometry. In all cases, the 
+coefficients are on the order of 1. 
+ 
+Average Fitting Parameters for Each Geometry 
+Geometry 
+Cone 
+Sphere 
+Rocket 
+D = 2 cm 
+D = 4 cm 
+D = 2 cm 
+D = 2 cm 
+C1 
+0.9 
+1.2 
+1.3 
+1.4 
+C2 
+0.8 
+0.9 
+0.9 
+0.9 
+ 
+Table 6: Fitting parameters for the analytical theory for standard atmospheric conditions on Earth, for each geometry. 
+ 
+To verify our simulations were based on realistic boundary conditions, we examined the streamline plots 
+generated in ANSYS Fluent’s results module, a sample of which is shown in Figure S8 below.  
+ 
+a 
+b 
+c 
+d 
+Figure S8: Velocity streamlines corresponding to the cone (a, c) and rocket (b, d) geometries simulations in 
+ANSYS, for a flow-through velocity of 1 m/s and atmospheric conditions corresponding to 0 km in altitude. Both 
+the cone and rocket have a characteristic dimension (D) of 5 cm. (c) and (d) denote a zoomed-in view of plots (a) 
+and (b), respectively. 
+a 
+b 
+c 
+Figure S7: Results from the altitude-dependent cone (4 cm diameter) simulations in ANSYS Fluent; each data 
+point corresponds to a different flow-through velocity, ranging from 10-6 m/s to 1 m/s, while plots (a), (b) and (c) 
+correspond to different cone lengths. 
+
+Velocity
+24.059
+18.044
+12.029
+6.015
+0.000
+[ms^-1]19:076
+14307
+4.769
+0.000
+[ms>-1]Reactionforcesforvariousflow-throughvelocities
+100
+ConeGeometry:Dia.=4cm, Len.=2cm
+ANSYSForce(Altitude:0km)
+FitC1=0.90,C2=0.99
+ANSYSForce (Altitude:40km)
+Fit:CI=1.42,C2=0.64
+ANSYSForce (Altitude:70km)
+Force (N)
+Fit:C1-1.48.C2-0.05
+10'5
+10-10
+10-6
+104
+102
+100
+Va(m/s)Reactionforcesforvariousflow-throughvelocities
+ConeGeometry:Dia.=4cm,Len.=5cm
+100
+ANSYSForce(Altitude:0km)
+Fit:CI=0.98,C20.98
+ANSYSForce(Altitude:40km)
+FitC=1.16,C2=0.73
+ANSYSForee (Altitude:70km)
+Fit:C1=1.24,C2=0.20
+10-5
+Force
+10-lo
+10-6
+104
+102
+100
+Va (m/s)Reactionforcesforvariousflow-throughvelocities
+ConeGeometry:Dia,=4 cm,Len,=10cm
+100
+ANSYSForce(Altitude:0km)
+Fit:C1-0.97,C2=0.95
+ANSYSForce (Altitude:40km)
+Fit:C1=1.11C2=0.83
+ANSYSForce(Altitude:70km)
+Fit:CI-1.19,C2-0.22
+10~5
+Force(
+10-lo
+106
+104
+102
+100Page 
+6 
+As expected, a jet of high-speed air exited the geometries as a result of the air flowing in through the porous 
+structures. Once the air left the geometry, it interacted with the walls of the outer air box by forming large 
+vortices, as anticipated for a fluid circulating in a contained box.  
+ 
+The next section of this document takes the force fitting parameters found from the ANSYS Fluent 
+simulations and focuses on MATLAB-based parametric optimization of our three different geometries. 
+ 
+ 
+2. MATLAB Code and Extension of Theoretical Framework 
+ 
+In this section of the supplementary information, we present the extension to 3D structures of the original 
+nanocardboard fluid mechanic theory developed by [R3]. The equations derived below were implemented 
+in a MATLAB code to perform a series of parametric studies that seek to optimize the geometric and porous 
+parameters of our three study geometries, a cone, a sphere and a rocket. More information about our code 
+can be found in our publicly available repository [R4].   
+ 
+2.1. Derivation of Equations 
+ 
+2.1.1 General Overview 
+ 
+For a general 3D porous structure, conservation of mass establishes that   
+ 
+ 
+𝐴𝑡𝑜𝑡𝑎𝑙𝑣𝑓𝑡 = 𝐴𝑜𝑢𝑡𝑣𝑜𝑢𝑡 . 
+(S2) 
+ 
+Here, 𝐴𝑡𝑜𝑡𝑎𝑙 represents the total surface area of the structure (as if the structure had no pores/channels) and 
+𝑣𝑓𝑡 is the flow-through velocity of the fluid across this surface. Similarly, 𝐴𝑜𝑢𝑡 corresponds to the area 
+covered by the outlet, while 𝑣𝑜𝑢𝑡 is the exit velocity of the fluid out of the structure. Adding Bernoulli’s 
+equation, we get the relationship that 
+ 
+ 
+𝑃𝑖𝑛 − 𝑃𝑜𝑢𝑡
+𝜌
+= ∆𝑃
+𝜌 = 𝑣𝑜𝑢𝑡
+2 − 𝑣𝑓𝑡
+2
+2
+ . 
+(S3) 
+ 
+In (S5), 𝑃𝑖𝑛 is the pressure right at the inlet of the structure, 𝑃𝑜𝑢𝑡 is the pressure right as the jet of fluid is 
+leaving the structure, located around the space close to 𝐴𝑜𝑢𝑡, while 𝜌 is the fluid density. This equation can 
+be rearranged to yield an expression for the pressure difference across both ends of the structure, resulting 
+in  
+ 
+ 
+∆𝑃 = 𝜌(𝑣𝑜𝑢𝑡
+2 − 𝑣𝑓𝑡
+2)
+2
+ . 
+(S4) 
+ 
+Assuming that the porosity of the 3D structure originates from using the nanocardboard geometry 
+developed by [R3] as the wall material, then we can model the mass flow rate of the fluid across one of the 
+structure’s pores (or more properly said, channels) using the following equation 
+ 
+ 
+𝑚̇ = −𝛼 ∗ ∆𝑃 +  𝛾 ∗ ∆𝑇 . 
+(S5) 
+ 
+In (S5), 𝛼 and 𝛾 represent two constants, which take the following forms1: 
+ 
+ 
+𝛼 = (𝛿
+6 + 1) (1 + 0.25
+√𝛿
+) 𝐴2𝐵𝛽∗
+𝐿
+ , 
+(S6) 
+ 
+and 
+ 
+𝛾 = (
+1.1
+1.5 + 𝛿) 𝐴2𝐵𝑃∗𝛽∗
+𝑇∗𝐿
+ . 
+(S7) 
+ 
+ 
+1 The variables 𝛼 and 𝛾 come from curve-fitting the data from by [R7] and transforming the non-dimensional flow rate equation into 
+a dimensional form again, with both pressure and temperature contributions. For more information, please see [R2]. 
+
+Page 
+7 
+Here, the variable 𝑃∗ denotes the average pressure2 between the two sides of the structure’s nanocardboard 
+wall,  𝑇∗ analogously describes the average temperature between both sides of the wall’s surface, while 𝛽∗ 
+is an inverse velocity parameter. Specifically, this last one is given by  
+ 
+ 
+𝛽∗ = √
+  𝑚
+2𝑘𝐵𝑇∗
+ , 
+(S8) 
+ 
+where 𝑘𝐵 is the Boltzmann constant (equal to 1.38 * 10-23 J/K), and m is the mass of an air molecule3. Lastly, 
+the parameter 𝛿 is the gas rarefaction coefficient, which [R7] defines as   
+ 
+ 
+𝛿 = √𝜋𝐴
+2𝜆 = √𝜋
+2𝐾𝑛 . 
+(S9) 
+ 
+In this expression, 𝜆 is the molecular mean free path, defined as the average distance traveled by a molecule 
+between collisions with other molecules, and Kn is the Knudsen number, which is characterized in terms 
+the of channel width. In essence, higher values of the 𝛿 parameter designates flows in the continuum regime, 
+while smaller values indicate flows taking place in the free molecular regime. As for the molecular mean 
+free path, mathematically it is usually expressed as 
+ 
+ 
+𝜆 = 𝜇(𝑇)
+𝑃(𝑇) √𝜋𝑘𝐵𝑇
+2𝑚 = 𝜇(𝑇)
+𝑃(𝑇) √𝜋𝑅𝑎𝑖𝑟𝑇
+2
+ , 
+(S10) 
+ 
+where 𝜇(𝑇) is the fluid’s viscosity and P(T) is the operating pressure, both given as a function of T, the 
+operating temperature. In addition, from equation (S9), we see the Knudsen number is defined as 
+ 
+ 
+𝐾𝑛 = 𝜆
+𝐴 . 
+(S11) 
+ 
+Additionally, as seen in Figure S9 below, the variables A and B characterize the nanocardboard channel’s 
+width and length, respectively, yielding a cross-sectional area of A x B. In addition, L denotes the channel’s 
+height. Note that in [R3], A is assumed to be much smaller than B. 
+ 
+After defining these variables and introducing the expression for the mass flow rate, 𝑚̇ , across one of 
+nanocardboard’s channels, then an equation can be derived for the average flow-through velocity across 
+the structure’s surface, which is simply described by 
+ 
+ 
+𝑣𝑓𝑡 = 𝜑𝑚̇
+𝜌𝐴𝐵 = 𝜑(−𝛼∆𝑃 +  𝛾∆𝑇)
+𝜌𝐴𝐵
+ . 
+(S12) 
+ 
+Here, 𝑚̇ /𝜌 is no other than the volumetric flow rate 𝑉̇ , while the term 𝜑 denotes the geometric fill factor, 
+which is defined in terms of 𝐴𝑖𝑛 (porous area) and 𝐴𝑡𝑜𝑡𝑎𝑙4, or the channel parameters, and takes the form 
+ 
+ 
+𝜑 =
+𝐴𝑖𝑛
+𝐴𝑡𝑜𝑡𝑎𝑙
+=
+𝐴𝐵𝑋
+(𝐴𝐵𝑋 + 𝑆𝐵𝑋) =
+𝐴
+(𝐴 + 𝑆) . 
+(S13) 
+ 
+The latter two equivalencies in (S13) originates from analyzing a single nanocardboard unit cell as opposed 
+to the full 3D structure. Indeed, as Figure S9 shows, the total cross-sectional area of the cell (if no channels 
+were present) is given by 
+ 
+ 
+𝐴𝑐𝑒𝑙𝑙 = (𝐴𝐵𝑋 + 𝑆𝐵𝑋) = (𝐴 + 𝑆)𝐵𝑋 , 
+(S14) 
+ 
+where the variable X is just the number of channels in a unit cell. 
+ 
+2 The value of this variable may be found from performing CFD simulations but will be simply approximated as the operating pressure. 
+3 The molar mass of air is 0.02896 kg/mol, so then the approximated mass of an air molecule would be 0.02896/(6.022*1023 ) 
+(Avogadro’s number), or 4.8089 * 10-26 kg.  
+4 This area is essentially the total 3D structure wall area if there were no channels present. This is analogous to 𝐴𝑐𝑒𝑙𝑙 in the single 
+nanocardboard unit cell. 
+
+Page 
+8 
+ 
+ 
+However, this number (X) is not arbitrarily chosen, and is dictated by A, B and S in the following way 
+ 
+ 
+𝑋 = 𝐵 − 𝑆
+𝑆 + 𝐴 . 
+(S15) 
+ 
+This expression considers the channel width A and spacing S as a unit, and tries to fit as many of those A 
++S units into the channel length B. Nonetheless, we need to consider an additional S for spacing against the 
+perpendicular channels. This can be seen more clearly in Figure S10 below, where the yellow bars represent 
+the A +S units, and as drawn, five of these fit in the length of B, after subtracting one S. 
+ 
+Overall, the flow-through velocity expression provided in (S12) is a step closer towards calculating the lift 
+force that a 3D structure could generate for a given combination of geometric and channel parameters. 
+However, computing lift will not be possible until we solve for 𝑣𝑜𝑢𝑡. Therefore, (S12) can be rearranged to 
+instead solve for another unknown, ∆𝑃 , and obtain  
+ 
+ 
+∆𝑃 = 𝛾∆𝑇
+𝛼
+− 𝑣𝑓𝑡𝜌𝐴𝐵
+𝛼𝜑
+ . 
+(S16) 
+ 
+Since both (S16) and (S4) from above provide two distinct expressions for the pressure difference, it is 
+possible to equate them, giving rise to yet another relationship between 𝑣𝑓𝑡 and 𝑣𝑜𝑢𝑡, giving 
+ 
+Figure S9: Main nanocardboard channel parameters. 
+Figure S10: Illustration of equation (S15), with the yellow bars showing the A + S units fitted into the channel length B. 
+
+Top
+Isometric
+View
+View
+Key
+Pi, T1
+Air
+A:ChannelWidth
+Trapped
+Side
+B:Channel Length
+A
+Air
+View
+S:Channel Spacing
+L: Channel Height
+P2, T2
+Air
+t: Alumina ThicknessTop
+View
+Key
+A: Channel Width
+B: Channel Length
+S:Channel SpacingPage 
+9 
+ 
+𝜌(𝑣𝑜𝑢𝑡
+2 − 𝑣𝑓𝑡
+2)
+2
+ = ∆𝑃 = 𝛾∆𝑇
+𝛼
+− 𝑣𝑓𝑡𝜌𝐴𝐵
+𝛼𝜑
+ . 
+(S17) 
+ 
+Rearranging this expression further, we get  
+ 
+ 
+𝑣𝑜𝑢𝑡
+2 = 2
+𝜌 (𝛾∆𝑇
+𝛼
+− 𝑣𝑓𝑡𝜌𝐴𝐵
+𝛼𝜑
+) + 𝑣𝑓𝑡
+2 . 
+(S18) 
+ 
+Now, recalling the conservation of mass relationship provided in (S2), it is possible to write 𝑣𝑓𝑡, the flow-
+through velocity across the channels, in terms of 𝑣𝑜𝑢𝑡 
+ 
+ 
+𝑣𝑓𝑡 = 𝐴𝑜𝑢𝑡
+𝐴𝑡𝑜𝑡𝑎𝑙
+𝑣𝑜𝑢𝑡 =  𝜑𝐴𝑜𝑢𝑡
+𝐴𝑖𝑛
+𝑣𝑜𝑢𝑡. 
+(S19) 
+ 
+Thus, (S19) can replace the 𝑣𝑓𝑡 term in (S18), leaving everything in terms of just 𝑣𝑜𝑢𝑡 
+ 
+ 
+𝑣𝑜𝑢𝑡
+2 = 2
+𝜌 (𝛾∆𝑇
+𝛼
+− 𝐴𝑜𝑢𝑡𝑣𝑜𝑢𝑡𝜌𝐴𝐵
+𝐴𝑡𝑜𝑡𝑎𝑙𝛼𝜑
+) + ( 𝐴𝑜𝑢𝑡
+𝐴𝑡𝑜𝑡𝑎𝑙
+)
+2
+𝑣𝑜𝑢𝑡
+2. 
+(S20) 
+ 
+Further manipulating (S20), we get the following quadratic  
+ 
+ 
+𝑣𝑜𝑢𝑡
+2 (1− ( 𝐴𝑜𝑢𝑡
+𝐴𝑡𝑜𝑡𝑎𝑙
+)
+2
+) + 𝑣𝑜𝑢𝑡 (2𝐴𝑜𝑢𝑡𝐴𝐵
+𝐴𝑡𝑜𝑡𝑎𝑙𝛼𝜑) − 2𝛾∆𝑇
+𝜌𝛼
+= 0 , 
+(S21) 
+ 
+which has precisely 𝑣𝑜𝑢𝑡 as its only unknown. The coefficients of this polynomial are 
+ 
+ 
+𝑎 = 1− ( 𝐴𝑜𝑢𝑡
+𝐴𝑡𝑜𝑡𝑎𝑙
+)
+2
+,
+���� = 2𝐴𝑜𝑢𝑡𝐴𝐵
+𝐴𝑡𝑜𝑡𝑎𝑙𝛼𝜑 ,
+𝑐 = − 2𝛾∆𝑇
+𝜌𝛼  , 
+(S22) 
+ 
+making it a fairly straightforward process to solve for the roots of the equation, provided by 
+ 
+ 
+𝑣𝑜𝑢𝑡 =
+− (2𝐴𝑜𝑢𝑡𝐴𝐵
+𝐴𝑡𝑜𝑡𝑎𝑙𝛼𝜑) ± √(2𝐴𝑜𝑢𝑡𝐴𝐵
+𝐴𝑡𝑜𝑡𝑎𝑙𝛼𝜑)
+2
++ 8𝛾∆𝑇
+𝜌𝛼 (1− ( 𝐴𝑜𝑢𝑡
+𝐴𝑡𝑜𝑡𝑎𝑙)
+2
+)
+ 2 (1− ( 𝐴𝑜𝑢𝑡
+𝐴𝑡𝑜𝑡𝑎𝑙)
+2
+)
+ . 
+(S23) 
+ 
+One underlying advantage of this derivation was that it removed the need to know the pressure difference, 
+∆𝑃, while providing us with enough information to solve for 𝑣𝑜𝑢𝑡 and 𝑣𝑓𝑡. In the following sub-section, we 
+deliver more details on the heat conduction modeling across the nanocardboard’s thickness, which enabled 
+obtaining an expression for the temperature difference, ∆𝑇, necessary to solve for 𝑣𝑜𝑢𝑡 in (S23). 
+ 
+2.1.2 Heat Conduction Modeling 
+ 
+2.1.2.1 Full Analytical Derivation for ∆𝑻 
+ 
+In order to compute ∆𝑇, the temperature difference between both sides of the structure’s walls, we needed 
+to model the heat conduction across the structure’s thickness. We performed a heat energy balance that 
+considered heat transfer across three distinct cross-sectional areas: the channel’s column of air, across the 
+alumina thickness of the channel, and across the air trapped within the structure, as shown in Figure S11 
+below. As a result, we can let 𝑄𝑡, the total heat transfer, be 
+ 
+ 
+𝑄𝑡 = ∆𝑇
+𝑅𝑡1
++ ∆𝑇
+𝑅𝑡2
++ ∆𝑇
+𝑅𝑡3
+ , 
+(S24) 
+ 
+where the 𝑅𝑡1, 𝑅𝑡2 and 𝑅𝑡3 represent the thermal resistances under the three scenarios detailed above.  
+
+Page 10 
+ 
+For the first of these areas (A1), the column of air in the channel, we define its thermal resistance as 
+ 
+ 
+𝑅𝑡1 =
+𝐿
+𝑘𝑎𝑖𝑟𝐴1𝑋 =
+𝐿
+𝑘𝑎𝑖𝑟𝐴𝐵𝑋 , 
+(S25) 
+ 
+where 𝑘𝑎𝑖𝑟 is the thermal conductivity of air, L is as usual the channel height, and 𝐴𝐵𝑋 is the cross-sectional 
+area of the individual channels multiplied by the number of channels in a unit cell, as shown in Figure S9 
+above. Notice that  𝜅𝑎𝑖𝑟 is both temperature and pressure dependent, as the equation developed by [R10] 
+captures, specifically for the small MEMS scale: 
+ 
+ 
+𝜅𝑎𝑖𝑟 =
+𝜅0
+(1 + 0.00076𝑇
+𝑃𝐿
+)
+ . 
+(S26) 
+ 
+In this expression, 𝜅0 is the air conductivity at standard atmospheric conditions, normally quoted as 𝜅0 =
+0.024
+𝑊
+𝑚 𝐾. Another comparable and slightly more succinct model for the conductivity of air is from [R8]:  
+ 
+ 
+𝜅𝑎𝑖𝑟 =
+𝜅0
+(1 + 3.116𝜆
+𝐿
+)
+ 
+(S27) 
+ 
+As the pressure decrease, the mean free path eventually becomes comparable to the channel length, and the 
+effective conductivity starts to decrease below the continuum value. Both equations (S26) and (S27) yielded 
+very similar values for the conductivity of air as a function of the channel thickness L, although we used 
+Eq. S27 in the simulations.  
+ 
+Continuing with the heat conduction modeling, the corresponding expression for the thermal resistance 
+across the alumina thickness on the channels (area A2 in Figure S11) is given by 
+ 
+ 
+𝑅𝑡2 =
+𝐿
+𝑘𝑎𝑙𝑑𝐴2𝑋  =
+𝐿
+𝑘𝑎𝑙𝑑[(𝐴 + 2𝑡)(𝐵 + 2𝑡) − 𝐴𝐵]𝑋 , 
+(S28) 
+ 
+where [(𝐴 + 2𝑡)(𝐵 + 2𝑡) − 𝐴𝐵]𝑋 is the cross-sectional area occupied by the alumina thickness of the 
+channels, which is denoted as 𝑡. In (S28), 𝑘𝑎𝑙𝑑 is the thermal conductivity of alumina, which has a constant 
+value of 1.8
+𝑊
+𝑚 𝐾 [R2]. Lastly, the thermal resistance of the air trapped within the structure (area A3) is  
+ 
+ 
+𝑅𝑡3 = 𝐿 − 2𝑡
+𝑘𝑎𝑖𝑟𝐴3
+=
+𝐿 − 2𝑡
+𝑘𝑎𝑖𝑟 [𝐴𝐵
+𝜑 − (𝐴 + 2𝑡)(𝐵 + 2𝑡)] 𝑋
+ , 
+(S29) 
+ 
+Figure S11: Main nanocardboard cross-sectional areas for which thermal resistance is calculated. 
+
+Isometric
+View
+Top
+View
+Key
+Ai:Channelcross-sectionalarea
+A2:ChannelAluminathicknesscross-sectionalarea
+SectionCut
+As:Cross-sectional area oftrappedairwithinnanocardboardPage 11 
+where recall from (S13) that  
+𝐴𝐵𝑋
+𝜑  is the full area of the cell, from which we subtract the combined cross-
+sectional area of the channels with thickness 𝑡 of alumina. Now, performing an energy balance, the heat 
+flow through the structure’s walls must be equal to that from the absorbed irradiation of the sun, which in 
+this case is given by 
+ 
+𝑄𝑡  = 𝜀𝜓𝐼𝑠𝑢𝑛 (𝐴𝐵𝑋
+𝜑 ) (1 −  𝜑) . 
+(S30) 
+ 
+In equation (S30), 𝜀 denotes the absorption coefficient (approximated to 0.9 based-off the measurements 
+from [R3]), 𝜓 the proportion of absorbed optical flux dissipated upward through the nanocardboard (which 
+is assumed to be 0.5 or 50%), and 𝐼𝑠𝑢𝑛  the intensity of the sun at a particular altitude. In particular, this last 
+term can be modeled using the following equation 
+ 
+ 
+𝐼𝑠𝑢𝑛   = 1000 +  3.8ℎ , 
+(S31) 
+ 
+where the variable h refers to the elevation above sea level in kilometers. Notice that this expression returns 
+the sun’s intensity in units of Watts per meter square. Furthermore, in equation (S30),  
+(𝐴𝐵𝑋/𝜑)(1 −  𝜑) corresponds to the solid area of the nanocardboard, 𝐴𝑠𝑜𝑙𝑖𝑑, where the sun’s irradiation 
+is absorbed. In any case, (S24) through (S31) were combined to write a general expression for ∆𝑇, which 
+is summarized by 
+ 
+ 
+∆𝑇 = 𝑇2 − 𝑇1 =
+𝜀𝜓𝐼𝑠𝑢𝑛 (𝐴𝐵𝑋
+𝜑 ) (1 −  𝜑)
+1
+𝑅𝑡1 + 1
+𝑅𝑡2 + 1
+𝑅𝑡3
+= 
+ 
+=
+𝜀𝜓𝐼𝑠𝑢𝑛 (𝐴𝐵𝑋
+𝜑 ) (1 −  𝜑)
+𝑘𝑎𝑖𝑟𝐴𝐵𝑋
+𝐿
++ 𝑘𝑎𝑙𝑑[(𝐴 + 2𝑡)(𝐵 + 2𝑡) − 𝐴𝐵]𝑋
+𝐿
++
+𝑘𝑎𝑖𝑟 [𝐴𝐵
+𝜑 − (𝐴 + 2𝑡)(𝐵 + 2𝑡)] 𝑋
+𝐿 − 2𝑡
+ . 
+ 
+(S32) 
+ 
+In (S32), 𝑇1 and 𝑇2 represent the average temperatures outside and inside the 3D structure, respectively. 
+However, these might not necessarily be known beforehand, reason why calculating ∆𝑇 or  𝑇∗, the average 
+temperature between both sides of the surface, may not be as trivial. In particular, to compute  𝑇∗, we make 
+use of the fact that we know what ∆𝑇 is from (S32) and take the following expression  
+ 
+ 
+𝑇∗ = 𝑇1 + 𝑇2
+2
+= (𝑇2 − 𝑇1) + 2 ∗ 𝑇1
+2
+= ∆𝑇 + 2 ∗ 𝑇1
+2
+ . 
+(S33) 
+ 
+Here, notice that 𝑇1 is simply equal to the temperature corresponding to the particular operating conditions 
+(altitude, pressure, density) of the fluid. Overall, ∆𝑇 allows us to solve for 𝑇∗ (which is needed to compute 
+𝛾 and 𝛽∗ in (S7) and (S9), respectively) and the last part of the puzzle in (S23), the 𝑣𝑜𝑢𝑡 expression. 
+  
+2.1.2.2 Simplified Expression for ∆𝑻 in the limit of zero alumina thickness 
+ 
+Beyond the derivation provided in 1.2.1, notice that one could potentially also approximate ∆𝑇 through a 
+more simplified expression given by 
+ 
+ 
+∆𝑇~ 𝐿𝐼𝑠𝑢𝑛(1 −  𝜑)
+2𝜅𝑎𝑖𝑟
+ . 
+(S34) 
+ 
+The origin of (S34) comes from taking the limit as t, the alumina thickness, approaches zero, in equation 
+(S32). Indeed,  
+ 
+ 
+lim
+𝑡→0
+𝜀𝜓𝐼𝑠𝑢𝑛 (𝐴𝐵𝑋
+𝜑 ) (1 −  𝜑)
+𝑘𝑎𝑖𝑟𝐴𝐵𝑋
+𝐿
++ 𝑘𝑎𝑙𝑑[(𝐴 + 2𝑡)(𝐵 + 2𝑡) − 𝐴𝐵]𝑋
+𝐿
++
+𝑘𝑎𝑖𝑟 [𝐴𝐵
+𝜑 − (𝐴 + 2𝑡)(𝐵 + 2𝑡)] 𝑋
+𝐿 − 2𝑡
+ 
+ 
+(S35) 
+
+Page 12 
+= lim
+𝑡→0
+𝐿𝜀𝜓𝐼𝑠𝑢𝑛 (𝐴𝐵𝑋
+𝜑 ) (1 −  𝜑)
+𝑘𝑎𝑖𝑟𝐴𝐵𝑋 + 𝑘𝑎𝑙𝑑[(𝐴 + 2𝑡)(𝐵 + 2𝑡) − 𝐴𝐵]𝑋 + 𝑘𝑎𝑖𝑟 [𝐴𝐵
+𝜑 − (𝐴 + 2𝑡)(𝐵 + 2𝑡)] 𝑋
+ 
+ 
+= lim
+𝑡→0
+𝐿𝜀𝜓𝐼𝑠𝑢𝑛 (𝐴𝐵𝑋
+𝜑 ) (1 −  𝜑)
+𝑘𝑎𝑖𝑟𝐴𝐵𝑋 + 𝑘𝑎𝑙𝑑[𝐴𝐵 − 𝐴𝐵]𝑋 + 𝑘𝑎𝑖𝑟 [𝐴𝐵
+𝜑 − 𝐴𝐵] 𝑋
+ 
+ 
+= lim
+𝑡→0
+𝐿𝜀𝜓𝐼𝑠𝑢𝑛 (𝐴𝐵𝑋
+𝜑 ) (1 −  𝜑)
+𝑘𝑎𝑖𝑟𝐴𝐵𝑋 + 𝑘𝑎𝑖𝑟 𝐴𝐵𝑋
+𝜑
+− 𝑘𝑎𝑖𝑟𝐴𝐵𝑋
+=
+𝐿𝜀𝜓𝐼𝑠𝑢𝑛 (𝐴𝐵𝑋
+𝜑 ) (1 −  𝜑)
+𝑘𝑎𝑖𝑟 𝐴𝐵𝑋
+𝜑
+ 
+ 
+= 𝐿𝜀𝜓𝐼𝑠𝑢𝑛(1 −  𝜑)
+𝑘𝑎𝑖𝑟
+ . 
+ 
+ 
+Furthermore, letting 𝜀 = 1 and 𝜓 = 0.5, then (S37) indeed becomes equation (S34) from above. As 
+evidenced by its compressed form, using (S36) to approximate ∆𝑇 simplifies the process of solving for the 
+flow-through velocity, 𝑣𝑓𝑡. This is especially true if we were to also neglect the pressure term, assuming its 
+contribution is negligible. As a result, the mass flow rate from (S5) can be re-written as 
+ 
+ 
+𝑚̇ ~𝛾 ∗ ∆𝑇 . 
+(S36) 
+ 
+This helps reduce the flow-through velocity expression to 
+ 
+ 
+𝑣𝑓𝑡 = 𝜑𝑚̇
+𝜌𝐴𝐵 = 𝜑 𝛾∆𝑇
+𝜌𝐴𝐵 = 𝜑 𝛾
+𝜌𝐴𝐵
+𝐿𝐼𝑠𝑢𝑛(1 −  𝜑)
+2𝜅𝑎𝑖𝑟
+ . 
+(S37) 
+ 
+Even this expression can be further simplified by reducing the 𝛾 term from (S7) to  
+ 
+ 
+𝛾~ 1.1𝐴2𝐵𝑃∗𝛽∗
+𝛿𝑇∗𝐿
+= 1.1𝐴2𝐵𝑃𝛽∗
+𝑇𝐿𝐴√𝜋/(2𝜆) = 2.2𝜆𝐴𝐵𝑃
+√𝜋𝑇𝐿
+√
+  𝑚
+2𝑘𝐵𝑇 . 
+(S38) 
+ 
+From the ideal gas law, we have that 𝑃 = 𝜌𝑅𝑎𝑖𝑟𝑇, so the pressure term can be replaced in (S38) to obtain 
+ 
+ 
+𝛾~ 2.2𝜆𝐴𝐵𝜌𝑅𝑎𝑖𝑟𝑇
+√𝜋𝑇𝐿
+√
+  𝑚
+2𝑘𝐵𝑇 = 2.2𝜆𝐴𝐵𝜌𝑅𝑎𝑖𝑟
+√𝜋𝐿
+√
+  𝑚
+2𝑘𝐵𝑇 . 
+(S39) 
+ 
+Combining equations (S37) and (S39), we resultant expression turns out as 
+ 
+ 
+𝑣𝑓𝑡 = 𝜑 
+𝜌𝐴𝐵
+𝐿𝐼(1 −  𝜑)
+2𝜅𝑎𝑖𝑟
+2.2𝜆𝐴𝐵𝜌𝑅���𝑖𝑟
+√𝜋𝐿
+√
+  𝑚
+2𝑘𝐵𝑇 = 1.1𝜑𝐼(1 −  𝜑)𝜆𝑅𝑎𝑖𝑟
+𝜅𝑎𝑖𝑟
+√
+  𝑚
+2𝑘𝐵𝑇𝜋 
+(S40) 
+ 
+Now, recall that the average molecular velocity is equal to 
+ 
+ 
+𝑣𝑎𝑣𝑔 = √8𝑅𝑎𝑖𝑟𝑇
+𝜋
+ , 
+(S41) 
+ 
+and the relationship between viscosity and velocity, as provided by [R6], is equal to  
+ 
+ 
+𝜇 = 𝜆𝜌𝑣𝑎𝑣𝑔
+2
+ . 
+(S42) 
+ 
+
+Page 13 
+Hence, combining both (S41) and (S42) and solving for 𝜆, we obtain an expression which can be 
+incorporated in (S40) to yield 
+ 
+ 
+𝑣𝑓𝑡 = 1.1𝜑𝐼(1 −  𝜑)𝑅𝑎𝑖𝑟
+𝜅𝑎𝑖𝑟
+𝜇
+𝑃 √𝜋𝑘𝐵𝑇
+2𝑚 √
+  𝑚
+2𝑘𝐵𝑇𝜋 = 1.1𝜑𝐼(1 −  𝜑)𝑅𝑎𝑖𝑟
+𝜅𝑎𝑖𝑟
+𝜇
+𝑃 √  𝑚𝜋𝑘𝐵𝑇
+4𝑘𝐵𝑇𝜋𝑚 
+ 
+= 1.1𝜑𝐼(1 −  𝜑)𝑅𝑎𝑖𝑟
+𝜅𝑎𝑖𝑟
+𝜇
+𝑃 √ 1
+4 = 1.1𝜑𝐼(1 −  𝜑)𝑅𝑎𝑖𝑟
+2𝜅𝑎𝑖𝑟
+𝜇
+𝑃 . 
+(S43) 
+ 
+Now, according to [R6], the conductivity of air can be often approximated as 𝜅𝑎𝑖𝑟 =
+2𝜇𝐶𝑣′
+𝑀
+= 2𝜇𝐶𝑣, where 
+M is the molar mass of air and 𝐶𝑣′ is the specific heat capacity at constant volume, in units of J/k mol. Thus, 
+equation (S43) can further simplify into  
+  
+ 
+𝑣𝑓𝑡 = 1.1𝜑𝐼(1 −  𝜑)𝑀𝑅𝑎𝑖𝑟
+4𝜇𝐶𝑣′
+𝜇
+𝑃 = 1.1𝜑𝐼(1 −  𝜑)𝑅𝑎𝑖𝑟
+4𝑃𝐶𝑣
+= 1.1𝜑(1 −  𝜑)𝑅𝑎𝑖𝑟
+4𝐶𝑣
+𝐼
+𝑃 = 𝐶 𝐼
+𝑃 . 
+(S44) 
+ 
+where the constant C is simply given by 
+ 
+ 
+𝐶 = 1.1𝜑(1 −  𝜑)𝑅𝑎𝑖𝑟
+4𝐶𝑣
+= 1.1 ∗ 0.5 ∗ (1 −  0.5) ∗ 0.287
+4 ∗ 0.718
+= 0.0275 . 
+(S45) 
+ 
+Hence, what these series of derivations shows is that it is possible to approximate and obtain order-of-
+magnitude estimations of the flow-through velocity by using 
+ 
+ 
+𝑣𝑓𝑡 = 0.0275 𝐼
+𝑃 . 
+(S46) 
+ 
+2.2. Lift-Force Calculations and Temperature-dependencies 
+ 
+Once we knew how to calculate 𝑣𝑓𝑡 and 𝑣𝑜𝑢𝑡 using the equations derived above (whether it is in the 
+simplified or full analytical form), we used the following equation to calculate the lift forces produced by 
+each geometry, as outlined in the ANSYS simulations section at the beginning of this document: 
+ 
+ 
+∑𝐹 = 𝐶1 ∗ 8 ∗ 𝜇 ∗ 𝐷 ∗ 𝑣𝑓𝑡 + 𝐶2 ∗ 𝜌 ∗ 𝐴𝑜𝑢𝑡 ∗ 𝑣𝑜𝑢𝑡
+2 . 
+(S47) 
+ 
+Here, R is the characteristic radius of the geometry (usually the inlet radius), while 𝜇 is the viscosity and 𝜌 
+the fluid density. In addition, C1 and C2 are the geometry dependent coefficients summarized in Table 6. 
+ 
+As the derivation of equations above evidences, all of the geometric (𝐴𝑡𝑜𝑡𝑎𝑙 and 𝐴𝑜𝑢𝑡) and channel (A, B, 
+L, S, t) variables are present in (S23), meaning that it was possible to construct parametric studies exploring 
+the dependency of 𝑣𝑓𝑡, and consequentially lift, on all of these. Notice, all of these variables were largely 
+independent of each other, making it possible to modify each separately. However, some other parameters 
+within (S23), such as 𝐼𝑠𝑢𝑛,  density 𝜌, and air viscosity 𝜇, were actually dependent on temperature, which 
+in turn was also altitude dependent. As a result, in order to accurately calculate the flow-through velocities 
+𝑣𝑓𝑡  experienced by a 3D geometry in a range of altitudes, we needed to derive expressions for 
+approximating the air temperature, air pressure, air viscosity and air density as a function of altitude itself.  
+ 
+2.2.1 Temperature-dependent Relations 
+ 
+We developed the relations characterizing the dependency between temperature and the fluid variable in 
+question by using standard atmospheric5 empirical data and fitting equations to it. For instance, for the data 
+describing the dependency between air temperature and altitude, we fit both a 6th, 10th and 15th order 
+ 
+5 The specific standard atmospheric data was taken from the following three websites: 
+https://www.engineeringtoolbox.com/standard-atmosphere-d_604.html | https://www.pdas.com/atmosTable1SI.html 
+https://www.pdas.com/bigtables.html 
+
+Page 14 
+polynomial, as Figure S12 to the below shows. Overall, the 15th order polynomial provided the best 
+empirical fit, which was why we decided to use it for the rest of this analysis. However, one interesting 
+aspect of this fit was that we actually fitted at the inverse of the temperature, the reason for which will 
+become clearer in the derivation of the altitude-pressure dependency. In any case, equation (S48) below 
+shows this explicit relation, with h (the altitude) being in kilometers, and all terms in the column added.  
+ 
+ 
+𝑇−1(ℎ) =
+−4.592 ∗ 10−29
+4.023 ∗ 10−27
+1.491 ∗ 10−23
+−7.942 ∗ 10−21
+2.021 ∗ 10−18
+−3.152 ∗ 10−16
+3.271 ∗ 10−14
+−2.332 ∗ 10−12
+1.150 ∗ 10−10
+−3.862 ∗ 10−09
+8.525 ∗ 10−08
+−1.150 ∗ 10−06
+8.154 ∗ 10−06
+−2.283 ∗ 10−05
+9.912 ∗ 10−05
+3.473 ∗ 10−03
+∙
+ℎ15
+ℎ14
+ℎ13
+ℎ12
+ℎ11
+ℎ10
+ℎ9
+ℎ8
+ℎ7
+ℎ6
+ℎ5
+ℎ4
+ℎ3
+ℎ2
+ℎ1
+1 .
+ 
+(S48) 
+ 
+Having derived the empirical relation between temperature (its inverse) and altitude, it was possible to 
+determine a similar expression for pressure. In essence, the differential equation describing the pressure-
+altitude relationship is given by 
+ 
+ 
+𝑑𝑃(ℎ) = −𝑔 ∗ 𝜌(ℎ) ∗ 𝑑ℎ , 
+(S49) 
+ 
+where 𝑔 is the gravitational constant on earth, and 𝜌(ℎ) the density of air at a particular altitude h. Using 
+the ideal gas law, 𝜌(ℎ) can be substituted to yield the following expression for the above differential in 
+equation (S49) 
+ 
+ 
+𝑑𝑃(ℎ) = −𝑔 ∗
+𝑃(ℎ)
+𝑅𝑎𝑖𝑟 ∗ 𝑇(ℎ) ∗ 𝑑ℎ , 
+(S50) 
+ 
+where now 𝑅𝑎𝑖𝑟 is the ideal gas constant of air and is equal to 287 𝐽/𝑘𝑔 ∗ 𝑚3.  Easily enough, one can 
+utilize the technique of separation of variables to obtain that 
+ 
+ 
+𝑑𝑃(ℎ)
+𝑃(ℎ) =
+−𝑔
+𝑅𝑎𝑖𝑟 ∗ 𝑇(ℎ) ∗ 𝑑ℎ , 
+(S51) 
+ 
+which leaves all of the pressure terms on one side, and the rest on the other. As a result, it is possible to see 
+with more clarity why the above polynomial fit was done for the inverse of temperature. Indeed, equation 
+(S51) can be equivalently written as 
+ 
+ 
+𝑑𝑃(ℎ)
+𝑃(ℎ) = −𝑔 ∗ 𝑇−1(ℎ)
+𝑅𝑎𝑖𝑟
+∗ 𝑑ℎ . 
+(S52) 
+ 
+This expression can be easily integrated to obtain the following logarithm: 
+ 
+ 
+ln(𝑃) = −𝑔
+𝑅𝑎𝑖𝑟
+∗ ∫ 𝑇−1(ℎ) ∗ 𝑑ℎ . 
+(S53) 
+ 
+Letting 𝜁(ℎ) = ∫ 𝑇−1(ℎ) ∗ 𝑑ℎ be a placeholder for the integral of the inverse temperature polynomial and 
+C be simply a constant of integration, we obtain that 
+ 
+ 
+ln(𝑃(ℎ)) = −𝑔
+𝑅𝑎𝑖𝑟
+∗ 𝜁(ℎ) + 𝐶 . 
+(S54) 
+ 
+Figure S12: Modeled temperature dependency on altitude. 
+
+Fitsto altitude-dependentT
+5.5+10~3
+5
+4.5
+4
+3.5
+Data
+3
+Order6polynomial
+Order10polynomial
+Order15polynomial
+2.5
+0
+20
+40
+60
+80
+100
+120
+A/titude(km)Page 15 
+Figure S13: Modeled pressure dependency on altitude. 
+Now, in order to remove the logarithm from the pressure, we can raise both sides of the expression to the 
+Euler’s number power, and get 
+ 
+𝑃(ℎ) = 𝑒
+−𝑔
+𝑅𝑎𝑖𝑟∗𝜁(ℎ)+𝐶 . 
+(S55) 
+ 
+After applying exponent rules, (S55) decomposes into the product given by 
+ 
+ 
+𝑃(ℎ) = 𝑒𝐶 ∗ 𝑒
+−𝑔
+𝑅𝑎𝑖𝑟∗𝜁(ℎ) , 
+(S56) 
+ 
+and can be further simplified, upon application of boundary conditions, into  
+ 
+ 
+𝑃(ℎ) = 101300 𝑃𝑎 ∗ 𝑒
+−𝑔
+𝑅𝑎𝑖𝑟∗𝜁(ℎ) , 
+   
+(S57) 
+ 
+which takes the following full form: 
+ 
+𝑃(ℎ) = 101300 𝑃𝑎 ∗ exp 
+[
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+−𝑔
+𝑅𝑎𝑖𝑟
+∗
+(
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+−2.870 ∗ 10−30
+2.682 ∗ 10−28
+1.064 ∗ 10−24
+−6.109 ∗ 10−22
+1.684 ∗ 10−19
+−2.865 ∗ 10−17
+3.271 ∗ 10−15
+−2.591 ∗ 10−13
+1.437 ∗ 10−11
+−5.518 ∗ 10−10
+1.421 ∗ 10−08
+−2.299 ∗ 10−07
+2.0385 ∗ 10−06
+−7.608 ∗ 10−06
+4.955 ∗ 10−05
+3.473 ∗ 10−03
+∙
+ℎ16
+ℎ15
+ℎ14
+ℎ13
+ℎ12
+ℎ11
+ℎ10
+ℎ9
+ℎ8
+ℎ7
+ℎ6
+ℎ5
+ℎ4
+ℎ3
+ℎ2
+ℎ1 )
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+]
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ . 
+(S58) 
+ 
+ 
+ 
+As Figure S13 above shows, the agreement of this equation with the empirical data is very reasonable, 
+especially below 80 km altitude. Above 80 km, the atmosphere is no longer well mixed, has increasing 
+concentrations of atomic oxygen, and the simple ideal gas law we used above no longer applies. For this 
+reason, the results that will be presented below correspond to altitudes below 80 km. 
+  
+The next step was modelling the air density dependency on altitude. With expressions for T(h) and P(h) 
+above, we could use the ideal gas law to write 
+ 
+ 
+Finally, the last dependency that remained to be defined was the air viscosity and altitude relation. To that 
+end, we could use Sutherland’s law, which relates viscosity and temperature through the following equation: 
+ 
+ 
+𝜇(ℎ) = 𝜇𝑟𝑒𝑓 ∗ (𝑇(ℎ)
+𝑇𝑟𝑒𝑓
+)
+1.5
+∗ (
+𝑇𝑟𝑒𝑓 + 𝑆
+𝑇(ℎ) + 𝑆) , 
+(S60) 
+ 
+where 𝜇𝑟𝑒𝑓 is the reference dynamic viscosity and 𝑇𝑟𝑒𝑓 the reference temperature. In this work, for air, at 
+𝑇𝑟𝑒𝑓 = 20 𝐶, we have that 𝜇𝑟𝑒𝑓 = 0.000018205 𝑃𝑎 ∗ 𝑠. Finally, S is a constant, known as Sutherland’s 
+temperature, which is given by 110.4 K. 
+ 
+ 
+ 
+ 
+ 
+ 
+𝜌(ℎ) =
+𝑃(ℎ)
+𝑅𝑎𝑖𝑟 ∗ 𝑇(ℎ) . 
+(S59) 
+
+DerivedEquationforPressure
+106
+Data
+DerivedEquation
+104
+(Pa)
+Pressure
+100
+102
+104
+0
+20
+40
+60
+80
+100
+120
+A/titude (km)Page 16 
+2.2.2 Payload Calculations 
+ 
+Once all of the required equations and relationships were derived, it was possible to calculate 𝑣𝑓𝑡 and 𝑣𝑜𝑢𝑡 
+for a specific set of geometric and channel parameters defining unique 3D structures. By calculating these 
+velocities, we determined the total force produced by each geometry, as outlined by equation (S47), from 
+which it was possible to perform some payload estimates. However, in order to obtain the payload estimates, 
+it was paramount to first determine the surface areas of each one of the 3D geometries in question, the 
+reason being that density of these structure was defined in areal terms as opposed to volumetric terms. As 
+was mentioned in the main paper, this work considered a truncated cone, truncated sphere, and a rocket, 
+and their defining equations are shown in Table 7 below.  
+ 
+Main Geometrical Area Definitions 
+Area 
+Truncated Cone 
+Truncated Sphere 
+Rocket 
+𝐴𝑡𝑜𝑡𝑎𝑙 
+𝜋 (𝐷
+2)
+2
++ 𝜋 (𝐷
+2)ℎ2 − 𝜋𝑟(ℎ2 − ℎ1) 
+𝜋(𝐷2 − 2𝑟ℎ) 
+2𝜋𝑟(𝑟 + 𝐷) 
+𝐴𝑜𝑢𝑡 
+𝜋𝑟2 
+𝐴𝑖𝑛 
+𝜑𝐴𝑡𝑜𝑡𝑎𝑙 
+𝐴𝑠𝑜𝑙𝑖𝑑 
+(1 − 𝜑)𝐴𝑡𝑜𝑡𝑎𝑙 
+Special 
+Variables 
+ℎ1 = √(𝐷
+2 − 𝑟)
+2
++ 𝐷2 
+ℎ2 = √(𝐷
+2)
+2
++ ℎ3 
+ℎ3 =
+𝐷2
+(𝐷 − 2𝑟) 
+ℎ = (𝐷
+2) − √(𝐷
+2)
+2
+− 𝑟2 
+N/A 
+ 
+Table 7: Area definitions used across this work for the cone, sphere and rocket. Notice that here, the variable ℎ3 
+follows from using similar triangles analysis, and letting ℎ3/(D/2) = D/(D/2 – r). For all three geometries, the variable 
+D represents the overall scale of the structure while r their outlet radius. Notice that 𝐴𝑖𝑛 is the porous area, while 
+𝐴𝑠𝑜𝑙𝑖𝑑 is the solid area in which the sun’s irradiance is absorbed, and it follows that 𝐴𝑡𝑜𝑡𝑎𝑙 = 𝐴𝑠𝑜𝑙𝑖𝑑 + 𝐴𝑖𝑛. 
+ 
+As a result, having defined these surface areas (using the parameters established in Figure S1), we 
+calculated the mass of our three 3D structures. In particular, since the cross-sectional area of a channel is 
+simply 𝐴𝐵, then one can define the number of channels as the following integer floor: 
+ 
+ 
+𝑛𝑐ℎ𝑎𝑛𝑛𝑒𝑙𝑠 = ⌊𝐴𝑖𝑛
+𝐴𝐵⌋ . 
+(S61) 
+ 
+The number of channels, 𝑛𝑐ℎ𝑎𝑛𝑛𝑒𝑙𝑠, is an important parameter, given that now it is possible to calculate the 
+volume of the structure that is occupied by the deposited alumina around each channel, which has thickness 
+t and relatively high density 𝜌𝑎𝑙𝑑 of 3950 kg/m3 [R9]. Indeed, similarly to equation (S28) above, we can 
+define this volume as 
+ 
+ 
+𝑉𝑎𝑙𝑑,𝑐ℎ𝑎𝑛𝑛𝑒𝑙𝑠 = 𝑛𝑐ℎ𝑎𝑛𝑛𝑒𝑙𝑠(𝐿 − 2𝑡)[(𝐴 + 2𝑡)(𝐵 + 2𝑡) − 𝐴𝐵] . 
+(S62) 
+ 
+Experimentally, it has already been found that the areal density of nanocardboard, 𝜎𝑛𝑐𝑏, is about 1 g/m2 
+[R5], but this corresponds to a value of L (nanocardboard thickness) equal to 50 μm.  However, in our 
+parametric studies, as we sweep through various values of L, especially those that are larger than 50 μm, 
+this areal density alone is not enough to estimate the weight of the structure. As a result, calculating the 
+volume of alumina around each of the channels is paramount, since the structure naturally becomes heavier 
+with increasing thickness. Hence, the overall mass of any one of these geometries will be given by 
+ 
+ 
+𝑚𝑔𝑒𝑜𝑚𝑒𝑡𝑟𝑦 = 𝜎𝑔𝑒𝑜𝑚(𝐴𝑠𝑜𝑙𝑖𝑑 − 𝐴𝑖𝑛) + 𝜌𝑎𝑙𝑑𝑉𝑎𝑙𝑑,𝑐ℎ𝑎𝑛𝑛𝑒𝑙𝑠 , 
+(S63) 
+ 
+where this expression accounts both for the areal density (𝜎𝑔𝑒𝑜𝑚) and the increases in the amount of the 
+deposited alumina as a result of changes in the wall thickness L. Thus, the net lift produced by the geometry 
+is simply given by subtracting the structure’s weight from the force expression in (S47), or  
+ 
+
+Page 17 
+ 
+𝐿𝑖𝑓𝑡𝑛𝑒𝑡 = 𝐹 − 𝑔𝑚𝑔𝑒𝑜𝑚𝑒𝑡𝑟𝑦 . 
+(S64) 
+ 
+While we know from simulations what 𝜎𝑔𝑒𝑜𝑚 is, notice that it is also possible to use our equations and a 
+series of approximations to obtain a theoretical upper bound for this value. In essence, we can start by 
+letting the force be equal to the expression below 
+ 
+ 
+𝐹 = 𝑚̇ 𝑣𝑜𝑢𝑡 = (𝐴𝑖𝑛𝜌𝑎𝑖𝑟𝑣𝑓𝑡)𝑣𝑜𝑢𝑡 = (𝐴𝑖𝑛
+𝑃
+𝑅𝑎𝑖𝑟𝑇 𝑣𝑓𝑡)𝑣𝑜𝑢𝑡 , 
+(S65) 
+ 
+which incorporates mass flow rate and the ideal gas law. Now, recall that equation (S4) provides an 
+expression relating 𝑣𝑓𝑡 and 𝑣𝑜𝑢𝑡, while (S46) provides a simplified approximation for 𝑣𝑓𝑡. As a result, 
+taking a conservative approach that lets 𝑣𝑜𝑢𝑡 = 0.2𝑣𝑎𝑣𝑔, a fifth of the average molecular velocity of a gas, 
+shown in (S41) above, and incorporating (S2) and (S46), it is possible to re-write (S68) to obtain  
+ 
+ 
+𝐹 = 𝐴𝑖𝑛
+𝑃
+𝑅𝑎𝑖𝑟𝑇 𝑣𝑓𝑡0.2√8𝑅𝑎𝑖𝑟𝑇
+𝜋
+= 
+= 0.0055𝐴𝑖𝑛
+𝑃
+𝑅𝑎𝑖𝑟𝑇
+𝐼
+𝑃 √8𝑅𝑎𝑖𝑟𝑇
+𝜋
+= 0.0055𝐴𝑖𝑛
+𝐼
+𝑅𝑎𝑖𝑟𝑇 √8𝑅𝑎𝑖𝑟𝑇
+𝜋
+ . 
+ 
+(S66) 
+ 
+Upon further simplification, equation (S69) reduces to  
+ 
+ 
+𝐹 = 0.0055𝐴𝑖𝑛
+𝐼
+𝑅𝑎𝑖𝑟𝑇 √8𝑅𝑎𝑖𝑟𝑇
+𝜋
+= 0.0055√8
+𝜋 𝐴𝑖𝑛𝐼√
+1
+𝑅𝑎𝑖𝑟𝑇 . 
+(S67) 
+ 
+Thus, the maximum areal density that can be entertained by these 3D structures can be approximated by 
+ 
+ 
+𝜎𝑔𝑒𝑜𝑚 =
+𝐹
+𝐴𝑖𝑛𝑔 = 0.0055√8
+𝜋
+𝐼
+𝑔 √
+1
+𝑅𝑎𝑖𝑟𝑇 = 𝐾𝐼√
+1
+𝑅𝑎𝑖𝑟𝑇 = 0.016 
+𝐼
+𝑣𝑎𝑣𝑔𝑔  , 
+(S68) 
+ 
+where 𝐾 =
+0.0055
+𝑔
+√
+8
+𝜋 = 0.0009  and 𝑣𝑎𝑣𝑔 = √8𝑅𝑎𝑖𝑟𝑇/𝜋  ≈ 400  m/s is the average velocity of air 
+molecules. Upon inserting the parameters, we find that 𝜎𝑔𝑒𝑜𝑚 can have an average value of 0.004 kg/m2, 
+four times of what the areal density of nanocardboard typically is in experiments. The main paper provides 
+additional areal density calculations based off from the parametric studies (detailed below) as well as cloud 
+plots denoting the maximum areal density for each of the study geometries. They are generally of the same 
+order of magnitude as the estimate (S68). 
+ 
+2.2.3 Parametric Studies 
+ 
+In this section, we provide four tables that accompany the presentation of the results shown in the main 
+paper. In essence, Table 8 both summarizes the chosen optimization ranges and discretization for the 
+variables that were varied (A, L and r) and specifies the values that the remaining variables (B, N, X, S and 
+t) took. Similarly, Table 9 through Table 11 present the results for the performed parametric optimization 
+on the three geometries, detailing the specific combination of A, L and r that first, yielded the maximum 
+payload capabilities and second, achieved flight at the lower altitude. In addition, Table 9 through Table 
+11 also provide the areal density of each structure for when the maximum payload was achieved. Notice 
+that this process was repeated for multiple values of D, as to explore the dependency of the overall 
+optimization results with the scale of the geometries. 
+ 
+Parametric Optimization Variables 
+Variable 
+Range 
+Truncated Cone 
+Truncated Sphere 
+Rocket 
+Discretization 
+𝐴 
+Min. 
+10 nm 
+80 equally spaced points  
+(log scale) 
+Max. 
+5 mm 
+𝐿 
+Min. 
+1  μm 
+80 equally spaced points  
+
+Page 18 
+ 
+Table 8: Main values used across the various variables during the parametric optimization. As can be seen, the search 
+range for the optimal A, L and r was discretized in all three cases in 100 points, following a log scale. Changing the 
+granularity of the discretization or the bounds of the search range did not significantly modify the results seen in Table 
+9 through Table 11 below. 
+ 
+ 
+Table 9: Combinations of A, L and r that returned the spheres capable of carrying the greatest payload and achieving 
+flight at the lowest altitude, for various values of D, as specified in Figure S1. 
+ 
+ 
+Table 10: Combinations of A, L and r that returned the cones capable of carrying the greatest payload and achieving 
+flight at the lowest altitude, for various values of D, as specified in Figure S1. 
+ 
+ 
+Max. 
+1 cm 
+(log scale) 
+𝑟 
+Min. 
+rmin =  D/20 (see Table 9 through Table 12) 
+80 equally spaced points  
+(log scale) 
+Max. 
+rmax = D/2.01 (see Table 9 through Table 12) 
+Altitude 
+Min. 
+0 km 
+17 equally spaced points  
+(5 km intervals) 
+Max. 
+80 km 
+ 
+𝐵 
+10𝐴 
+𝑁 
+1 sun 
+𝑋 
+𝐵 − 𝑆
+𝑆 + 𝐴  
+𝑆 
+𝐴 
+𝑡 
+50 nm  
+Parametric Optimization Results – Various Sphere Sizes 
+Variable 
+Case 
+D = 2 cm 
+D = 0.1 m 
+D = 0.5 m 
+D = 1 m 
+D = 2 m 
+D = 5 m 
+rmin =  D/20, rmax = D/2.01, with a discretization of 80 points (log scale) 
+𝐴 
+Max. Payload 
+0.90 mm 
+0.90 mm 
+0.90 mm 
+0.90 mm 
+0.90 mm 
+0.90 mm 
+Min. Altitude 
+0.13 mm 
+0.13 mm 
+0.20 mm 
+0.20 mm 
+0.20 mm 
+0.20 mm 
+𝐿 
+Max. Payload 
+0.91 mm 
+0.91 mm 
+0.91 mm 
+0.91 mm 
+0.91 mm 
+0.91 mm 
+Min. Altitude 
+0.14 mm 
+0.14 mm 
+0.21 mm 
+0.21 mm 
+0.21 mm 
+0.21 mm 
+𝑟 
+Max. Payload 
+9.95 mm 
+4.07 cm 
+19.05 cm 
+36.85 cm 
+73.70 cm 
+1.84 m 
+Min. Altitude 
+4.05 mm 
+1.89 cm 
+10.82 cm 
+21.63 cm 
+43.27 cm 
+1.08 m 
+ 
+Max.  
+Payload 
+Payload (mg) 
+8.34 
+79.11  
+1 445 
+5 526 
+21 612  
+133 242 
+Altitude (km) 
+80  
+80  
+80  
+80  
+80 
+80  
+A. Density (g/m2) 
+25.48 
+7.81  
+5.91 
+5.64  
+5.54 
+5.49 
+Sphere Area (m2) 
+0.0007  
+0.025 
+0.64 
+2.63 
+10.52 
+65.82  
+𝐴𝑡𝑜𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 ratio 
+2.22 
+4.77 
+5.68 
+6.17 
+6.17 
+6.17 
+ 
+Min. 
+Altitude 
+Payload (mg) 
+0.24 
+0.58  
+223.94 
+872.33 
+3 442  
+21 339  
+Altitude (km) 
+55 
+55  
+60 
+60 
+60 
+60 
+𝐴𝑡𝑜����𝑎𝑙/𝐴𝑜𝑢𝑡 ratio 
+23.34 
+26.96 
+20.30 
+20.32 
+20.31 
+20.38 
+Parametric Optimization Results – Various Cone Sizes 
+Variable 
+Case 
+D = 2 cm 
+D = 0.1 m 
+D = 0.5 m 
+D = 1 m 
+D = 2 m 
+D = 5 m 
+rmin =  D/20, rmax = D/2.01, with a discretization of 80 points (log scale) 
+𝐴 
+Max. Payload 
+0.90 mm 
+0.90 mm 
+0.90 mm 
+0.90 mm 
+0.90 mm 
+0.90 mm 
+Min. Altitude 
+0.13 mm 
+0.35 mm 
+0.35 mm 
+0.35 mm 
+0.35 mm 
+0.35 mm 
+𝐿 
+Max. Payload 
+0.91 mm 
+0.91 mm 
+0.91 mm 
+0.91 mm 
+0.91 mm 
+0.91 mm 
+Min. Altitude 
+0.14 mm 
+0.36 mm 
+0.36 mm 
+0.36 mm 
+0.36 mm 
+0.36 mm 
+𝑟 
+Max. Payload 
+9.95 mm 
+4.97 cm 
+24.86 cm 
+49.73 cm 
+99.45 cm 
+2.49 m 
+Min. Altitude 
+4.05 mm 
+2.39 cm 
+11.56 cm 
+23.12 cm 
+46.25 cm 
+1.16 m 
+ 
+Max.  
+Payload 
+Payload (mg) 
+7.96 
+101.26 
+2 043 
+7 929 
+31 228 
+193 348 
+Altitude (km) 
+80  
+80  
+80  
+80  
+80 
+80  
+A. Density (g/m2) 
+11.59 
+6.61 
+5.61  
+5.48 
+5.42  
+5.38 
+Cone Area (m2) 
+0.0016  
+0.039 
+0.98 
+3.92 
+15.67  
+97.97 
+𝐴𝑡𝑜𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 ratio 
+5.04 
+5.05 
+5.05 
+5.04 
+5.04 
+5.03 
+ 
+Min. 
+Altitude 
+Payload (mg) 
+0.18 
+10.12  
+208.65  
+812.97  
+3 209  
+ 19 892 
+Altitude (km) 
+55 
+60 
+60 
+60 
+60 
+60 
+𝐴𝑡𝑜𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 ratio 
+23.97 
+17.75 
+18.84 
+18.84 
+18.83 
+18.72 
+
+Page 19 
+ 
+Table 11: Combinations of A, L and r that returned the rockets capable of carrying the greatest payload and achieving 
+flight at the lowest altitude, for various values of D, as specified in Figure S1. 
+ 
+The results from these tables are discussed in greater detail in the main paper. However, there are four 
+important points to highlight. First, changing D (the scaling of the overall geometries) did not affect 
+significantly the optimal channel parameters A and L that yielded the maximum payload capabilities and 
+achieved flight at the lowest altitude. Secondly, the obtained maximum areal densities were similar across 
+the three geometries (as seen in Figure S14 (a) below) and had average values of 9.31 g/m2, 6.68 g/m2 and 
+6.96 g/m2, for the sphere, cone, and rocket, respectively. Notice that these are above the theoretical order-
+of-magnitude estimation for the upper limit of 4 g/m2 in (S71). Thirdly, the optimized 𝐴𝑡𝑜𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 ratios 
+for the three geometries were relatively invariant across the various values of D and the two missions (max. 
+payload and minimum altitude). For instance, for the maximum payload optimization, 𝐴𝑡𝑜𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 
+averaged 5.20, 5.04, and 6.02 for the sphere, cone and rocket, respectively, while for the minimum altitude 
+case, this ratio averaged 21.94, 19.49 and 28.65, respectively. Lastly, for a given surface area, the amount 
+of payload that each geometry could carry was comparable, as can be seen in Figure S14 (b) below. As a 
+result, 1 m2 of a porous and geometrically optimized cone has a similar maximum payload capability than 
+1 m2 of an optimized rocket and sphere.  
+Finally, Figure S15 through Figure S20 present cloud plots that permit visualizing the results from the 
+parametric studies, in particular how different combinations of A, L and r enabled geometries with various 
+altitude (a), payload (b) and areal density (c) capabilities. These plots correspond to the D = 10 cm and D 
+= 10 m cone, sphere and rocket, and are accompanied with illustrations of the optimized geometries that 
+achieved flight at minimum altitude (d) and carried the most payload (e). These figures were generated by 
+discretizing the search ranges of A, L and r in 500 equally spaced, and the results from the optimized 
+geometries are shown in Table 12 through Table 14). Despite the increase in discretization points (from 80 
+to 500) in each dimension, the results were comparable. 
+Parametric Optimization Results – Various Rocket Sizes 
+Variable 
+Case 
+D = 2 cm 
+D = 10 cm 
+D = 0.5 m 
+D = 1 m 
+D = 2 m 
+D = 5 m 
+rmin =  D/20, rmax = D/2.01, with a discretization of 80 points (log scale) 
+𝐴 
+Max. Payload 
+0.90 mm 
+0.90 mm 
+0.90 mm 
+0.90 mm 
+0.90 mm 
+0.90 mm 
+Min. Altitude 
+0.092 mm 
+0.13 mm 
+0.13 mm 
+0.13 mm 
+0.13 mm 
+0.13 mm 
+𝐿 
+Max. Payload 
+0.91 mm 
+0.91 mm 
+0.91 mm 
+0.91 mm 
+0.91 mm 
+0.91 mm 
+Min. Altitude 
+0.094 mm 
+0.14 mm 
+0.14 mm 
+0.14 mm 
+0.14 mm 
+0.14 mm 
+𝑟 
+Max. Payload 
+9.95 mm 
+4.97 cm 
+24.86 cm 
+49.73 cm 
+99.45 cm 
+2.49 m 
+Min. Altitude 
+1.00 mm 
+0.94 cm 
+4.12 cm 
+8.24 cm 
+15.94 cm 
+0.40 m 
+ 
+Max.  
+Payload 
+Payload (mg) 
+9.51 
+127.59  
+2 639 
+10 281  
+40 573 
+251 516  
+Altitude (km) 
+80  
+80  
+80  
+80  
+80  
+80  
+A. Density (g/m2) 
+11.60 
+6.89  
+5.95 
+5.83 
+5.77 
+5.74 
+Rocket Area (m2) 
+0.0019 
+0.047 
+1.17  
+4.68  
+18.71  
+117.12 
+𝐴𝑡𝑜𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 ratio 
+6.02 
+6.02 
+6.02 
+6.02 
+6.02 
+6.02 
+ 
+Min. 
+Altitude 
+Payload (mg) 
+0.03 
+1.54 
+8.29 
+18.57  
+45.37  
+175.67  
+Altitude (km) 
+45 
+55  
+55  
+55 
+55  
+55  
+𝐴𝑡���𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 ratio 
+42 
+23.28 
+26.27 
+26.27 
+27.09 
+27 
+Figure S14: D against Areal Density (a) and Surface Area against Payload (b) for the 3D geometries. 
+a 
+b 
+
+Max.PayloadagainstGeometrySurfaceArea
+100
+Max.Payload (kg)
+10
+Sphere
+-Cone
+Rocket
+9-01
+10-4
+102
+100
+102Max.Areal Density against characteristic D
+25
+Sphere
+Cone
+20
+Rocket
+15
+10
+102
+10-1
+100
+101
+D (m)Page 20 
+ 
+ 
+Comparison of D = 10 cm and D = 10 m Cone Geometries 
+Case 
+A 
+L 
+r 
+Surface 
+Area (m2) 
+𝑨𝒕𝒐𝒕𝒂𝒍/
+𝑨𝒐𝒖𝒕 ratio 
+Payload 
+(mg) 
+Altitude 
+(km) 
+Discretization of 500 points  
+D = 
+10 cm 
+Min. Altitude 
+0.15 mm 
+0.16 mm 
+1.94 cm 
+0.03 
+25.92 
+0.52 
+55 
+Max. Payload 
+1.24 mm 
+1.25 mm 
+4.97 cm 
+0.04 
+5.05 
+102.31 
+80 
+D = 
+10 m 
+Min. Altitude 
+0.21 mm 
+0.22 mm 
+2.36 m 
+317.52 
+18.16 
+95 288 
+60 
+Max. Payload 
+1.24 mm 
+1.25 mm 
+4.97 m 
+391.56 
+5.05 
+780 408 
+80 
+ 
+Table 12: Combinations of A, L and r that returned the optimal cone geometries described in Figure S15 and Figure 
+S16 above. 
+ 
+Figure S15: Minimum Altitude (a), Maximum Payload (b) and Areal Density (c) plots for the D = 10 cm Cone 
+Geometry. Here, the geometry that was able to levitate payload at minimum altitude (0.52 mg at 55 km) is shown in 
+(d), while that which was able to levitate the maximum payload (102.31 mg at 80 km) is shown in (e). 
+ 
+a 
+c 
+b 
+d 
+e 
+Figure S16: Minimum Altitude (a), Maximum Payload (b) and Areal Density (c) plots for the D = 10 m Cone 
+Geometry. Here, the geometry that was able to levitate payload at minimum altitude (95 288 mg at 60 km) is shown 
+in (d), while that which was able to levitate the maximum payload (780 408 mg at 80 km) is shown in (e). 
+ 
+a 
+c 
+b 
+d 
+e 
+
+0.045
+0.04
+0.035
+0.03
+日
+Aerial Densities: Cone Geometry
+0.025
+5.8g/m²99%percentile density
+0.02
+4.49g/m~190%percentile density
+2.84g/m150%percentile density
+2.35g/m|25%percentiledensity
+-01
+10-3
+102
+A [m]
+104
+L [m]0.04
+0.03
+[m]
+MinimumAltitudes:ConeGeometry
+0.02
+55km
+60 km
+65km
+70km
+10~2
+104
+103
+102
+A [m]
+104
+L[m]0.045
+0.04
+0.035
+0.03
+Maximum Payloads:Cone Geometry
+101.29mg/99%max.payload
+0.025
+92.08 mg/90% max.payload
+51.15 mg/50% max.payload
+30.69mg|30%max.payload
+0.02
+10~
+104
+103
+102
+A [m]
+L[m]r=1.94cm
+r=4.97cmAerial Densities:ConeGeometry
+4.85g/m199%percentile density
+2
+3.88g/m²90%percentile density
+2.58g/m50%percentiledensity
+2.19g/m25%percentile density
+102
+10-3
+103
+10-2
+A[m]
+o1
+104
+L [m]3
+[u]
+MinimumAltitudes:Cone Geometry
+2
+60km
+65 km
+70km
+75km
+102
+102
+A [m]
+-01
+104
+L[m]4.5
+4.
+3.5
+3
+MaximumPayloads:ConeGeometry
+772603.92mg/99%max.payload
+2.53
+702367.2mg/90%max.payload
+390204mg/50%max.payload
+2
+234122.4mg/30%max.payload
+102
+10-3
+104
+103
+102
+A[m]
+104
+L [m]r=2.36m
+r=4.97mPage 21 
+ 
+Comparison of D = 10 cm and D = 10 m Rocket Geometries 
+Case 
+A 
+L 
+r 
+Surface 
+Area (m2) 
+𝑨𝒕𝒐𝒕𝒂𝒍/
+𝑨𝒐𝒖𝒕 ratio 
+Payload 
+(mg) 
+Altitude 
+(km) 
+Discretization of 500 points  
+D = 
+10 cm 
+Min. Altitude 
+0.11 mm 
+0.12 mm 
+0.50 cm 
+0.001 
+>100 
+0.01 
+50 
+Max. Payload 
+1.24 mm 
+1.25 mm 
+4.97 cm 
+0.05 
+6.02 
+129.56 
+80 
+D = 
+10 m 
+Min. Altitude 
+0.15 mm 
+0.16 mm 
+0.87 m 
+59.39 
+24.98 
+2132.57 
+55 
+Max. Payload 
+1.24 mm 
+1.25 mm 
+4.97 m 
+467.23 
+6.02 
+1021162 
+80 
+ 
+Table 13: Combinations of A, L and r that returned the optimal rocket geometries described in Figure S17 and Figure 
+S18 above. 
+Figure S17: Minimum Altitude (a), Maximum Payload (b) and Areal Density (c) plots for the D = 10 cm Rocket 
+Geometry. Here, the geometry that was able to levitate payload at minimum altitude (0.01 mg at 50 km) is shown in (d), 
+while that which was able to levitate the maximum payload (129.56 mg at 80 km) is shown in (e). 
+ 
+a 
+c 
+b 
+d 
+r = 5.00 mm 
+e 
+r = 4.97 cm 
+Figure S18: Minimum Altitude (a), Maximum Payload (b) and Areal Density (c) plots for the D = 10 m Rocket 
+Geometry. Here, the geometry that was able to levitate payload at minimum altitude (2 132.57 mg at 55 km) is shown 
+in (d), while that which was able to levitate the maximum payload (1 021 162 mg at 80 km) is shown in (e). 
+ 
+a 
+c 
+b 
+r = 0.87 m 
+r = 4.97 m 
+d 
+e 
+
+0.04
+0.03
+0.02
+AerialDensities:RocketGeometry
+6.64g/m²99%percentiledensity
+0.01
+5.78g/m|90%percentiledensity
+4.33g/m/50%percentiledensity
+3.36g/m²|25%percentile density
+102
+A [m]
+104
+103
+102
+104
+L [m]0.04
+0.03
+0.02
+[m]
+MinimumAltitudes:Rocket Geometry
+50km
+0.01
+55km
+60km
+65km
+102
+104
+102
+A[m]
+104
+L[m]0.045
+0.04
+0.035
+0.03
+MaximumPayloads:RocketGeometry
+128.26mg|99%max.payload
+0.025
+116.6mg/90%max.payload
+64.78mg|50% max.payload
+38.87mg|30%max.payload
+0.02
+102
+103
+10~
+10-3
+10~2
+A [m]
+104
+L[m]41
+3
+2
+[u]
+Aerial Densities:RocketGeometry
+5.18g/m²199%percentile density
+1.
+4.15g/m90%percentiledensity
+2.78g/m150%percentiledensity
+2.37g/m²25%percentiledensity
+102
+10-3
+104
+10-3
+10-2
+A [m]
+104
+L[m]4
+3
+2
+宜
+MinimumAltitudes:Rocket Geometry
+55km
+1
+60km
+65km
+70km
+102
+10-4
+102
+A [m]
+-01
+L [m]4.5
+43
+3.5,
+MaximumPayloads:RocketGeometry
+3
+1010951.02mg|99%max.payload
+919046.38mg|90%max.payload
+2.53
+510581.32mg|50%max.payload
+306348.79mg30%max.payload
+102
+103
+104
+10-3
+102
+A[m]
+104
+L [m]Page 22 
+ 
+Comparison of D = 10 cm and D = 10 m Sphere Geometries 
+Case 
+A 
+L 
+r 
+Surface 
+Area (m2) 
+𝑨𝒕𝒐𝒕𝒂𝒍/
+𝑨𝒐𝒖𝒕 ratio 
+Payload 
+(mg) 
+Altitude 
+(km) 
+Discretization of 500 points  
+D = 
+10 cm 
+Min. Altitude 
+0.15 mm 
+0.16 mm 
+1.93 cm 
+0.03 
+25.81 
+1.41 
+55 
+Max. Payload 
+1.03 mm 
+1.04 mm 
+4.02 cm 
+0.03 
+4.93 
+79.86 
+80 
+D = 
+10 m 
+Min. Altitude 
+0.15 mm 
+0.16 mm 
+1.90 m 
+302.22 
+26.66 
+831.92 
+55 
+Max. Payload 
+1.24 mm 
+1.25 mm 
+3.67 m 
+263.63 
+6.23 
+540 528 
+80 
+ 
+Table 14: Combinations of A, L and r that returned the optimal sphere geometries described in Figure S19 and Figure 
+S20 above. 
+Figure S19: Minimum Altitude (a), Maximum Payload (b) and Areal Density (c) plots for the D = 10 cm Sphere 
+Geometry. Here, the geometry that was able to levitate payload at minimum altitude (1.41 mg at 55 km) is shown in 
+(d), while that which was able to levitate the maximum payload (79.86 mg at 80 km) is shown in (e). 
+ 
+a 
+c 
+b 
+d 
+e 
+Figure S20: Minimum Altitude (a), Maximum Payload (b) and Areal Density (c) plots for the D = 10 m Sphere 
+Geometry. Here, the geometry that was able to levitate payload at minimum altitude (831.92 mg at 55 km) is shown 
+in (d), while that which was able to levitate the maximum payload (540 528 mg at 80 km) is shown in (e). 
+ 
+a 
+c 
+b 
+d 
+e 
+
+0.043
+0.03
+AerialDensities:SphereGeometry
+7.05g/m²|99% percentile density
+0.02
+5.13g/m²190%percentile density
+3.01g/m/50%percentiledensity
+2.43g/m²25%percentiledensity
+102
+103
+102
+A [m]
+10~4
+104
+10-3
+L[m]0.043
+0.03
+[m]
+MinimumAltitudes:SphereGeometry
+0.02
+55km
+60km
+65km
+70km
+102
+102
+A[m]
+-01
+104
+L [m]0.04
+MaximumPayloads:SphereGeometry
+79.06mg/99%max.payload
+71.87mg90%max.payload
+0.02
+39.93mg/50%max.payload
+23.96mg|30%max.payload
+102
+103
+102
+A [m]
+104
+104
+10-3
+L[m]r=1.90
+r=3.67
+m
+u4.5
+4.
+3.5
+3
+AerialDensities:SphereGeometry
+2.5
+5.03g/m199%percentile density
+2
+3.99g/m90%percentiledensity
+2.61g/m|50%percentile density
+2.2g/m²25%percentiledensity
+10~2
+10-3
+104
+10-3
+102
+A [m]
+10-4
+L[m]43
+m
+[m]
+MinimumAltitudes: Sphere Geometry
+2
+55km
+60km
+65km
+70km
+102
+102
+A [m]
+104
+104
+L[m]4.5
+4
+3.5
+E
+3
+MaximumPavloads: Sphere Geometry
+2.53
+535122.95mg/99%max:payload
+486475.41mg/90%max.payload
+23
+270264.11mg/50%max.payload
+162158.47mg/30%max.payload
+102
+10-
+104
+103
+102
+A [m]
+104
+L [m]r=1.93
+r=4.02
+cm
+cmPage 23 
+References 
+ 
+[R1] Azadi, Mohsen, George A. Popov, Zhipeng Lu, Andy G. Eskenazi, Avery Ji Won Bang, Matthew F. 
+Campbell, Howard Hu, and Igor Bargatin. "Controlled levitation of nanostructured thin films for sun-
+powered near-space flight." Science Advances 7, no. 7 (2021): eabe1127. 
+ 
+[R2] Cappella, Andrea, Jean‐Luc Battaglia, Vincent Schick, Andrzej Kusiak, Alessio Lamperti, Claudia 
+Wiemer, and Bruno Hay. "High Temperature Thermal Conductivity of Amorphous Al2 O 3 Thin Films 
+Grown by Low Temperature ALD." Advanced Engineering Materials 15, no. 11 (2013): 1046-1050. 
+ 
+[R3] Cortes, John, Christopher Stanczak, Mohsen Azadi, Maanav Narula, Samuel M. Nicaise, Howard Hu, 
+and Igor Bargatin. "Photophoretic levitation of macroscopic nanocardboard plates." Advanced Materials 32, 
+no. 16 (2020): 1906878. 
+ 
+[R4] Eskenazi, Andy, Tom Celenza, and Igor Bargatin. “MATLAB-fluid-flow-parametric-studies.” (2022) 
+https://github.com/andyeske/MATLAB-fluidflow-parametric-studies 
+ 
+[R5] Lin, Chen, Samuel M. Nicaise, Drew E. Lilley, Joan Cortes, Pengcheng Jiao, Jaspreet Singh, Mohsen 
+Azadi et al. "Nanocardboard as a nanoscale analog of hollow sandwich plates." Nature communications 9, 
+no. 1 (2018): 1-8. 
+ 
+[R6] O'Neal Jr, Cleveland, and Richard S. Brokaw. "Relation between thermal conductivity and viscosity 
+for some nonpolar gases." The Physics of Fluids 5, no. 5 (1962): 567-574. 
+ 
+[R7] Sharipov, Felix, and Vladimir Seleznev. "Data on internal rarefied gas flows." Journal of Physical 
+and Chemical Reference Data 27, no. 3 (1998): 657-706. 
+ 
+[R8] Teagan, William P., and George S. Springer. "Heat‐Transfer and Density‐Distribution Measurements 
+between Parallel Plates in the Transition Regime." The Physics of Fluids 11, no. 3 (1968): 497-506. 
+ 
+[R9] Wagiman, Abdullah, Mohammad Sukri Mustapa, Mohd Amri Lajis, Shazarel Shamsudin, Mahmod 
+Abd Hakim, and Rosli Asmawi. "Effect of Thermally Formed Alumina on Density of AlMgSi Alloys 
+Extrudate Recycled Via Solid State Technique." Journal of Advanced Research in Fluid Mechanics and 
+Thermal Sciences 87, no. 2 (2021): 137-144. 
+ 
+[R10] Wu, H., S. Grabarnik, A. Emadi, G. De Graaf, and R. F. Wolffenbuttel. "Characterization of thermal 
+cross-talk in a MEMS-based thermopile detector array." Journal of Micromechanics and 
+Microengineering 19, no. 7 (2009): 074022. 
+

SdAyT4oBgHgl3EQfuPmB/content/tmp_files/2301.00609v1.pdf.txt

@@ -0,0 +1,2307 @@
+Generalized Uncertainty Principle Impact on Nonextensive Black Hole Thermodynamics
+Ilim Çimdiker,1, ∗ Mariusz P. Da¸browski,1, 2, 3, † and Hussain Gohar1, ‡
+1Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland
+2National Centre for Nuclear Research, Andrzeja Sołtana 7, 05-400 Otwock, Poland
+3Copernicus Center for Interdisciplinary Studies, Szczepa´nska 1/5, 31-011 Kraków, Poland
+(Dated: January 3, 2023)
+The effect of the generalized uncertainty principle (GUP) on nonextensive thermodynamics ap-
+plied to black holes, as well as the sparsity of radiation at different temperatures associated with
+each nonextensive entropy, is investigated. We examine the Rényi, Tsallis-Cirto, Kaniadakis, Sharma
+Mittal, and Barrow entropies, temperatures, and heat capacities and show that, in each case, due to
+GUP corrections, the temperature and entropy have finite values, implying that the final state of the
+black hole is a remnant at the end of the evaporation process and that the sparsity of the radiation at
+each temperature depends on the mass of the black hole. We also find that GUP reduces the value of
+the sparsity parameter for each case as compared to the sparsity parameter at Hawking temperature,
+which is always constant throughout the evaporation.
+I.
+INTRODUCTION
+Black holes emit radiation due to the Hawking evap-
+oration process, and therefore, there is an established
+concept of Hawking temperature [1] and Bekenstein
+entropy [2] connected with the black hole horizon.
+The black hole evaporation process operates within the
+purview of quantum field theory, and one of its more in-
+triguing aspects may be that it appears to indicate a non-
+unitary evolution, which gives rise to the well-known is-
+sue of the information loss paradox [3–5]. In this regard,
+black holes behave like thermodynamic objects, and the
+laws of black hole thermodynamics [6–10] are analogous
+to the conventional thermodynamic laws. The thermo-
+dynamics of black holes have been extensively studied
+and used in a variety of cosmological and gravitational
+applications [11–20].
+Entropy measures how difficult it is for an outside ob-
+server to get information about the underlying structure
+of the system. This is a clear reflection of the macro-
+scopic features that result from the quantum statisti-
+cal mechanics that govern the behavior of quantum mi-
+crostates. For the case of black holes, there is no defi-
+nition of Bekenstein entropy in quantum statistical me-
+chanics and it only relies on Hawking’s area theorem
+[21], therefore, it would be required to have a complete
+theory of quantum gravity in order to fully comprehend
+the origin of this entropy and the nature of microstates
+in the case of black holes. Therefore, we rely on the defi-
+nition of Bekenstein entropy for black holes. For the case
+of a Schwarzschild black hole with mass M, the Hawk-
+ing temperature TH and Bekenstein entropy SB are given
+by [1, 2]
+TH =
+¯hκ
+2πkBc , SB = kBc3A
+4G¯h
+,
+(1)
+∗ [email protected]
+† [email protected]
+‡ [email protected]
+where ¯h, G, kB, and c are the reduced Planck constant,
+the Newton gravitational constant, the Boltzmann con-
+stant, and the speed of light, respectively. The area A of
+the event horizon is defined as A = 4πr2
+h in the above
+equation (1), where rh = 2GM/c2 is the Schwarzschild
+radius and κ = c4/4πGM is the surface gravity defined
+on the event horizon of the Schwarzschild black hole.
+The core assumption of Gibbs thermodynamics and
+statistical mechanics is that entropy is extensive and ad-
+ditive. Nonextensive statistical mechanics, such as Tsal-
+lis nonextensive statistical mechanics [22–31], is the out-
+come of removing this assumption.
+The assumption
+of the extensive nature of entropy is connected to ig-
+noring the long-range forces between thermodynamic
+sub-systems. Since the size of the system exceeds the
+range of the interaction between the system’s compo-
+nents, Gibbs thermodynamics ignores these long-range
+forces. Because of this, the total entropy of a composite
+system equals the sum of the entropies of the individ-
+ual subsystems and entropy grows with the size of the
+system. However, long-range forces are important in
+various unique thermodynamic systems. For instance,
+if we think of a black hole as a (3 + 1) dimensional ob-
+ject, it is vital to note that Bekenstein entropy scales with
+the area and is thus regarded as a nonextensive quan-
+tity [32–38]. Furthermore, because of the area scaling,
+Bekenstein entropy is nonadditive.
+Therefore, Gibbs
+thermodynamics or statistical mechanics may not be the
+appropriate choice for studying the thermodynamics of
+black holes. In order to understand the nonextensive
+and nonadditive nature of Bekenstein entropy, several
+extensions [22, 39–44] of standard Gibbs thermodynam-
+ics have been applied to black holes and cosmologi-
+cal horizons [45–70]. One of the main proposals is the
+Tsallis-Cirto’s black hole entropy definition [32], which
+makes the black entropy extensive and compatible with
+the Legendre structure. Rényi entropy [39], being a mea-
+sure of entanglement, is another definition of entropy
+applied to black holes and cosmological horizons which
+is nonextensive, but additive (by assumption). There
+arXiv:2301.00609v1  [gr-qc]  2 Jan 2023
+
+2
+have been some other nonextensive forms of entropy
+suggested such as the Sharma-Mittal entropy [40, 41]
+as a generalization of Rényi entropy, the Kaniadakis en-
+tropy [42] which takes inspiration from Lorentz group
+transformations and the Barrow entropy [44] which is
+based on a hypothetical fractal structure of black hole
+horizon as a result of quantum fluctuations.
+Due to the prevalence of quantum gravity effects, it is
+anticipated that the semiclassical technique would fail
+during the last phases of Hawking evaporation. There is
+currently no satisfactory theory of quantum gravity that
+enables us to completely explain that regime, despite the
+development of several quite diverse proposals [71–77].
+Investigating the phenomenological consequences of an
+underlying theory of quantum gravity is one technique
+to explore the quantum gravity effects at those scales.
+The generalized uncertainty principle (GUP) [76–79] is
+one approach that has the benefit of being sufficiently
+generic to be compatible with several quantum gravity
+theories. The Bekenstein entropy and Hawking temper-
+ature of a black hole in its last phases of evaporation
+are modified within this framework [73].
+Because of
+these modifications, black holes do not entirely evapo-
+rate during the evaporation process, and the final state
+of the black hole is a remnant of the order of Planck mass
+Sparsity [80–91] is an important feature of Hawking
+radiation. It is defined as the average time between the
+emission of successive quanta over the timescales set by
+the energies of the emitted quanta. It was shown that
+Hawking radiation is very sparse during the black hole
+evaporation process [84], which is one of the key char-
+acteristics that distinguish it from black-body radiation.
+However, it has been found that when GUP corrections
+are incorporated [87–89], the sparsity decreases toward
+the late stages of evaporation. When nonextensivity is
+considered in the context of Rényi temperature [90], the
+Rényi radiation is initially not sparse, but as evaporation
+progresses, it begins to become sparse and eventually
+approaches the case of Hawking radiation.
+In this paper, we are interested in exploring the GUP
+modifications to the nonextensive entropies and corre-
+sponding thermodynamic quantities in Rényi, Tsallis-
+Cirto, Sharma-Mittal, Kaniadakis, and Barrow nonex-
+tensive statistics. Furthermore, the sparsity of the radia-
+tion is analyzed at different temperatures corresponding
+to different nonextensive entropies.
+The following is the outline of the paper. In Sec. II, we
+introduce the notion of GUP and apply it to the case of
+standard thermodynamic black hole quantities. In Sec.
+III, we introduce nonextensive entropies and accompa-
+nying nonextensive thermodynamic quantities, as well
+as GUP modifications to nonextensive black hole ther-
+modynamics. Finally, in Sec. IV, we summarize and
+discuss our findings.
+II.
+GUP AND BLACK HOLE THERMODYNAMICS
+A.
+Generalized Uncertainty Principle
+One common aspect of several quantum gravity the-
+ories is that they all predict a minimum measurable
+length [77, 92].
+For example, the notion of minimal
+length is defined in string theory as the string length
+[72, 93], in loop quantum gravity [74] it is the expec-
+tation value of the length operator, and this notion
+can also be developed by the phenomenological aspects
+coming from black hole physics [77]. Because of the ap-
+pearance of a minimum length at the Planck scale in var-
+ious quantum gravity approaches, it has been proposed
+that the Heisenberg Uncertainty Principle (HUP)
+∆x0∆p ≥ ¯h, or ∆x0 ∼
+¯h
+∆p
+(2)
+where ∆x0 and ∆p are position and momentum uncer-
+tainties can be modified when gravitational interaction
+is introduced. The simplest argument for the modifica-
+tion of HUP within the framework of Newtonian theory
+is that there is a gravitational acceleration⃗a of an electron
+due to photon of mass E/c2 [73], where E is the pho-
+ton energy and r is the photon-electron distance, which
+reads
+⃗a = ¨⃗r = − G(E/c2)
+r2
+⃗r
+r,
+(3)
+and the interaction takes place in a characteristic region
+of length L ∼ r and in characteristic time t ∼ L/c. Then,
+the velocity acquired by an electron ∆v is
+∆v ∼ GE
+c2r2
+L
+c ,
+(4)
+and the (extra due to gravity) distance ∆x1 it is shifted
+reads
+∆x1 ∼ GE
+c2r2
+L2
+c2 ∼ G∆p
+c3
+= c∆p
+4Fmax
+= l2
+p
+∆p
+¯h ,
+(5)
+where lp =
+√
+G¯h/c3 is the Planck length, and Fmax =
+c4/4G is the maximum force [94–97]. Extra uncertainty (5)
+adds to the standard HUP uncertainty of position ∆x0 as
+in (2) giving
+∆x = ∆x0 + ∆x1 ∼
+¯h
+∆p + l2
+p
+∆p
+¯h ,
+(6)
+leading to the generalized uncertainty principle (GUP)
+∆x∆p ≥ ¯h
+�
+1 +
+l2
+p
+¯h2 (∆p)2
+�
+.
+(7)
+Taking an algebraic point of view, GUP can be derived
+from the deformed commutation relation between the
+
+3
+position operator ˆx and the momentum operator ˆp such
+that
+[ ˆx, ˆp] = i¯h f ( ˆp),
+(8)
+where f ( ˆp) is a general function of momentum operator
+ˆp and there exist different proposed functions for f ( ˆp).
+In order to make the function f ( ˆp) compatible with (7),
+following the literature, we choose
+f ( ˆp) = 1 + α
+l2
+p
+¯h2 ˆp2,
+(9)
+where we the introduce GUP parameter α – a dimen-
+sionless parameter predicted to be an order of unity, but
+in reality bounded by different experiments and obser-
+vations to be much larger than that [98–102]. By intro-
+ducing α, the equation (10), now, reads as
+∆x∆p ≥ ¯h
+�
+1 + α
+l2
+p
+¯h2 (∆p)2
+�
+.
+(10)
+1.
+GUP Modified Hawking Temperature and Bekenstein Entropy
+An interesting application of (10) to black hole physics
+is the modification to the Hawking temperature, which
+can be derived by solving it for ∆p, which gives
+∆p = ∆x ¯h
+αl2p
+�
+�1 ±
+�
+1 −
+αl2p
+(∆x)2
+�
+� .
+(11)
+We consider the + sign in (11), following the discus-
+sion in [87]. Considering the minimum position uncer-
+tainty near the event horizon of the Schwarzschild black
+hole as ∆x = 2lp = 4GM/c2, where lp is taken as the
+Schwarzschild radius rh, the GUP modified Hawking
+temperature TGUP reads
+TGUP =
+m2
+pc2
+8πkBM
+�
+���
+4
+2 +
+�
+4 − α
+m2p
+M2
+�
+��� .
+(12)
+By introducing a correction term due to GUP, K(α, M),
+TGUP can be written in terms of TH and K, such that
+TGUP = TH(M)K(α, M),
+(13)
+where the GUP correction term is defined as
+K(α, M) =
+4
+2 +
+�
+4 − α
+m2p
+M2
+.
+(14)
+This provides us with a more compact form of TGUP,
+which will be used in the next sections for GUP mod-
+ifications to the thermodynamic quantities. Using the
+Clausius relation, the GUP modified Bekenstein entropy
+SGUP in terms of SB and the correction term K(α, M) can
+be written as
+SGUP = SB
+K − απkB
+2
+ln
+� 4M
+m0K
+�
+,
+(15)
+where m0 is a dimensionful constant of unit mass, which
+is introduced in order to make the logarithm dimension-
+less. Note that in the limit α → 0, the correction term K
+goes to one, and hence TGUP and SGUP reduce to TH and
+SB. The plots of (12) and (15) are given in Figs. 1 and 2.
+It is important to mention that all the plots in the pa-
+per, unless explicitly stated, are given in natural units
+¯h = c = G = 1 and also with the GUP parameter α = 1.
+TH
+TGUP,α=1
+TGUP,α=-1
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+M
+T
+Figure 1. Temperature vs mass for the Hawking temperature
+TH and the GUP corrected temperature with positive and neg-
+ative values of α. Threshold with positive α for mass lies at the
+remnant mass M2r = (α/4)m2p (cf. formula (16)).
+SB
+SGUP,α=1
+SGUPα=-1
+0.0
+0.5
+1.0
+1.5
+2.0
+0
+10
+20
+30
+40
+50
+M
+S
+Figure 2. Entropy vs mass for the Hawking temperature and
+GUP corrected temperatures with positive and negative val-
+ues of α. The threshold for mass lies at the remnant mass given
+by M2r = (α/4)m2p.
+It is interesting to note that, for real physical situa-
+tions, the equation (14) gives a bound on the mass which
+reads: M2 ≥ αm2
+p/4. This means that for positive values
+of α, the black hole evaporation stops when the mass of
+the black hole reaches some critical value of mass
+Mr =
+√αmp
+2
+= 2lp
+√α
+c2
+Fmax,
+(16)
+which is called the black hole remnant mass. Therefore,
+we can say that the final state of the black hole evapora-
+tion is a remnant having the mass Mr. In fact, without
+
+4
+a well-defined quantum gravity theory, we cannot pre-
+dict what happens if the mass of a black hole is smaller
+than this critical value. For the critical mass value Mr,
+the formulas (12) and (15) for TGUP and SGUP, give the
+temperature Tr and the entropy Sr for the remnant as
+[90]
+Tr =
+mpc2
+2πkB
+√α, Sr = παkB
+2
+�
+1 − ln
+�√αmp
+m0
+��
+,
+(17)
+provided that α > 0. For α < 0 in (14), we have a smooth
+correction function defined for all black hole mass val-
+ues.
+In this case, the black hole continues to radiate
+slowly and yields an infinite lifetime [89]. When M ap-
+proaches zero, interestingly, the temperature is still fi-
+nite, and for this case, in [103], it is referred to as a rem-
+nant with zero rest mass.
+2.
+GUP Modified Heat Capacity
+In order to investigate the GUP modifications to the
+heat capacity of a black hole with mass M, we use the
+definition of heat capacity C, which reads
+C = −S′2(M)
+S′′(M) ,
+(18)
+where S is the black hole entropy and prime and dou-
+ble prime denote the first and second derivative with
+respect to the mass M. For the case of Schwarzschild
+black hole, we have (denoting C as CSc)
+CSc = −8πkB
+M2
+m2p
+,
+(19)
+and we can see that it is negative for all mass values.
+This means that the Schwarzschild black hole is thermo-
+dynamically unstable. In order to introduce GUP cor-
+rections, we introduce the quantity
+βGUP =
+1
+kBTGUP
+,
+(20)
+which after using (12) gives
+S′
+GUP(M)
+kBc2
+= βGUP = β
+K ,
+(21)
+where β = 1/kBTH is the inverse Hawking tempera-
+ture. Differentiating βGUP once more, and using equa-
+tions (18) and (21), we obtain the GUP modified heat
+capacity CGUP, which can be written as (cf. Fig. 3)
+CGUP = CSc
+�2 − K
+K2
+�
+.
+(22)
+This means that the GUP corrections still yield a nega-
+tive heat capacity for M > Mr, and when the black hole
+mass approaches the critical mass Mr, we have K = 2
+and interestingly, we get the zero heat capacity for the
+remnant. In such a case, from the thermodynamic point
+of view, a small amount of heat would then increase the
+temperature of the remnant by an infinite amount.
+CSc
+CGUP,α=1
+CGUP,α=-1
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+-30
+-25
+-20
+-15
+-10
+-5
+0
+M
+C
+Figure 3. Specific heat capacity of the Hawking radiation for
+GUP corrected black holes. For positive α, there is a remnant
+with zero heat capacity.
+3.
+GUP Modified Sparsity of Hawking Radiation
+One of the most important aspects of Hawking radia-
+tion is that it is extremely sparse as compared to black-
+body radiation. The sparsity can be defined by using the
+parameter η [84, 87, 90],
+η = C
+g
+�
+λ2
+t
+Ae f f
+�
+,
+(23)
+where C is a dimensionless constant associated with dif-
+ferent physical cases [84], g is the spin degeneracy fac-
+tor of the particle, λt = 2π¯hc/kBT is the thermal wave-
+length in terms of the temperature T and Ae f f = 27A/4
+[80, 84] is the effective area with A being the horizon
+area for the case of black holes. For the Schwarzschild
+black hole, one can find the thermal wavelength λt by
+taking T = TH = 1/kBβ as
+λt = 2π¯hc
+kBTH
+= 2π¯hcβ,
+(24)
+and the sparsity parameter for the Hawking radiation
+reads [84]
+ηH = 64π3
+27
+≈ 73.38,
+(25)
+which is constant and is much greater than one. Note
+that for standard black body radiation, the value of η
+is less than one. This implies that the sparsity param-
+eter clearly differentiates the Hawking radiation from
+blackbody radiation. One can obtain the GUP effects on
+the sparsity by replacing the Hawking temperature with
+the GUP corrected temperature TGUP given by (12) [87].
+However, it is assumed that GUP also modifies the black
+hole horizon area [87, 90]. Thus, it is logical to take the
+effective area that GUP modifies. In fact, the GUP mod-
+ifications to A can be derived from the equation (15) by
+writing it as
+SGUP = kBc3AGUP
+4¯hG
+,
+(26)
+
+5
+ηH
+ηGUP,α=1
+ηGUP,α=-1
+0
+1
+2
+3
+4
+0
+20
+40
+60
+80
+100
+M
+η
+Figure 4. Sparsity of Hawking vs GUP corrected black holes in
+natural units. For positive values of α, we observe that sparsity
+decreases when a black hole is near the final evaporation state.
+where the GUP modified area AGUP reads
+AGUP = A
+K − απl2
+p ln
+� 16A
+A0K2
+�
+,
+(27)
+and A0 = 16πm2
+0G2/c4 is a constant having the dimen-
+sion of area. Note that in [90], corrections are only in the
+first order of α, while in the above equation (27) the area
+is corrected to all orders in α. Now, the GUP modified
+sparsity can be found by replacing T by TGUP and A by
+AGUP in (23), which now reads
+ηGUP = ηH
+K2
+�
+A
+AGUP
+�
+.
+(28)
+Interestingly, GUP modified sparsity ηGUP, depends on
+the mass of the black hole and the GUP parameter α.
+For the negative values of α, the sparsity parameter in-
+creases as M goes to zero. For the positive values of
+α, the sparsity parameter decreases below the values of
+sparsity for the Hawking radiation until it reaches the
+critical mass Mr. In Fig. 4, we can see that the GUP
+corrected sparsity is not a constant and it increases first
+before M approaches Mr for α > 0 and then it decreases
+to finite value when M approaches to Mr. For the case
+of α < 0, first, it decreases, and then it goes to plus in-
+finity when M approaches zero. It is due to the fact that
+A/AGUP > 1 for α > 0 and ηH/K2 turns back the spar-
+sity from a maximum value to a constant value, which
+is less than ηH. Therefore, we can clearly see the effects
+of GUP on sparsity due to TGUP and AGUP as depicted
+in Fig. 4. Similarly, A/AGUP < 1 for α < o and K goes
+to zero when M approaches zero, therefore, sparsity de-
+creases first, and then it goes to infinity. Note that in
+[89], the GUP corrected area is not taken into account,
+therefore, there is no bump in the sparsity parameter.
+III.
+GUP AND NONEXTENSIVE BLACK HOLE
+THERMODYNAMICS
+A.
+Tsallis Nonextensive Entropy
+Entropy plays a significant role in Gibbs thermody-
+namics or statistical mechanics.
+It is extensive and
+adheres to the additive composition rule.
+However,
+Gibbs statistical mechanics ignores long-range forces.
+On the other hand, there are some physical systems for
+which Gibbs thermodynamics cannot be the appropri-
+ate choice to apply [24] since they are subject to long-
+range forces.
+Important examples are the some self-
+gravitating systems such as black holes, since for them
+long-range forces play significant role. For that reason
+Constantino Tsallis in Refs. [22, 24] generalized the con-
+ventional Gibbs entropy for nonextensive systems in or-
+der to encompass and address this issue. Tsallis entropy
+ST was one of the earliest proposals to extend Gibbs en-
+tropy and the suggested new form of it reads
+ST = −kB ∑
+i
+[p(i)]q lnq p(i),
+(29)
+where p(i) is the probability distribution defined on a
+set of microstates Ω, with the parameter q determining
+the degree of nonextensivity, and we consider it positive
+to ensure the concavity of Sq. The q-logarithmic function
+lnq p is given by
+lnq p = p1−q − 1
+1 − q
+,
+(30)
+where, in the limit q → 1, Tsallis entropy Sq given by
+(29), reduces to Gibbs entropy SG
+SG = −kB ∑
+i
+p(i) ln p(i).
+(31)
+In fact, the Tsallis entropy (29) satisfies quite general,
+nonadditive composition rule of the following form
+ST 12 = ST 1 + ST 2 + λ
+kB
+ST 1ST 2,
+(32)
+for a composite system ”12”, made up of two subsys-
+tems ”1” and ”2”. In above equation, we have defined a
+new nonextensivity parameter λ = 1 − q.
+B.
+Rényi Entropy
+The Rényi entropy [39], a measure of entanglement
+in quantum information that is additive and preserves
+event independence, is another important generaliza-
+tion of the Gibbs-Shannon entropy. It is defined as
+SR = kB
+ln ∑i pq(i)
+1 − q
+.
+(33)
+
+6
+It is important that SR can be written in terms of ST by
+using the formal logarithm approach [30], and both en-
+tropies are related as follows
+SR = kB
+λ ln[1 + λ
+kB
+ST ].
+(34)
+It is interesting to mention here that SR is the equilib-
+rium entropy which corresponds to an equilibrium tem-
+perature TR defined from the equilibrium condition by
+maximizing the Tsallis entropy (32), which is given by
+[53]
+TR = (1 + λ
+kB
+ST ) 1
+kBβ.
+(35)
+Here, kBβ = ∂ST /∂U, where U is the internal energy of
+the nonextensive system.
+1.
+Rényi black hole Entropy and Temperature
+For the case of a Schwarzschild black hole, assuming
+that the Bekenstein entropy SB is just the Tsallis entropy
+ST , and replacing internal energy U with the mass of
+the black hole M in equations (34) and (35), the Rényi
+entropy can be defined on the horizon of a black hole as
+[33–37]
+SR = kB
+λ ln[1 + λ
+kB
+SB],
+(36)
+and the associated Rényi temperature reads
+TR = (1 + λ
+kB
+SB)TH.
+(37)
+Furthermore, we can write down the GUP corrected
+Rényi entropy using GUP corrected Bekenstein entropy
+as follows [90] (cf. Fig. 5)
+SRgup = kB
+λ ln
+�
+1 + λ
+kB
+(SGUP)
+�
+,
+(38)
+and corresponding GUP modified Rényi temperature
+TRgup can be written as (cf. Fig. 6)
+TRgup =
+�
+1 + λ
+kB
+(SGUP)
+�
+KTH.
+(39)
+The Rényi entropy increases logarithmically (for 0 <
+λ < 1), whereas the Bekenstein entropy (λ → 0) in-
+creases quadratically, as shown in Fig. 5. Furthermore,
+for the GUP corrections, the Rényi black holes do not
+completely evaporate; rather, evaporation stops at the
+critical mass Mr, leaving a remnant with finite entropy
+and temperature as the Rényi black hole’s final state.
+Using (37) and (39), we can write the inverse Rényi
+temperature parameters, βR and βRgup, which will fur-
+ther be used in calculating the heat capacities, such that
+kBβR = S′
+B(M)/c2
+1 + λ
+kB SB
+=
+kBβ
+1 + λ
+kB SB
+,
+(40)
+λ=0
+λ=0.5
+λ=1
+λ=0
+λ=0.5
+λ=1
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+0
+2
+4
+6
+8
+10
+M
+SR
+Figure 5.
+Rényi entropy SR of a black hole vs its mass M.
+Dashed lines represent GUP corrected cases, λ → 0 limit is
+the Bekenstein-Hawking case.
+λ=0
+λ=0.5
+λ=1
+λ=0
+λ=0.5
+λ=1
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+0.0
+0.5
+1.0
+1.5
+2.0
+M
+TR
+Figure 6. Rényi temperature TR of a black hole vs its mass M.
+Dashed lines represent GUP corrected cases, λ → 0 limit is the
+Bekenstein-Hawking case.
+and the GUP-corrected inverse Rényi temperature reads
+kBβRgup = S′
+GUP(M)/c2
+1 + λ
+kB SGUP
+=
+kBβGUP
+1 + λ
+kB SGUP
+.
+(41)
+One may determine the characteristic length scale LR
+for λ [49, 50, 52], which reveals the impact of nonexten-
+sive parameter λ in SR and SRgup, and in TR and TRgup.
+As a result, it can be concluded that below this charac-
+teristic length scale LR, the Rényi temperature behaves
+like TH, and that above LR, the nonextensive effects in-
+crease and TR grows linearly with M. The precise value
+for the length scale is found in the following subsection.
+2.
+Heat Capacity for the Rényi black hole
+In order to investigate the thermodynamic stability of
+Rényi black holes, we define the heat capacity CR of the
+Rényi black hole as
+CR = −S′2
+R(M)
+S′′
+R(M) .
+(42)
+
+7
+Inserting (40) and (41) into (42), the heat capacity for the
+non-GUP case reads
+CR =
+CSc
+1 + λ
+kB SB + λ
+kB CSc
+,
+(43)
+and for the GUP case, we have
+CRgup =
+CGUP
+1 + λ
+kB SGUP + λ
+kB CGUP
+.
+(44)
+We plot the heat capacity in Fig.
+(7), where we can
+λ=0
+λ=0.5
+λ=1
+λ=0
+λ=0.5
+λ=1
+0.0
+0.5
+1.0
+1.5
+2.0
+-10
+-5
+0
+5
+10
+M
+CR
+Figure 7. Heat capacity CR of a Rényi black hole vs its mass M.
+Dashed lines represent GUP corrected cases, λ → 0 limit is the
+Bekenstein-Hawking case.
+see that L differentiates two regions for non-GUP and
+GUP cases. In order to understand the behavior of CR in
+both regions, we find LR in terms of λ from the singular
+points of equation (43) for the case Schwarzschild black
+hole. We find, for the non-GUP case
+λ = −
+kB
+[SB + CSc] =
+m2
+p
+4πM2 ,
+(45)
+and for the GUP case, we have
+λ = −
+kB
+[SGUP + CGUP]
+(46)
+≈
+m2
+p
+4πM2 +
+3αm4
+p
+64πM4 +
+αm4
+p log
+�
+4M
+mp
+�
+32πM4
+by ignoring the higher order terms in α. This means that
+for the non-GUP case, we define the mass scale
+Mc =
+mp
+2
+√
+πλ
+,
+(47)
+which differentiates the two regions and can be further
+used to define the characteristic length scale LR, which
+can be written as
+LR = 2lp
+√
+πλ,
+(48)
+where we have defined LR = GMc/c2. For the GUP
+case, we would expect the characteristic length scale
+LRgup ≈ LR + α f (λ) by using equation (47), where f is
+a function of the nonextensivity parameter λ. However,
+we can not solve it exactly, and it again shows the effects
+of α and λ for the values of M greater than the GUP cor-
+rected mass scale. Interestingly, for the non-GUP case,
+the heat capacity is positive for the values greater than
+this scale, and below this scale, black holes have neg-
+ative heat capacity. This means that black holes with
+higher masses than Mc are thermodynamically stable
+and with masses lower than Mc, they are unstable. Note
+that, if we exclude quantum gravity effects, LR should
+be greater than lp. This puts a numerical constraint on
+the nonextensive parameter λ > 1/4π and this can also
+be derived by considering Mc > mp by excluding the
+quantum gravity effects. In [49, 50, 52], the authors de-
+rived this constraint as λ > 1/π because they consid-
+ered LR = 2GMc/c2 as characteristic length scale for λ,
+where the extra 2 in LR is motivated by Schwarzschild
+radius rh = 2GM/c2. We believe that the proper way to
+introduce the length or mass scale for λ should be irre-
+spective of the definition which is motivated by rh.
+3.
+Sparsity of the Rényi Radiation
+In order to calculate the sparsity of Rényi radiation,
+we replace T with TR in (23), and so the sparsity param-
+eter ηR reads
+ηR =
+ηH
+[1 + λ
+kB SB]2 .
+(49)
+Replacing T with TRgup and using GUP modified area
+AGUP in equation (23), the GUP modified sparsity pa-
+rameter ηRgup reads
+ηRgup =
+ηGUP
+[1 + λ
+kB SGUP]2 .
+(50)
+From (49), we conclude that the sparsity parameter ηR
+λ=0
+λ=0.5
+λ=1
+λ=0
+λ=0.5
+λ=1
+0.0
+0.5
+1.0
+1.5
+2.0
+0
+20
+40
+60
+80
+M
+ηR
+Figure 8.
+Sparsity ηR of a Rényi blackhole vs its mass M.
+Dashed lines represent GUP corrected cases, λ → 0 limit is
+the Bekenstein-Hawking case.
+depends on both the mass of the black hole and the
+nonextensivity parameter λ.
+From Fig.
+(8), we can
+easily see that the radiation is not sparse initially and
+then, at the final stages of the evaporation, the sparsity
+
+8
+grows, reaching the value of ηH, when M approaches to
+zero. For the GUP case, initially, the behavior of spar-
+sity is similar to the non-GUP case, however, when M
+approaches Mr, it has a finite value which is much less
+than the sparsity of Hawking radiation for the non-GUP
+and GUP cases. Again, we can see the bump before M
+reaches Mr, which is due to the effect of GUP correc-
+tions to the Rényi temperature and GUP corrections to
+the area.
+C.
+Tsallis-Cirto Black Hole Entropy
+Tsallis-Cirto black hole entropy [32] is based on key
+principles of Gibbs thermodynamics. First, the entropy
+must be extensive and additive, and second, the entropy
+and associated temperature for a thermodynamic sys-
+tem must satisfy the Legendre structure. For the case
+of black holes, if we rely on the definition of Beken-
+stein entropy, then black holes are considered to be
+two-dimensional thermodynamic objects since Beken-
+stein entropy scales with area and Bekenstein entropy
+and Hawking temperature fulfill the Legendre struc-
+ture. However, if we consider a black hole as a (3 + 1)
+dimensional thermodynamic object, then the Bekenstein
+entropy is thought to be nonextensive due to its area
+scaling and also because it follows a nonadditive com-
+position rule S12 = S1 + S2 + 2√S1
+√S2 (see e.g. [90]),
+whereas Gibbs statistical mechanics or thermodynam-
+ics is based on the extensive and additive properties of
+the entropy. This indicates that Bekenstein entropy vio-
+lates a key principle of classical Gibbs thermodynamics
+and that new definitions of entropy and temperature for
+black holes are required in order to comply with the fun-
+damental principles of thermodynamics in the case of
+(3 + 1)-dimensional black holes. Therefore, Tsallis and
+Cirto proposed the following entropy definition [32, 38].
+Sδ
+kB
+=
+�SB
+kB
+�δ
+,
+(51)
+where δ > 0 is a real parameter and it follows the com-
+position rule for a composite thermodynamic system,
+which is given by
+Sδ12 = kB
+��Sδ1
+kB
+�1/δ
++
+�Sδ2
+kB
+�1/δ�δ
+.
+(52)
+In this context, the SB is additive, and Sδ is nonadditive.
+For δ = 3/2, Sδ is proportional to the volume for the
+case of the Schwarzschild black hole, and so it is an ex-
+tensive quantity. The corresponding Tsallis-Cirto tem-
+perature can be written by using the Clausius relation
+[53]
+Tδ = TH
+δ
+�SB
+kB
+�1−δ
+,
+(53)
+and it scales with 1/M2 for δ = 3/2, i.e., Tδ ∝ 1/M2, for
+the case of Schwarzschild black hole. GUP corrections
+to the Tsallis-Cirto black hole entropy can be obtained
+by the GUP corrected Bekenstein entropy SGUP given
+by (15) into (51), which results in
+Sδgup
+kB
+=
+�SGUP
+kB
+�δ
+,
+(54)
+and the corresponding GUP-modified Tsallis-Cirto tem-
+perature can be derived from the Clausius relation, giv-
+ing
+Tδgup = TGUP
+δ
+�SGUP
+kB
+�1−δ
+.
+(55)
+From the Figs.
+(9) and (10), it shows that the evap-
+δ=0.4
+δ=0.7
+δ=1.5
+δ=0.4
+δ=0.7
+δ=1.5
+0.0
+0.5
+1.0
+1.5
+2.0
+0
+5
+10
+15
+20
+M
+Sδ
+Figure 9. Tsallis-Cirto entropy ST of a black hole vs its mass
+M. Dashed lines represent GUP-corrected cases in this figure
+oration process stops at the critical value Mr for the
+Tsallis-Cirto case when GUP corrections are included.
+This means that the final state of the black hole for the
+Tsallis-Cirto case is also a remnant with finite entropy
+and temperature. Generally, for the non-GUP case, the
+parameter δ plays a significant role. For δ > 1/2, the
+Tsallis-Cirto entropy behaves similarly to Bekenstein en-
+tropy and increases exponentially with mass, whereas
+for δ < 1/2, it increases with mass sub-linearly. For
+δ = 1/2, the entropy depends linearly on mass, and
+in this case, Tsallis-Cirto temperature becomes constant.
+Furthermore, the behavior of the Tsallis temperature is
+similar to the Hawking temperature for δ > 1/2 while
+for δ < 1/2, the behavior is completely different for the
+non-GUP case and, interestingly, it behaves like Rényi
+temperature for the GUP-corrected case. Note that, un-
+like λ parameter of the Rényi entropy, δ is not associated
+with the length scale for the non-GUP case. On the other
+hand, introducing GUP corrections to Tsallis-Cirto en-
+tropy, one can define a characteristic length scale for δ
+as well.
+1.
+Heat Capacity for Tsallis-Cirto black holes
+Following the previous subsection, the heat capacity
+for the Tsallis-Cirto case can be written in terms of Csc,
+
+9
+δ=0.4
+δ=0.7
+δ=1.5
+δ=0.4
+δ=0.7
+δ=1.5
+0.0
+0.5
+1.0
+1.5
+2.0
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+M
+Tδ
+Figure 10.
+Temperature Tδ vs the mass M for Tsallis-Cirto
+black hole entropy. Dashed lines correspond to a GUP case.
+and SB
+Cδ = CSc
+�
+SB
+SB − (δ − 1)CSc
+�
+,
+(56)
+where for the Schwarzschild black hole, we have CSc =
+−2SB. For δ = 1/2, we have infinite heat capacity for
+all masses. For δ < 1/2, we have positive heat capac-
+ity values and negative heat capacity for δ > 1/2. This
+means that black holes are thermodynamically stable for
+δ < 1/2, and unstable for δ > 1/2. For the GUP correc-
+δ=0.4
+δ=0.7
+δ=1.5
+δ=0.4
+δ=0.7
+δ=1.5
+0.0
+0.5
+1.0
+1.5
+2.0
+-20
+-10
+0
+10
+20
+M
+Cδ
+Figure 11. Heat Capacity Cδ for Tsallis-Cirto black hole en-
+tropy. Dashed lines correspond to a GUP case.
+tions, we can write the GUP-corrected heat capacity as
+Cδgup = CGUP
+�
+SGUP
+SGUP − (δ − 1)CGUP
+�
+.
+(57)
+Note that from equations (15) and (22), we have
+−2SGUP ̸= CGUP, therefore, we can find an associated
+characteristic length scale Lδgup for the δ parameter, for
+which, we have two regions, which corresponds to pos-
+itive and negative values of GUP corrected heat capac-
+ities. The length scale Lδgup can be found by using the
+singular points of the above equation (57) for δ, which is
+given by
+δ = SGUP
+CGUP
++ 1.
+(58)
+One could solve the above equation (58) for mass M,
+which gives Lδgup as a function of δ. However, it is ana-
+lytically not possible. One may use the perturbative ap-
+proach to solve the equation for M and define the corre-
+sponding length scale or mass scale. From the Figs. (9)
+and (11), for δ < 1/2, and below Lδgup, the GUP cor-
+rected Tsallis-Cirto entropy behaves like SR and it gives
+positive GUP modified heat capacity for the GUP case.
+For values δ > 1/2, Lδgup does not exist as (58) yields
+imaginary numbers. Thus, it gives negative heat capac-
+ity, implying that GUP-corrected Tsallis black holes are
+thermodynamically stable for δ < 1/2, and unstable for
+δ > 1/2.
+2.
+Sparsity of the Tsallis-Cirto Radiation
+By following the previous subsection, and using the
+Tsallis-Cirto temperature, we can write the sparsity pa-
+rameter ηδ for Tsallis-Cirto radiation as
+ηδ = ηHδ2
+�SB
+kB
+�2δ−2
+,
+(59)
+and the GUP-corrected sparsity ηδgup, by using (23) and
+(55), it can be written as
+ηδgup = ηGUPδ2
+�SGUP
+kB
+�2δ−2
+.
+(60)
+Fig. (12) depicts the sparsity vs. mass relationship. For
+δ=0.8
+δ=1
+δ=1.1
+δ=0.8
+δ=1
+δ=1.1
+0.0
+0.5
+1.0
+1.5
+2.0
+0
+50
+100
+150
+200
+M
+ηδ
+Figure 12.
+Sparsity ηδ for Tsallis-Cirto black hole entropy.
+Dashed lines correspond to a GUP case.
+the Tsallis-Cirto temperature, the sparsity scales with
+M4δ−4. Again, the value of δ, significantly changes the
+behavior of the sparsity. It should be noted that the spar-
+sity parameter is now affected by mass as well as δ and
+the GUP-parameter α. In the non-GUP case, ηδ = ηH
+for δ = 1. When δ > 1, the value of ηδ is initially very
+high and approaches zero at the end of the black hole
+evaporation. This means that, initially, the Tsallis-Cirto
+radiation is highly sparse, and during the final stages of
+evaporation, it is not sparse at all. In this way, for δ < 1,
+Tsallis-Cirto radiation is initially not sparse, but at the
+end of the evaporation, it is extremely sparse with the
+sparsity parameter infinite. For the GUP case, initially,
+the behavior is the same as for the non-GUP case, but
+
+10
+when the mass approaches the order of Planck mass,
+i.e., the remnant mass Mr, the sparsity parameter de-
+creases to some finite values for each case. Note that all
+these finite values of sparsity parameters are less than
+the standard sparsity parameter ηH.
+D.
+Sharma-Mittal Entropy
+Sharma-Mittal (SM) is an entropic form [40, 104] that
+generalizes the Rényi and Tsallis entropies. It is defined
+as
+SSM = 1
+R
+�
+�
+�
+W
+∑
+i=1
+p1−λ
+i
+� R
+λ
+− 1
+�
+�
+(61)
+where R is another free parameter that is introduced in
+SM entropy. Under the equiprobability condition of the
+states [69], the above equation (61) reduces to
+SSM = kB
+R
+�
+(1 + λ
+kB
+ST)R/λ − 1
+�
+,
+(62)
+where R → λ limit yields the Tsallis entropy, and R → 0
+yields Rényi entropy. The Sharma-Mittal entropy obeys
+the same general nonextensive composition rule (32).
+Assuming that the Bekenstein entropy SB is the same
+as the Tsallis entropy ST , we can write SSM for the case
+of a Schwarzschild black hole as
+SSM = kB
+R
+�
+(1 + λ
+kB
+SB)R/λ − 1
+�
+,
+(63)
+and replacing SGUP with ST in equation (62), the GUP
+corrected SM entropy SSMgup reads as
+SSMgup = kB
+R
+�
+(1 + λ
+kB
+SGUP)R/λ − 1
+�
+.
+(64)
+The corresponding temperatures can be found by using
+the Clausius relation, as
+TSM = TH(1 + λ
+kB
+SB)1− R
+λ ,
+(65)
+and the GUP corrected SM temperature TSMgup reads as
+TSMgup = TGUP(1 + λ
+kB
+SGUP)1− R
+λ .
+(66)
+We can now define the inverse temperature parameters
+for GUP and non-GUP cases by using the above equa-
+tions (65) and (66), which are given, for the non-GUP
+case, as
+βSM = S′
+SM
+kBc2 = β(1 + λ
+kB
+SB)
+R
+λ −1,
+(67)
+and for the GUP case, as
+βSMgup =
+S′
+SMgup
+kBc2
+= βGUP(1 + λ
+kB
+SGUP)
+R
+λ −1.
+(68)
+R=0.2
+R=0.6
+R=0.9
+R=0.2
+R=0.6
+R=0.9
+0.0
+0.5
+1.0
+1.5
+2.0
+0
+10
+20
+30
+40
+50
+M
+SSM
+Figure 13.
+Plot of the Sharma-Mittal entropy for λ = 0.7.
+Dashed lines correspond to a GUP case.
+Since SM entropy is the generalization of the Tsallis and
+Rényi entropy, the behavior of the temperature and the
+entropy are similar to that of SB and SR and TH and TR
+for different values of Sharma-Mittal parameter R. Also,
+the black hole does not evaporate in this case as well,
+and the evaporation process stops at Mr, leaving the fi-
+nal state of the black hole as a remnant having finite en-
+tropy and temperature. The plots of SM entropy and
+temperature are given in Figs. 13 and 14.
+R=0.2
+R=0.6
+R=0.9
+R=0.2
+R=0.6
+R=0.9
+0.0
+0.5
+1.0
+1.5
+2.0
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+M
+TSM
+Figure 14. Sharma-Mittal temperature for λ = 0.7. Dashed
+lines correspond to a GUP case.
+1.
+Heat Capacity for Sharma-Mittal Black Holes
+By following the previous subsections, we can calcu-
+late the heat capacity CSM for the SM black holes as
+CSM =
+CSc(1 + λ
+kB SB)
+R
+λ
+(1 + λ
+kB SB) − λ
+kB CSc
+�
+R
+λ − 1
+� ,
+(69)
+and for the GUP SM black holes case, it reads as
+CSMgup =
+CGUP(1 + λ
+kB SGUP)
+R
+λ
+(1 + λ
+kB SGUP) − λ
+kB CGUP
+�
+R
+λ − 1
+� . (70)
+The plots of (69) and (70) are given in Fig. 15. Similarly
+as for the Rényi case, we define the characteristic length
+scale LSM in terms of λ and R by employing the singular
+
+11
+point of CSM. For the non-GUP case, we have such a
+singular point for
+λ = RCSc − kB
+CSc + SB
+.
+(71)
+From (71), we can easily define the following character-
+istic relation by solving it for M, which reads
+LSM = 2lp
+�
+π(λ − 2R),
+(72)
+where LSM = GMc/c2, and the mass scale Mc is defined
+as
+Mc =
+mp
+2
+�
+π(λ − 2R)
+.
+(73)
+Similarly, one can define LSMgup for the GUP case by
+using the following singular point at
+λ = RCGUP − kB
+CGUP + SGUP
+,
+(74)
+and solve it for M. Since the analytic solution is not pos-
+sible, one could use a perturbative approach to find the
+GUP corrections to LSM up to the first order in α. Note
+that R → 0 limit yields the LR for the Rényi case. For
+λ − 2R > 0 and M > Mc, the heat capacity is positive
+for both non-GUP and GUP cases, and for M < Mc,
+the heat capacity is negative for both non-GUP and GUP
+cases.
+R=0.2
+R=0.6
+R=0.9
+R=0.2
+R=0.6
+R=0.9
+0.0
+0.5
+1.0
+1.5
+2.0
+-30
+-20
+-10
+0
+10
+20
+30
+M
+CSM
+Figure 15. Heat capacity CSM for Sharma-Mittal entropy for
+λ = 0.7. Dashed lines correspond to a GUP case.
+2.
+Sparsity of the Sharma-Mittal Radiation
+The sparsity parameter ηSM can be derived by apply-
+ing the Sharma-Mittal temperature to (23), and reads
+ηSM = ηH(1 + λ
+kB
+SB)2( R
+λ −1),
+(75)
+and for the GUP case, substituting equations (66) and
+(27) in (23), the GUP modified sparsity parameter for the
+Sharma-Mittal radiation reads as
+ηSMgup = ηGUP(1 + λ
+kB
+SGUP)2( R
+λ −1).
+(76)
+R=0.45
+R=0.5
+R=0.6
+R=0.45
+R=0.5
+R=0.6
+R=0.3
+R=0.3
+0.0
+0.5
+1.0
+1.5
+2.0
+0
+100
+200
+300
+400
+500
+600
+M
+ηSM
+Figure 16. Sparsity for Sharma-Mittal entropy for λ = 0.4.
+Dashed lines correspond to a GUP case.
+The plots of the sparsity for SM (75) and SM GUP (76)
+cases are given in Fig. 16. The behavior of the sparsity
+parameter again depends on the Sharma-Mittal param-
+eter R in addition to the nonextensive parameter λ and
+also the GUP parameter α in the case of GUP corrections.
+For the values of λ and R, which satisfy the inequality
+λ + 2R > 0, the sparsity of the Sharma-Mittal radiation
+behaves like the sparsity of the Rényi radiation for both
+non-GUP and GUP cases. This means that, initially, the
+Sharma-Mittal radiation is not sparse, and at the end
+of the evaporation, its value approaches the value of
+Hawking’s case, i.e., ηH, for the non-GUP case. For the
+GUP case, when M approaches Mr, the Sharma-Mittal
+sparsity parameter approaches some finite value, which
+is less than ηH. For λ > R, initially, the Sharma-Mittal
+sparsity parameter is higher than ηH and its value ex-
+actly approaches ηH at the end of the evaporation, while
+for the case of GUP, it approaches to some finite value
+less than ηH. It is interesting to note that, for α > 0, the
+GUP modified sparsity parameter is always less than the
+standard Hawking case.
+E.
+Kaniadakis Entropy
+Kaniadakis entropy [42, 70] is a type of nonextensive
+entropy that results from the Lorentz transformation of
+special relativity. It is a single parameter deformation of
+Gibbs entropy in which The standard Gibbs entropy is
+generalized to the relativistic regime with the help of a
+new parameter K that is connected to the dimensionless
+rest energy of the various parts of a multibody relativis-
+tic system. The Kaniadakis entropy SK is defined as
+SK = kB logK Ω
+(77)
+where
+logK(Ω) = ΩK − Ω−K
+2K
+.
+(78)
+Considering SB = kB ln Ω, which means that the num-
+ber of microstates Ω for a black hole is proportional to
+
+12
+eSB/kB, the above equation (77) can be written in the fol-
+lowing form
+SK = kB
+K sinh
+�
+K SB
+kB
+�
+,
+(79)
+where we have used equation (78) for the sinh x function
+and used the relation Ω = eSB/kB. Replacing SB with
+SGUP, the GUP modified Kaniadakis entropy SKGUP
+reads as
+SKGUP = kB
+K sinh
+�
+K SGUP
+kB
+�
+.
+(80)
+Note that, in the limit K → 0, SK reduces to Gibbs en-
+K=0.1
+K=0.5
+K=0.9
+K=0.1
+K=0.5
+K=0.9
+0.0
+0.5
+1.0
+1.5
+2.0
+0
+20
+40
+60
+80
+100
+M
+SK
+Figure 17. Kaniadakis Entropy SK vs mass M. Dashed lines
+correspond to a GUP case.
+tropy. In Fig. (17), one can see the characteristic form of
+sine hyperbolic (sinh) function for different small val-
+ues of K which shows the similar behaviour like the
+Bekenstein entropy.
+As expected, for the GUP case,
+black holes do not evaporate completely and the final
+state of the black hole is a remnant like for the case of
+standard GUP modified Bekenstein-Hawking case. Fur-
+thermore, as K increases, the entropy increases sharply.
+By using the Clausius relation, the corresponding Kani-
+adakis black black hole temperature TK reads as
+TK = TH sech
+�
+K SB
+kB
+�
+,
+(81)
+and the GUP modified Kaniadakis temperature TKGUP
+can be written as
+TKgup = TGUP sech
+�
+K SGUP
+kB
+�
+.
+(82)
+By using (81) and (82), one can write the following in-
+verse temperature parameters βK as follows
+kBβK = kBβ cosh
+�
+K SB
+kB
+�
+,
+(83)
+and for the GUP case, βKGUP reads
+kBβKgup = kBβGUP cosh
+�
+K SGUP
+kB
+�
+,
+(84)
+K=0.1
+K=0.5
+K=0.9
+K=0.1
+K=0.5
+K=0.9
+0.0
+0.5
+1.0
+1.5
+2.0
+0.00
+0.05
+0.10
+0.15
+0.20
+M
+TK
+Figure 18. Kaniadakis temprature TK vs mass. Dashed lines
+correspond to a GUP case.
+which can further be used to find the heat capacities
+for Kaniadiakis black holes. Fig. (18) shows that Ka-
+niadakis temperature behaves as Hawking temperature
+with a slight change depending on the parameter K. For
+the GUP case, it stops at some finite value, when M ap-
+proaches to Mr during the final stages of the black hole
+evaporation process.
+1.
+Heat capacity for Kaniadakis Black Holes
+The heat capacities for Kaniadakis entropy can be
+calculated by following the previous subsections. For
+the non-GUP case, the heat capacity CK for Kaniadakis
+black hole reads as
+CK = CSc
+cosh2[K SB
+kB ]
+cosh[K SB
+kB ] − CSc sinh[K SB
+kB ]
+,
+(85)
+and for the GUP modified heat capacity, CKgup, it can
+written as
+CKgup = CGUP
+cosh2[K SGUP
+kB ]
+cosh[K SGUP
+kB ] − CGUP sinh[K SGUP
+kB ]
+. (86)
+From Fig. (19), one can easily notice the negative heat
+K=0.1
+K=0.5
+K=0.9
+K=0.1
+δ=0.5
+K=0.9
+0.0
+0.5
+1.0
+1.5
+2.0
+-100
+-80
+-60
+-40
+-20
+0
+M
+CK
+Figure 19. Kaniadakis heat capacity CK vs mass M. Dashed
+lines correspond to a GUP case.
+capacities for all values of K.
+This means that Kani-
+adakis black holes are thermodynamically unstable for
+all M.
+
+13
+K=0.1
+K=0.5
+K=0.7
+K=0.1
+K=0.5
+K=0.7
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+0
+50
+100
+150
+200
+250
+300
+M
+ηK
+Figure 20. Sparsity ηK for Kaniadakis radiation vs mass M
+of Kaniadakis black hole. Dashed lines correspond to a GUP
+case.
+2.
+Sparsity of the Kaniadakis Radiation
+The sparsity parameter ηK for the Kaniadakis radia-
+tion can be derived by applying (81) into (23), and reads
+ηK = ηH cosh2
+�
+K SB
+kB
+�
+,
+(87)
+and for the GUP modified sparsity parameter ηKGUP, we
+apply (82) and (27) into (23), to obtain
+ηKGUP = ηGUP cosh2
+�
+K SGUP
+kB
+�
+.
+(88)
+From Fig.
+(20), the sparsity parameter for the Kani-
+adakis case is always high from the beginning of the
+evaporation process as compared to the standard Beken-
+stein Hawking case. However, for the non-GUP case, ηK
+approaches to the value of ηH at the end of the evapo-
+ration. For the GUP case, again, it approaches to some
+finite value of sparsity when M approaches Mr, which
+is always less than the sparsity parameter ηH. Further-
+more, we see that increasing value of K directly results
+in sparser Kaniadakis radiation.
+F.
+Barrow entropy
+Barrow entropy [44] is an entropic form that has no
+statistical roots, but is closely tied to black hole hori-
+zon geometry.
+It is proposed to replace the smooth
+black hole horizon with a fractal of spheres known as
+a sphereflake. This structure is distinguished by its frac-
+tal dimension d f , where 3 ≥ d f ≥ 2, and results in an
+effective horizon area of r+d f , where r+ is the horizon
+radius. As a result, in this scenario, the horizon area is
+modified, yielding Barrow entropy as below SBarrow
+SBarrow = kB
+� A
+Ap
+�1+ ∆
+2
+(89)
+where A is the horizon area, Ap is the Planck area, and
+∆ is the parameter directly tied to the fractal dimension
+d f through ∆ = d f − 2. In this form, ∆ can take values
+between 0 and 1, and ∆ → 1 limit yields maximally frac-
+tal structure, where the horizon area effectively behaves
+like a 3−dimensional volume, while ∆ → 0 limit yields
+the well-known Bekenstein area law where no fractal-
+ization occurs. Although Barrow entropy offers a dif-
+ferent picture in the geometrical sense, in its essence,
+it has the same form as Tsallis-Cirto entropy. We can
+see that they are equivalent by making the following
+parametrization in Tsallis-Cirto entropy [105]
+δ → 1 + ∆
+2
+(90)
+Thus, qualitatively, both entropic forms yield the same
+temperatures and heat capacities as a function of black
+hole mass. Similarly, the Tsallis-Cirto entropy limit ∆ =
+1 (δ = 3/2 for Sδ) yields an extensive, but still nonaddi-
+tive entropy for black holes.
+IV.
+SUMMARY AND DISCUSSION
+We have investigated the nonextensive thermody-
+namics of black holes, the impact of the generalized
+uncertainty principle on nonextensive thermodynamics
+quantities, and the sparsity and GUP-modified sparsity
+of the radiation in the nonextensive scenario. We have
+found that all nonextensive black hole entropies and as-
+sociated temperatures have finite values at the end of
+the black hole evaporation process due to GUP modifi-
+cations, indicating the existence of a remnant at the end
+of the evaporation. This means that black holes do not
+evaporate fully in the nonextensive setup as well. We
+have also investigated the sparsity parameter in each
+nonextensive configuration. Despite the fact that the be-
+havior of the sparsity parameter varies for each nonex-
+tensive scenario, GUP consistently lowers the radiation
+sparsity in all circumstances toward the end of the evap-
+oration process.
+Even though multiple nonextensive
+scenarios have the same temperatures and entropic pro-
+files, we have demonstrated that the sparsity parameter
+can be used to distinguish between them.
+We have introduced GUP and GUP-corrected thermo-
+dynamic parameters and have revised otherwise well-
+known GUP corrected quantities to a better form in
+which the two crucial limits - the extensivity limit for
+λ → 0 and the HUP limit for α → 0 - are easily iden-
+tified. Even though GUP corrections on Rényi entropy
+in black hole thermodynamics have been researched in
+the literature, we presented a full discussion of it in or-
+der to help readers distinguish between various sorts of
+nonextensive scenarios. Additionally, we have provided
+non-perturbative results for each quantity, with a focus
+on the Rényi sparsity parameter, which rises (as shown
+by the "bump" in Fig. (8)) before the value of the rem-
+nant mass. This is because it is assumed that the area
+can change as a result of the GUP-modified Bekenstein
+entropy, which is explicitly shown in (28). This indi-
+
+14
+cates that AGUP as well as TGUP have an impact on the
+sparsity parameter. Furthermore, we have introduced
+black hole mass scale Mc = mp/2
+√
+πλ for the nonexten-
+sive parameter λ for the Rényi black hole quantities and
+we defined corresponding characteristic length for λ in
+terms of Mc, i.e. LR = GMc/c2 = 2lp
+√
+πλ. We have
+shown that, for M > Mc, the heat capacity is positive
+and hence black holes in Rényi scenario are thermody-
+namically stable, while for M < Mc, the heat capacity is
+negative and SR and TR behave like Bekenstein entropy
+SB and Hawking temperature TH, hence unstable black
+holes.
+Similarly, we have also analyzed the thermodynamic
+black hole quantities associated with Tsallis-Cirto black
+hole entropy.
+Particularly, we have focused on GUP
+corrections and the sparsity of the Tsallis-Cirto radia-
+tion. We have shown that, when GUP corrections are
+included, Tsallis-Cirto entropy and associated temper-
+ature have a finite value, and this proves that the fi-
+nal state of the black hole is also a remnant with finite
+entropy and temperature. It is interesting to note that
+the Tsallis-Cirto parameter δ plays a significant role. We
+have found that, for δ > 1/2, Tsallis-Cirto entropy and
+temperature behave similarly to Bekenstein entropy and
+Hawking temperature, and hence have negative heat ca-
+pacity. For the GUP case, Tsallis-Cirto temperature be-
+haves like Rényi temperature and has positive heat ca-
+pacity for δ < 1/2. This means that, in this framework,
+we must have δ < 1/2 for thermodynamic stability of
+black holes. In this way, we have shown that the Tsallis-
+Cirto sparsity parameter is very high during the start of
+the evaporation for δ > 1, but it approaches zero at the
+the end of the black hole evaporation. On the contrary,
+for δ < 1, we have shown that the Tsallis-Cirto radi-
+ation is not sparse during the start of the evaporation,
+but at the end of the evaporation, the sparsity parame-
+ter becomes infinite and hence shows the highly sparse
+Tsallis-Cirto radiation. The behavior of the GUP case is
+initially the same as that of the non-GUP case, but as the
+mass approaches the order of Planck mass, i.e., Mr, the
+Tsallis-Cirto sparsity parameter for each case reduces to
+some finite values. It should be noted that all of these fi-
+nite sparsity parameter values are less than the sparsity
+parameter ηH for the standard Hawking case.
+We have also shown that the behavior of the tempera-
+ture and the entropy for the Sharma-Mittal case is com-
+parable to that of SB and SR and TH and TR for differ-
+ent values of the Sharma-Mittal parameter R since the
+Sharma-Mittal entropy is the extension of the Tsallis and
+Rényi entropy. Also, in this instance, the black hole does
+not evaporate, and the evaporation process stops at Mr,
+leaving the black hole in its ultimate state as a remnant
+of mass Mr with finite entropy and temperature. We
+have analysed the sparsity of the Sharma-Mittal radia-
+tion and compared it with the standard Hawking case.
+We have found that the sparsity of the Sharma-Mittal ra-
+diation behaves similarly to the Rényi radiation in both
+non-GUP and GUP instances for values of λ and R that
+fulfill the condition λ − 2R > 0.
+This indicates that
+the Sharma-Mittal radiation is initially not sparse and
+that by the end of the evaporation, its value approaches
+that of Hawking’s scenario, or ηH, for the non-GUP case.
+When M approaches Mr for the GUP case, the Sharma-
+Mittal sparsity parameter approaches a finite value that
+is smaller than ηH. For the case, R > λ, we have shown
+that the Sharma-Mittal sparsity parameter is initially
+larger than ηH and its value exactly approaches ηH by
+the end of the evaporation whereas for the case of GUP,
+it approaches a finite value that is smaller than ηH. It is
+noteworthy to notice that, for α > 0, the GUP modified
+sparsity parameter is always lower than the standard
+Hawking case. Moreover, we have also introduced the
+characteristic mass scale, Mc = mp/2
+�
+π(λ − 2R), for
+the Sharma-Mittal scenario and also, defined the corre-
+sponding characteristic length scale LSM = GMc/c2 =
+2lp
+�
+π(λ − 2R). We have shown that, for M > Mc with
+λ − 2R > 0, the black holes are thermodynamically sta-
+ble in the Sharma-Mittal scenario for both GUP and non-
+GUP cases, while for M < Mc, black holes are thermo-
+dynamically unstable.
+We have also examined the Kaniadakis thermody-
+namic black hole quantities, and the results demonstrate
+that, with a little variation depending on the parame-
+ter K, Kaniadakis entropy and temperature behave sim-
+ilarly to Bekenstein entropy and Hawking temperature.
+In the case of the GUP, both quantities reach a finite
+value as black hole mass approaches Mr during the late
+stages of the black hole evaporation process. It results in
+negative heat capacity for all values of K, indicating that
+Kaniadakis black holes are thermodynamically unstable
+for all values of black hole mass. Furthermore, in con-
+trast to the typical Hawking example, the sparsity pa-
+rameter for the Kaniadakis instance is consistently high
+from the beginning of the evaporation process. For the
+non-GUP example, however, ηK approaches the value of
+ηH at the end of the evaporation. In the GUP situation,
+it approaches some finite value of sparsity when M ap-
+proaches Mr, which is always smaller than the sparsity
+parameter ηH. Additionally, it is clear that a rise in the
+value of K causes the Kaniadakis radiation to become
+sparser.
+Finally, our short look onto the Barrow entropy has
+proven its equivalence (though in a restricted range of
+parameters) to the Tsallis-Cirto entropy. In view of that,
+all the discussion of termodynamical quantities for Bar-
+row entropy should be the same as for Tsallis-Cirto.
+ACKNOWLEDGMENTS
+The work of I.C. and M.P.D. was supported by
+the Polish National Science Centre grant No.
+DEC-
+2020/39/O/ST2/02323.
+
+15
+[1] S. W. Hawking, Nature 248, 30 (1974).
+[2] J. D. Bekenstein, Phys. Rev. D 7, 2333 (1973).
+[3] S. W. Hawking, Phys. Rev. D 14, 2460 (1976).
+[4] P. Chen, Y. C. Ong, and D.-h. Yeom, Phys. Rept. 603, 1
+(2015), arXiv:1412.8366 [gr-qc].
+[5] W. G. Unruh and R. M. Wald, Rept. Prog. Phys. 80,
+092002 (2017), arXiv:1703.02140 [hep-th].
+[6] J. M. Bardeen, B. Carter, and S. W. Hawking, Commun.
+Math. Phys. 31, 161 (1973).
+[7] G. W. Gibbons and M. J. Perry, Proc. Roy. Soc. Lond. A
+358, 467 (1978).
+[8] S. W. Hawking and D. N. Page, Commun. Math. Phys.
+87, 577 (1983).
+[9] S. W. Hawking, Phys. Rev. D 13, 191 (1976).
+[10] S. W. Hawking, Commun. Math. Phys. 43, 199 (1975),
+[Erratum: Commun.Math.Phys. 46, 206 (1976)].
+[11] T. Jacobson, Phys. Rev. Lett. 75, 1260 (1995), arXiv:gr-
+qc/9504004.
+[12] E. P. Verlinde, JHEP 04, 029 (2011), arXiv:1001.0785 [hep-
+th].
+[13] T. Padmanabhan, Mod. Phys. Lett. A 25, 1129 (2010),
+arXiv:0912.3165 [gr-qc].
+[14] D. Kubiznak and R. B. Mann, Can. J. Phys. 93, 999 (2015),
+arXiv:1404.2126 [gr-qc].
+[15] M. Cvetic, G. W. Gibbons, D. Kubiznak, and C. N. Pope,
+Phys. Rev. D 84, 024037 (2011), arXiv:1012.2888 [hep-th].
+[16] M. M. Caldarelli, G. Cognola,
+and D. Klemm, Class.
+Quant. Grav. 17, 399 (2000), arXiv:hep-th/9908022.
+[17] R.-G. Cai and S. P. Kim, JHEP 02, 050 (2005), arXiv:hep-
+th/0501055.
+[18] P. C. W. Davies, Proc. Roy. Soc. Lond. A 353, 499 (1977).
+[19] B. P. Dolan, “Where Is the PdV in the First Law
+of Black Hole Thermodynamics?”
+(INTECH, 2012)
+arXiv:1209.1272 [gr-qc].
+[20] D. A. Easson, P. H. Frampton, and G. F. Smoot, Phys.
+Lett. B 696, 273 (2011), arXiv:1002.4278 [hep-th].
+[21] S. W. Hawking, Phys. Rev. Lett. 26, 1344 (1971).
+[22] C. Tsallis, J. Statist. Phys. 52, 479 (1988).
+[23] C. Tsallis, R. Mendes, and A. Plastino, Physica A: Statis-
+tical Mechanics and its Applications 261, 534 (1998).
+[24] C. Tsallis, Introduction to Nonextensive Statistical Mechan-
+ics: Approaching a Complex World (Springer New York,
+NY, 2009).
+[25] S. Abe, S. Martınez, F. Pennini, and A. Plastino, Physics
+Letters A 281, 126 (2001).
+[26] S. Abe and A. K. Rajagopal, Europhysics Letters (EPL)
+55, 6 (2001).
+[27] S. Abe, Physical Review E 63 (2001), 10.1103/phys-
+reve.63.061105.
+[28] T. S. Biró and P. Ván, Physical Review E 83 (2011),
+10.1103/physreve.83.061147.
+[29] M. Nauenberg, Phys. Rev. E 67, 036114 (2003).
+[30] T. S. Biró and P. Ván, Phys. Rev. E 83, 061147 (2011).
+[31] A. S. Parvan and T. S. Biro, Phys. Lett. A 340, 375 (2005),
+arXiv:hep-ph/0407131.
+[32] C. Tsallis and L. J. L. Cirto, Eur. Phys. J. C 73, 2487 (2013),
+arXiv:1202.2154 [cond-mat.stat-mech].
+[33] T. S. Biró and V. G. Czinner, Phys. Lett. B 726, 861 (2013),
+arXiv:1309.4261 [gr-qc].
+[34] V. G. Czinner, Int. J. Mod. Phys. D 24, 1542015 (2015).
+[35] V. G. Czinner and H. Iguchi, Universe 3, 14 (2017).
+[36] V. G. Czinner and H. Iguchi, Phys. Lett. B 752, 306 (2016),
+arXiv:1511.06963 [gr-qc].
+[37] V. G. Czinner and H. Iguchi, Eur. Phys. J. C 77, 892 (2017),
+arXiv:1702.05341 [gr-qc].
+[38] C. Tsallis, Entropy 22, 17 (2019).
+[39] A. Rényi, Acta Mathematica Academiae Scientiarum
+Hungaricae 10, 193–215 (1959).
+[40] B. D. Sharma and D. P. Mittal, J.Comb.Inf.Syst.Sci. 2, 122
+(1977).
+[41] B. D. Sharma and D. P. Mittal, J. Math. Sci 10, 28.
+[42] G.
+Kaniadakis,
+Phys.
+Rev.
+E
+66,
+056125
+(2002),
+arXiv:cond-mat/0210467.
+[43] G.
+Kaniadakis,
+Phys.
+Rev.
+E
+72,
+036108
+(2005),
+arXiv:cond-mat/0507311.
+[44] J. D. Barrow, Physics Letters B 808, 135643 (2020).
+[45] S. Nojiri, S. D. Odintsov, and V. Faraoni, Phys. Rev. D
+104, 084030 (2021), arXiv:2109.05315 [gr-qc].
+[46] S. Nojiri, S. D. Odintsov, and V. Faraoni, Phys. Rev. D
+105, 044042 (2022), arXiv:2201.02424 [gr-qc].
+[47] S. Nojiri, S. D. Odintsov, and V. Faraoni, Int. J. Geom.
+Meth. Mod. Phys. 19, 2250210 (2022), arXiv:2207.07905
+[gr-qc].
+[48] S. Nojiri, S. D. Odintsov, and T. Paul, Phys. Lett. B 825,
+136844 (2022), arXiv:2112.10159 [gr-qc].
+[49] C. Promsiri, E. Hirunsirisawat, and W. Liewrian, Phys.
+Rev. D 102, 064014 (2020), arXiv:2003.12986 [hep-th].
+[50] C. Promsiri, E. Hirunsirisawat, and W. Liewrian, Phys.
+Rev. D 104, 064004 (2021), arXiv:2106.02406 [hep-th].
+[51] L.
+Tannukij,
+P.
+Wongjun,
+E.
+Hirunsirisawat,
+T. Deesuwan,
+and C. Promsiri, Eur. Phys. J. Plus
+135, 500 (2020), arXiv:2002.00377 [gr-qc].
+[52] R.
+Nakarachinda,
+E.
+Hirunsirisawat,
+L.
+Tannukij,
+and P. Wongjun, Phys. Rev. D 104, 064003 (2021),
+arXiv:2106.02838 [gr-qc].
+[53] I. Çimdiker, M. P. Da¸browski,
+and H. Gohar, (2022),
+arXiv:2208.04473 [gr-qc].
+[54] C. Promsiri, E. Hirunsirisawat,
+and R. Nakarachinda,
+Phys. Rev. D 105, 124049 (2022), arXiv:2204.13023 [hep-
+th].
+[55] R. Nakarachinda,
+C. Promsiri,
+L. Tannukij,
+and
+P. Wongjun, (2022), arXiv:2211.05989 [gr-qc].
+[56] E. N. Saridakis, Phys. Rev. D 102, 123525 (2020),
+arXiv:2005.04115 [gr-qc].
+[57] M. P. Dabrowski and V. Salzano, Phys. Rev. D 102, 064047
+(2020), arXiv:2009.08306 [astro-ph.CO].
+[58] S. Nojiri, S. D. Odintsov,
+and V. Faraoni,
+(2022),
+arXiv:2208.10235 [gr-qc].
+[59] N.
+Komatsu,
+Eur.
+Phys.
+J.
+C
+77,
+229
+(2017),
+arXiv:1611.04084 [gr-qc].
+[60] N. Komatsu and S. Kimura, Phys. Rev. D 93, 043530
+(2016), arXiv:1511.04364 [gr-qc].
+[61] R. C. Nunes, E. M. Barboza, Jr., E. M. C. Abreu, and J. A.
+Neto, JCAP 08, 051 (2016), arXiv:1509.05059 [gr-qc].
+[62] Y. Liu, (2022), arXiv:2203.01814 [gr-qc].
+[63] A. Majhi, Phys. Lett. B 775, 32 (2017), arXiv:1703.09355
+[gr-qc].
+[64] G. G. Luciano and M. Blasone, Phys. Rev. D 104, 045004
+(2021), arXiv:2104.00395 [hep-th].
+[65] S. Di Gennaro and Y. C. Ong, (2022), arXiv:2205.09311
+[gr-qc].
+
+16
+[66] S. Di Gennaro, H. Xu,
+and Y. C. Ong,
+(2022),
+arXiv:2207.09271 [gr-qc].
+[67] M. Asghari and A. Sheykhi, Eur. Phys. J. C 82, 388 (2022),
+arXiv:2110.00059 [gr-qc].
+[68] E. M. C. Abreu and J. A. Neto, Phys. Lett. B 835, 137565
+(2022), arXiv:2207.13652 [gr-qc].
+[69] A. Sayahian Jahromi, S. A. Moosavi, H. Moradpour, J. P.
+Morais Graça, I. P. Lobo, I. G. Salako,
+and A. Jawad,
+Phys. Lett. B 780, 21 (2018), arXiv:1802.07722 [gr-qc].
+[70] N. Drepanou, A. Lymperis, E. N. Saridakis,
+and
+K. Yesmakhanova, Eur. Phys. J. C 82, 449 (2022),
+arXiv:2109.09181 [gr-qc].
+[71] S. Carlip, Rept. Prog. Phys. 64, 885 (2001), arXiv:gr-
+qc/0108040.
+[72] K. Konishi, G. Paffuti, and P. Provero, Phys. Lett. B 234,
+276 (1990).
+[73] R. J. Adler and D. I. Santiago, Mod. Phys. Lett. A 14, 1371
+(1999), arXiv:gr-qc/9904026.
+[74] C. Rovelli, Phys. Rev. Lett. 77, 3288 (1996), arXiv:gr-
+qc/9603063.
+[75] K. A. Meissner, Class. Quant. Grav. 21, 5245 (2004),
+arXiv:gr-qc/0407052.
+[76] F. Scardigli, Phys. Lett. B 452, 39 (1999), arXiv:hep-
+th/9904025.
+[77] S.
+Hossenfelder,
+Living
+Rev.
+Rel.
+16,
+2
+(2013),
+arXiv:1203.6191 [gr-qc].
+[78] M. Maggiore, Phys. Lett. B 319, 83 (1993), arXiv:hep-
+th/9309034.
+[79] S. B. Giddings, Phys. Rev. D 46, 1347 (1992), arXiv:hep-
+th/9203059.
+[80] D. N. Page, Phys. Rev. D 13, 198 (1976).
+[81] D. N. Page, Phys. Rev. D 14, 3260 (1976).
+[82] D. N. Page, Phys. Rev. D 16, 2402 (1977).
+[83] S. Schuster, Black Hole Evaporation: Sparsity in Analogue
+and General Relativistic Space-Times, Ph.D. thesis, Victoria
+U., Wellington (2018), arXiv:1901.05648 [gr-qc].
+[84] F. Gray, S. Schuster, A. Van-Brunt, and M. Visser, Class.
+Quant. Grav. 33, 115003 (2016), arXiv:1506.03975 [gr-qc].
+[85] S. Schuster, Class. Quant. Grav. 38, 047002 (2021),
+arXiv:1910.07256 [gr-qc].
+[86] A. Paul and B. R. Majhi, Int. J. Mod. Phys. A 32, 1750088
+(2017), arXiv:1601.07310 [gr-qc].
+[87] A. Alonso-Serrano, M. P. Da¸browski,
+and H. Gohar,
+Phys. Rev. D 97, 044029 (2018).
+[88] A. Alonso-Serrano, M. P. Da¸browski, and H. Gohar, Int.
+J. Mod. Phys. D 27, 1847028 (2018), arXiv:1805.07690 [gr-
+qc].
+[89] Y. C. Ong, JHEP 10, 195 (2018), arXiv:1806.03691 [gr-qc].
+[90] A. Alonso-Serrano, M. P. Da¸browski,
+and H. Gohar,
+Phys. Rev. D 103, 026021 (2021), arXiv:2009.02129 [gr-qc].
+[91] Z.-W. Feng, X. Zhou, S.-Q. Zhou, and D.-D. Feng, An-
+nals Phys. 416, 168144 (2020), arXiv:1808.09958 [gr-qc].
+[92] D. Amati, M. Ciafaloni, and G. Veneziano, Phys. Lett. B
+216, 41 (1989).
+[93] A. Kempf, G. Mangano, and R. B. Mann, Phys. Rev. D
+52, 1108 (1995), arXiv:hep-th/9412167.
+[94] C. Schiller, International Journal of Theoretical Physics
+44, 1629 (2005).
+[95] J.
+D.
+Barrow
+and
+G.
+W.
+Gibbons,
+Monthly
+No-
+tices of the Royal Astronomical Society 446, 3874
+(2014),
+https://academic.oup.com/mnras/article-
+pdf/446/4/3874/9388063/stu2378.pdf.
+[96] M. P. Da¸browski and H. Gohar, Physics Letters B 748, 428
+(2015).
+[97] Y. C. Ong, Phys. Lett. B 785, 217 (2018), arXiv:1809.00442
+[gr-qc].
+[98] D. Gao and M. Zhan, Phys. Rev. A 94, 013607 (2016),
+arXiv:1607.04353 [gr-qc].
+[99] Z.-W. Feng, S.-Z. Yang, H.-L. Li, and X.-T. Zu, Phys. Lett.
+B 768, 81 (2017), arXiv:1610.08549 [hep-ph].
+[100] P. Bosso, S. Das, and R. B. Mann, Phys. Lett. B 785, 498
+(2018), arXiv:1804.03620 [gr-qc].
+[101] D. Gao, J. Wang, and M. Zhan, Phys. Rev. A 95, 042106
+(2017), arXiv:1704.02037 [gr-qc].
+[102] S. Giardino and V. Salzano, Eur. Phys. J. C 81, 110 (2021),
+arXiv:2006.01580 [gr-qc].
+[103] P. Jizba, H. Kleinert, and F. Scardigli, Phys. Rev. D 81,
+084030 (2010), arXiv:0912.2253 [hep-th].
+[104] M. Masi, Physics Letters A 338, 217 (2005).
+[105] E. M. C. Abreu and J. A. Neto, Phys. Lett. B 810, 135805
+(2020), arXiv:2009.10133 [gr-qc].
+

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+arXiv:2301.13327v1  [math.OC]  30 Jan 2023
+Optimization Over the Pareto Front of Nonconvex
+Multi-objective Optimal Control Problems
+C. Yal¸cın Kaya∗
+Helmut Maurer†
+February 1, 2023
+Abstract
+Simultaneous optimization of multiple objective functions results in a set of trade-off, or
+Pareto, solutions. Choosing a, in some sense, best solution in this set is in general a challenging
+task: In the case of three or more objectives the Pareto front is usually difficult to view, if not
+impossible, and even in the case of just two objectives constructing the whole Pareto front
+so as to visually inspect it might be very costly. Therefore, optimization over the Pareto (or
+efficient) set has been an active area of research. Although there is a wealth of literature
+involving finite dimensional optimization problems in this area, there is a lack of problem
+formulation and numerical methods for optimal control problems, except for the convex case.
+In this paper, we formulate the problem of optimizing over the Pareto front of nonconvex
+constrained and time-delayed optimal control problems as a bi-level optimization problem.
+Motivated by existing solution differentiability results, we propose an algorithm incorporating
+(i) the Chebyshev scalarization, (ii) a concept of the essential interval of weights, and (iii) the
+simple but effective bisection method, for optimal control problems with two objectives. We
+illustrate the working of the algorithm on two example problems involving an electric circuit
+and treatment of tuberculosis and discuss future lines of research for new computational
+methods.
+Key words: Multi-objective optimization, Optimal control, Optimization over Pareto
+front, Optimization over efficient set, Numerical methods, Rayleigh problem, Tu-
+berculosis, Time-delay problems.
+1
+Introduction
+We continue our study of optimal control problems where one wishes to minimize simul-
+taneously a number of conflicting objective functionals. These problems are referred to as
+multi-objective optimal control problems and can be expressed in the following concise form:
+(P)
+min
+(x,u,tf)∈X (ϕ1(x(tf), tf), . . . , ϕr(x(tf), tf)) .
+The constraint or the feasible set X in Problem (P) involves a system of differential equations
+(DEs) in the state and control variables x(·) and u(·), respectively, over a time horizon [0, tf].
+The set X also typically involves point and path equality and inequality constraints. The
+DEs and constraints in X might even include time delays in the variables x(·) and u(·). It is
+∗Mathematics, UniSA STEM, University of South Australia, Mawson Lakes, S.A. 5095, Australia. E-mail:
+[email protected] .
+†Institut f¨ur Numerische und Angewandte Mathematik, Westf¨alische Wilhelms-Universit¨at M¨unster,
+M¨unster, Germany. E-mail: [email protected] .
+
+Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer
+2
+worth noting that although each of the objective functionals ϕi(x(tf), tf), i = 1, . . . , r, in (P)
+above constitutes the so-called Mayer form, other forms (Bolza and Lagrange) can easily be
+converted into this form conveniently. Therefore, the general model in (P) caters for a wide
+range of conflicting objectives; for instance, minimization of the energy, the terminal time,
+the deviations from a reference state trajectory, or the uncertainty in measurements, to name
+just a few.
+Broadly speaking, the simultaneous or Pareto minimization in Problem (P) is the process of
+finding a compromise solution, referred to as a Pareto minimum, where the value of some cost
+cannot be improved (i.e., reduced) further, without making the value of some other cost worse
+(i.e., higher). One typical example is the case when one wants to minimize simultaneously
+the fuel expenditure of an airplane travelling from one given city to another and the time the
+airplane takes for this travel: A shorter travel time often requires a higher fuel consumption.
+The set of all such compromise or trade-off solutions form the Pareto set in the optimization
+space, or the Pareto front in the value space. Pareto set and Pareto front are also commonly
+referred to as the efficient set and the efficient front, respectively1.
+The authors of this paper have studied in [27] the problem of constructing the Pareto
+front of Problem (P) involving ODEs and constraints of general form. They discussed and
+demonstrated that for the nonconvex optimal control problems like the one in Problem (P),
+it is better to use the so-called weighted Chebyshev-norm scalarization (or just Chebyshev
+scalarization) to guarantee that the whole Pareto front can be constructed, instead of using
+the traditional weighted-sum scalarization, i.e., a convex combination of the objective func-
+tionals. They discretized the scalarized problem directly and utilized large-scale optimization
+software (the AMPL–Ipopt suite [23,46]) to find the Pareto fronts of two constrained optimal
+control problems as examples, one involving tumour anti-angiogenesis and the other a fed-
+batch bioreactor, by means of what they called a scalarize–discretize–then–optimize approach.
+This approach is in contrast with the other existing discretize–scalarize–then–optimize ap-
+proach (see e.g. [28–30,39]) which scalarizes the discretized problem rather than the original
+(continuous-time) problem.
+An additional benefit of the Chebyshev scalarization is also reported and illustrated in [27]:
+One can compute the whole Pareto front by using only those weights of the objective func-
+tionals within what they name as the essential subinterval of weights, instead of the whole
+interval. Having to compute fewer Pareto solutions over a smaller number of grid points in a
+subinterval is obviously a computational advantage. For further details and an extensive list
+of references on multi-objective optimal control the reader is referred to [27]. Other relevant
+studies on the topic in more recent years have appeared in [13,16].
+Apart from certain trivial or special cases, the Pareto front consists of infinitely many
+solutions to choose from. When a discrete approximation of the front is found the number
+of solutions to choose from is still relatively large since the approximate front is required to
+be accurate enough. Making a decision as to which Pareto solution in the front is the most
+suitable (to the needs of a practitioner) is often very hard for the following reasons.
+• In the case of three or more objectives, the Pareto front might be difficult (if not
+impossible) to view and to carry out a visual inspection (or “eyeballing”) for a decision.
+• Even with two objectives, a visual inspection alone may not be enough to choose a
+desirable solution.
+• Constructing the whole Pareto front might just be too costly a thing to do numerically.
+1These and other definitions will be given in more precise terms in Section 2.
+
+Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer
+3
+Motivated by these drawbacks, minimization of an additional (single) objective function over
+the Pareto front has been of great interest to many researchers over the past decades—see,
+for example, [2, 3, 5, 6, 14, 15, 25, 26, 31, 41, 47]. Despite this rich collection of works, to the
+knowledge of the authors, it was not before the reference [7] that optimization over the Pareto
+front was studied and a numerical method proposed for convex multi-objective optimal control
+problems. In the current paper, we extend the works in [7,27] to nonconvex multi-objective
+optimal control problems and propose a numerical method for carrying out optimization over
+the Pareto front.
+We set the optimal control problem as a bi-level optimization problem as in [7]: One has
+to minimize a master objective functional subject to the minimization of a scalarization of
+Problem (P). The lower level problem uses the Chebyshev scalarization as in [27], as opposed
+to the weighted-sum scalarization in [7]. The problems we consider is in much more general
+form in this paper: We consider nonconvex instead of convex problems compared to [7] and
+we consider problems with time-delay instead of those without time delay compared to [27].
+Just to re-iterate, [27] only proposes a technique to construct the Pareto front, otherwise it
+does not carry out optimization over the Pareto front.
+As the optimization technique over the Pareto front, we propose the simplest possible
+technique, namely the bisection method, over the set of weights for the bi-objective problem,
+which are the parameters of the lower level optimal control problem. Even in this simplest
+case, it is necessary to obtain derivatives with respect to the weight, for which we employ
+difference approximations.
+However, is it guaranteed that these derivatives exist?
+This
+question is answered by [32, 33, 36, 37] which studied the differentiability of a solution of a
+parametric optimal control problem with respect to the parameters. We add a discussion
+concerning these studies in the paper.
+The main algorithm first finds the essential interval of weights over which the first step of
+the bisection method is taken to find a new subinterval. Then the subsequent steps of the
+bisection method are carried out until the stopping criterion is met.
+The algorithm is illustrated on two challenging numerical examples: the Rayleigh problem,
+which comes from an electric circuit, and a compartmental optimal control model for tuber-
+culosis. In the first problem there are constraints on the control variables, and the second
+problem not only has constraints on the two control variables but also time delays on both
+the control and state variables.
+The paper is organized as follows. In Section 2, we introduce the multi-objective optimal
+control problem, discuss scalarization, introduce the problem of optimization over the Pareto
+front, and elaborate on solution differentiability. In Section 3, we first define and explain
+the essential interval of weights, and then introduce the bisection method for our problem
+and provide the detailed algorithm. In Section 4, we illustrate the algorithm on two example
+optimal control problems. Finally, in Section 5, we provide concluding remarks.
+2
+Problem Statement and Preliminaries
+2.1
+Multi-objective optimal control problem
+We consider the following general multi-objective optimal control problem (similar to that
+in [27] but made look slightly more general here) to underlie our study on minimization over
+its Pareto front. The ensuing notation and definitions can also be found in [27] but given
+
+Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer
+4
+here for completeness as well as convenience.
+(OCP)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+min
+(ϕ1(x(tf), tf), . . . , ϕr(x(tf), tf))
+subject to
+˙x(t) = f(x(t), u(t), t) ,
+for a.e. t ∈ [0, tf] ,
+θ(x(0), x(tf), tf) = 0 ,
+�θ(x(0), x(tf ), tf) ≤ 0 ,
+C(x(t), u(t), t) ≤ 0 ,
+for a.e. t ∈ [0, tf] ,
+S(x(t), t) ≤ 0 ,
+for all t ∈ [0, tf] ,
+where r ∈ {2, 3, 4, . . .} is fixed, the state variable x ∈ W 1,∞(0, tf; IRn), ˙x := dx/dt, and
+the control variable u ∈ L∞(0, tf; IRm), with x(t) := (x1(t), . . . , xn(t)) ∈ IRn and u(t) :=
+(u1(t), . . . , um(t)) ∈ IRm. The functions ϕi : IRn × IR+ → IR, f : IRn × IRm × IR+ → IRn,
+θ : IRn × IRn × IR+ → IRp1, �θ : IRn × IRn × IR+ → IRp2, C : IRn × IRm × IR+ → IRp3, and
+S : IRn × IR+ → IRp4, are continuous in their arguments. In this problem, tf is either fixed
+or free. Here, L∞(0, tf; IRm) corresponds to the space of essentially bounded, measurable
+functions equipped with the essential supremum norm. Furthermore, W 1,∞(0, tf; Rn) is the
+Sobolev space consisting of functions x : [0, tf] → Rn whose first derivatives lie in L∞.
+Assume that ϕi(x(tf), tf) ≥ 0, for all i = 1, . . . , r. Note that this assumption can easily be
+met by adding a large enough positive number to each objective functional.
+Note that Problem (OCP) is in general a nonsmooth problem, because it does not require
+differentiability of the objective functionals or the constraints. Moreover, although we have
+stated Problem (OCP) in very broad terms, it can further be generalized, for example by
+adding multi-point constraints, partial differential equations, time delays, etc. In other words,
+although Problem (OCP) is already in a more general form than what one usually encounters
+in applications, it can be further made look more general.
+Of the possible extensions mentioned above, time delays in the state and control vari-
+ables, for instance, can be incorporated into Problem (OCP) by replacing the ODEs in
+Problem (OCP) with
+˙x(t) = f(x(t), x(t − dx), u(t), u(t − du), t) ,
+for a.e. t ∈ [0, tf] ,
+(1a)
+x(t) = x0(t) ,
+for all t ∈ [−dx, 0) ,
+(1b)
+u(t) = u0(t) ,
+for all t ∈ [−du, 0) ,
+(1c)
+where dx, du > 0 are the time delays in the state and control variables, respectively.
+For technical convenience, let tf ≤ tmax
+f
+, where tmax
+f
+> 0 is some constant. Next, we define
+the feasible set, X ⊂ W 1,∞(0, tf; IRn) × L∞(0, tf; IRm) × IR+, such that
+X := {(x, u, tf) : ˙x(t) = f(x(t), x(t − dx), u(t), u(t − du), t) ,
+for a.e. t ∈ [0, tf] ;
+x(t) = x0(t) ,
+for all t ∈ [−dx, 0];
+u(t) = u0(t) ,
+for all t ∈ [−du, 0) ;
+θ(x(0), x(tf), tf) = 0 ; �θ(x(0), x(tf ), tf) ≤ 0 ;
+C(x(t), u(t), t) ≤ 0 , for a.e. t ∈ [0, tf]; S(x(t), t) ≤ 0, for all t ∈ [0, tf]} .
+Note that, for the case of time delays in the state and control variables, we have included
+Equations (1a)–(1c) instead of the ODEs
+˙x(t) = f(x(t), u(t), t) in the set X.
+Define the vector of objective functionals, ϕ(x(tf), tf) := (ϕ1(x(tf), tf), . . . , ϕr(x(tf), tf)).
+The triplet (x∗, u∗, t∗
+f) ∈ X is said to be a Pareto minimum if there exists no (x, u, tf) ∈ X
+
+Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer
+5
+such that ϕ(x(tf), tf) ̸= ϕ(x∗(t∗
+f), t∗
+f) and
+ϕi(x(tf), tf)) ≤ ϕi(x∗(t∗
+f), t∗
+f) ,
+for all i = 1, . . . , r .
+On the other hand, (x∗, u∗, t∗
+f) ∈ X is said to be a weak Pareto minimum if there exists no
+(x, u, tf) ∈ X such that
+ϕi(x(tf), tf)) < ϕi(x∗(t∗
+f), t∗
+f) ,
+for all i = 1, . . . , r .
+The set of all the Pareto and weak Pareto minima is said to be the Pareto set. On the other
+hand, the set of all vectors of objective functional values at the Pareto and weak Pareto min-
+ima is said to be the Pareto front (or the efficient set) of Problem (OCP) in the r-dimensional
+objective value, or outcome, space. Note that the coordinates of a point in the Pareto front
+are simply ϕi(x∗(t∗
+f), t∗
+f), i = 1, . . . , r. Obviously, when r = 2 the Pareto front is in general a
+curve; and when r = 3 the Pareto front is in general a surface.
+2.2
+Scalarization
+In [27], to compute a solution of Problem (OCP), the following single-objective problem (Pw),
+i.e., scalarization, was employed.
+(Pw)
+min
+(x,u,tf)∈X max{w1 ϕ1(x(tf), tf), . . . , wr ϕr(x(tf), tf)} ,
+where wi, i = 1, . . . , r, are referred to as weights, with the vector of weights w defined
+as w := (w1, . . . , wr) ∈ IRr, such that �r
+i=1 wi = 1.
+Problem (Pw) is referred to as the
+weighted Chebyshev problem (or Chebyshev scalarization) because of the weighted Chebyshev
+norm, maxi |wi ϕi(x(tf), tf)| = maxi wi ϕi(x(tf), tf), appearing in the objective. This type of
+scalarization is typically used for nonconvex multi-objective finite-dimensional optimization
+problems, as opposed to the weighted sum scalarization which is effective for convex problems
+but not the nonconvex ones—see, for example, [38].
+Define the set of weights
+Y :=
+�
+w ∈ IRr |
+r
+�
+i=1
+wi = 1
+�
+.
+The following theorem was originally presented in [27, Theorem 1] for the case when there
+was no delay in the state and control variables. It still holds with the set X modified with
+the delayed state equations.
+Theorem 1 (Bijection between sets of weights and Pareto minima [27]) The triplet
+(x∗, u∗, t∗
+f) is a weak Pareto minimum of (OCP) if, and only if, (x∗, u∗, t∗
+f) is a solution of
+(Pw) for some w1, . . . , wr > 0.
+Remark 1 Suppose that Z ⊂ X denotes the Pareto set, namely the set of all Pareto minima
+of (OCP). Then Theorem 1 establishes that there is a bijection between the set of weights
+Y and the Pareto set Z. This implies that by solving (Pw) for all w ∈ Y , one can obtain
+the whole Pareto set Z and in turn get the Pareto front. With numerical computations on
+the other hand, one would of course carry out some discretization of the weight space Y and
+typically get a discrete approximation of the Pareto front. The bijection between Y and Z
+will also help us devise our algorithm for optimization over the Pareto front.
+✷
+
+Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer
+6
+An ideal cost ϕ∗
+i , i = 1, . . . , r, associated with Problem (Pw) is the optimal value of the
+optimal control problem,
+min
+(x,u,tf )∈X ϕi(x(tf), tf) .
+(2)
+Let (x, u, tf) be a minimizer of the single-objective problem in (2). Then ϕ∗
+i := ϕi(x(tf), tf)
+and we also define ϕj := ϕj(x(tf), tf), for j ̸= i and j = 1, . . . , r.
+In the case when ϕ∗
+i is negative, one can simply add a large enough positive number to the
+ith objective, to make the objective positive. In general, it is useful to add a positive number
+to each objective in order to obtain an even spread of the Pareto points approximating the
+Pareto front – see for example [21] for further discussion and geometric illustration. To serve
+this purpose, it is common practice to define the so-called utopian objective values.
+A utopian objective vector associated with Problem (OCP) is given as β∗ := (β∗
+1, . . . , β∗
+r),
+with β∗
+i := ϕ∗
+i − ηi and ηi > 0 for all i = 1, . . . , r. Problem (Pw) can then be equivalently
+written as
+min
+(x,u,tf)∈X max{w1 (ϕ1(x(tf), tf) − β∗
+1), . . . , wr (ϕr(x(tf), tf) − β∗
+r)} .
+In the case when the objective functionals and the constraints in Problem (OCP) are
+differentiable in their arguments, it is worth reformulating Problem (Pw) using a standard
+technique from mathematical programming in the following (smooth) form.
+(OCPw)
+
+
+
+
+
+
+
+
+
+
+
+
+
+min
+α≥0
+(x,u,tf )∈X
+α
+subject to
+w1 (ϕ1(x(tf), tf) − β∗
+1) ≤ α ,
+...
+wr (ϕr(x(tf), tf) − β∗
+r) ≤ α .
+Problem (OCPw) is referred to as goal attainment method [38], as well as Pascoletti-Serafini
+scalarization [22]. We will solve Problem (OCPw) in an algorithm we present in the next
+section, for the two examples we want to study.
+We re-iterate that the “popular” weighted-sum scalarization, given below, fails to generate
+the “nonconvex parts” of a Pareto front.
+(Pws)
+min
+(x,u,tf)∈X
+r
+�
+i=1
+wi ϕi(x(tf), tf) .
+This deficiency is illustrated with a multi-objective optimal control problem, for example, in
+the fed-batch bioreactor problem in [27].
+2.3
+Optimization over the Pareto front
+The main task in this paper is to devise a numerical algorithm for solving the problem of
+decision making as to which Pareto point should be chosen.
+This obviously depends on
+the criterion a decision maker uses in making his/her choice. As pointed in Remark 1, the
+whole Pareto front can be parameterized in terms of the vector of weights w. Therefore,
+Problem (Pw), or equivalently (OCPw), can be regarded as a parametric optimal control
+problem, and it also makes sense to express the decision maker’s objective as the minimization
+of a function of w.
+
+Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer
+7
+Before going ahead with the statement of this problem, we re-write the variables of the
+optimal control problem, with a slight abuse of notation, as xw(t) := x(t, w), uw(t) := u(t, w),
+and tw
+f := tf(w) to emphasize their dependence on the vector of weights w.
+We call the decision maker’s objective function the master objective function, expressed by
+ϕ0(xw, uw, tw
+f ). With the weight vector w of the scalarization treated now as a variable, the
+problem of optimization over the Pareto front reduces to the problem of finding an optimal
+weight w∗. Then the corresponding Pareto minimum is a solution of Problem (OCPw∗).
+The problem of optimizing a master objective function over the Pareto front of (OCP)
+with r ≥ 2 objectives is nothing but a bilevel programming problem and can be written as
+(OPF)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+min
+w∈Y
+ϕ0(xw, uw, tw
+f )
+subject to
+min
+α≥0
+(x,u,tf )∈X
+α
+subject to w1 (ϕ1(x(tf, w), tf) − β∗
+1) ≤ α ,
+...
+wr (ϕr(x(tf, w), tf) − β∗
+r) ≤ α .
+Remark 2 The lower-level problem in (OPF) for some given w is simply Problem (OCPw).
+A solution of (OCPw) is nothing but a point in the Pareto set Z of (OCP) and is described
+by the triplet Zw := (x∗(t, w), u∗(t, w), t∗
+f(w)). Then the (whole) Pareto set can be expressed
+as Z = ∪w∈Y Zw. Now Problem (OPF) can equivalently be written as
+�
+min
+w∈Y
+ϕ0(xw, uw, tw
+f )
+subject to
+(xw, uw, tw
+f ) ∈ Zw .
+We note that the optimization variable of the upper-level problem is the “unknown” param-
+eter w. If the solution (x∗(t, w), u∗(t, w), t∗
+f(w)) of Problem (OCPw) is differentiable in the
+parameter w, then powerful differentiable optimization techniques can be employed in solving
+Problem (OPF) (or in a more concise form the above problem). This is what was done in [7]
+for convex multi-objective optimal control problems. In this paper, we are extending the
+work in [7] to the nonconvex setting by also incorporating the Chebyshev scalarization and
+the concept of essential interval of weights given in [27].
+✷
+2.4
+Solution differentiability
+We briefly review results on solution differentiability or C1-sensitivity of solutions to the
+following parametric optimal control problems depending on a parameter p ∈ P, where P is
+a Banach space:
+(OCP(p))
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+min x,u,p g(x(tf), tf, p)
+subject to ˙x(t) = �f(x(t), u(t), p) ,
+for a.e. t ∈ [0, tf] ,
+ψ(x(0), x(tf), tf, p) = 0 ,
+˜ψ(x(0), x(tf), tf, p) ≤ 0 ,
+˜C(x(t), u(t), p) ≤ 0 ,
+for a.e. t ∈ [0, tf] ,
+˜S(x(t), p) ≤ 0 ,
+for a.e. t ∈ [0, tf] .
+We note that problem (OCPw) is a special case of the parametric problem (OCP(p)) by
+simply taking the parameter as the weight, p = w, which then appears only in the terminal
+inequality constraints. The problem (OCP(p0)) corresponding to a reference parameter p0
+is considered as the nominal or unperturbed problem. It is assumed that a local solution
+
+Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer
+8
+(x0, u0) of the reference solution exists. Let p be a parameter in a neighbourhood of the
+nominal parameter p0 and denote the solution to (OCP(p)) by (x(t, p), u(t, p)). Dontchev
+and Hager [17] gave conditions under which the mapping p �→ (x(·, p), u(·, p)) is Lipschitz.
+Malanowski and Maurer [32, 33] and Maurer and Pesch [36, 37] investigated the solution
+differentiability or C1-sensitivity of the optimal solution.
+The authors derived conditions
+such that an optimal solution (x(·, p), u(·, p)) of the perturbed control problem OCP(p) exists
+for all parameters p in a neighborhood of p0 and, moreover, the solution (x(t, p), u(t, p)) is
+a C1 function with respect to both arguments (t, p). In broad descriptions, these conditions
+include certain smoothness of the functions in Problem (OCP1), satisfaction of the strict
+Legendre–Clebsch condition, uniqueness of the optimal control minimizing the Hamiltonian,
+nonsingularity of the Jacobian of an associated boundary-value problem, and boundedness
+of the symmetric solution of an associated Riccati ODE.
+Fixing an increment d ∈ P, the differentials
+zd(t, p0) = ∂x
+∂p(t, p0)d,
+vd(t, p0) = ∂u
+∂p(t, p0)d,
+satisfy a linear boundary value problem that contains only information obtained in the process
+of computing the unperturbed solution. The computations of these sensitivity differentials
+can also be performed by discretization methods applied to the parametric optimal control
+problem; see B¨uskens [11] and B¨uskens and Maurer [12]. The sensitivity differentials can be
+conveniently used in the minimization of a master function defined on the Pareto front; see
+Section 2.3.
+The above mentioned conditions for showing solution differentiability exclude optimal con-
+trol problems with control appearing linearly, since for this class of problems the strict
+Legendre-Clebsch condition does not hold. Here, optimal controls are combinations of bang-
+bang and singular arcs. In case of finitely many switching times and junction times with
+the boundary of a mixed control-state constraint or a pure state constraint, one can set up
+a finite-dimensional optimization problem, the Induced Optimization Problem, where the
+switching and junction times are optimized directly; see Maurer et al. [34] and Osmolovskii
+and Maurer [40].
+If second-order sufficient conditions hold for the Induced Optimization
+Problem (see [40]), one immediately obtains the result that the switching and junction times
+locally are differentiable functions of the parameter p.
+To our knowledge extensions of these results on solution differentiability to optimal control
+problems with control and state delays can not be found in the literature.
+3
+An Algorithm For Optimization Over the Pareto Front
+As discussed in Section 2.4, the results [36, Theorem 3.1] and [37, Theorem 5.1] lay the ground
+for devising and implementing numerical methods for solving Problem (OPF). Bonnel and
+Kaya propose in [7] a barrier method for convex bi-objective optimal control problems with
+pure control constraints.
+Their method relies on twice continuous differentiability of the
+solution (class C2) in the weight w, using the result in [36, Theorem 3.1].
+In this paper, we propose a bisection method also for the case of two objectives, which
+relies on the solution of Problem (OCPw) being of class C1 w.r.t. the weight w, and thus
+taking the result in [37, Theorem 5.1] as a basis.
+Although a mathematical justification
+of the applicability of our proposed method, i.e., solution differentiability, is given only for
+Problem (OCPw), the working of the method will also be illustrated on problems of more
+general class as in Problem (OCPw).
+
+Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer
+9
+PSfrag replacements
+Pareto front
+ϕ1
+ϕ2
+(ϕ∗
+1, ϕ2)
+(ϕ1, ϕ∗
+2)
+wf (ϕ1 − β∗
+1) = (1 − wf) (ϕ2 − β∗
+2)
+w0 (ϕ1 − β∗
+1) = (1 − w0) (ϕ2 − β∗
+2)
+(β∗
+1, β∗
+2)
+Figure 1: Determination of the essential subinterval of weights [w0, wf] [27].
+In the scalarized problem (OCPw) with two objectives (r = 2), by choosing w1 = w, and
+w2 = 1 − w, where w ∈ [0, 1], one can simply consider the single parameter w.
+3.1
+Essential interval of weights
+With the Chebyshev scalarization, it would usually be enough for the weight w to take values
+over a (smaller) subinterval [w0, wf] ⊂ [0, 1], with w0 > 0 and wf < 1, for the generation of
+the whole front. Figure 3.1 illustrates the geometry to compute the subinterval end-points,
+w0 and wf. In the illustration, the points (ϕ∗
+1, ϕ2) and (ϕ1, ϕ∗
+2) represent the boundary of
+the Pareto front. The equations of the “rays” which emanate from the utopia point (β∗
+1, β∗
+2)
+and pass through the boundary points are also shown. By substituting the boundary values
+of the Pareto curve into the respective equations, and solving each equation for w0 and wf
+one simply gets
+w0 =
+(ϕ∗
+2 − β∗
+2)
+(ϕ1 − β∗
+1) + (ϕ∗
+2 − β∗
+2)
+and
+wf =
+(ϕ2 − β∗
+2)
+(ϕ∗
+1 − β∗
+1) + (ϕ2 − β∗
+2) .
+(3)
+From the geometry depicted in Figure 3.1, as also discussed in [27], one can deduce that
+with every w ∈ [0, w0] the solution of (OCPw) will yield the same boundary point (ϕ1, ϕ∗
+2)
+on the Pareto front. Likewise with every w ∈ [wf, 1] the same boundary point (ϕ∗
+1, ϕ2) is
+generated. This observation justifies the avoidance of the weights w ∈ [0, w0) ∪ (wf, 1] in
+order not to keep getting the boundary points of the Pareto front, as otherwise one would
+end up wasting valuable computational effort and time.
+As a result of the above argument, the bisection method, implemented in the algorithm
+described in the next section, starts with the essential interval [w0, wf] rather than [0, 1]. It
+is worth re-iterating that our main concern here, unlike in [27], is not really to construct the
+Pareto front, but rather do a search (in this case using the bisection method) over the Pareto
+front, at the same time avoiding the task of constructing the front, so as to find in some sense
+the best solution point in the Pareto front.
+
+Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer
+10
+3.2
+Bisection method for solving Problem (OPF)
+The problem of finding a best point in the Pareto front/set has now been transformed into
+a problem of finding best w, by virtue of the surjection from the set of weights to the set of
+Pareto minima furnished by Theorem 1. This has resulted in Problem (OPF) and its concise
+form: Find some weight w ∈ [w0, wf] such that the master objective function ϕ0(xw, uw, tw
+f ) is
+minimized, where (xw, uw, tw
+f ) is found by solving (OCPw) for that w. For a simpler setting, it
+is helpful to define a function F : [0, 1] → IR+ representing the function we want to minimize
+over the Pareto front:
+F(w) := ϕ0(xw, uw, tw
+f ) ,
+(4)
+such that (xw, uw, tw
+f ) solves (OCPw). In other words, an evaluation of the function F(·) at
+w requires the solution of Problem (OCPw) with that w.
+Problem (OPF) can now be re-written in an even more concise form as
+min
+w∈[w0,wf] F(w) ,
+(5)
+where F(·) is evaluated as in (4). In [7], a log-barrier method is proposed and implemented
+to solve (5), with an underlying convex and smooth optimal control problem with no state
+constraints for which the solution can be assumed to be of class C2, and so Newton-like
+methods are used with heuristic barrier parameter updates. For the general form we have in
+Problem (OCP), which is nonconvex and has state constraints, we assume that the solution
+is of class C1. As elaborated in Section 2.4, under certain regularity conditions which can
+in many cases be checked, this assumption is guaranteed to hold. Therefore we apply the
+bisection method [10] as an effective and simple approach to solving (5) in the case of this
+paper.
+Albeit elementary and standard, a statement of the optimality conditions in the fact below
+will be useful in formulating a computational algorithm later in this section.
+Fact 1 Consider the minimization problem in (5) with F(·) of class C1.
+(a) The interior point w∗ ∈ (w0, wf) is a strict local minimizer of F(·) if, and only if,
+F ′(w∗) = 0 ,
+(6)
+and, for arbitrarily small ε > 0,
+F ′(w∗ − ε) < 0
+and
+F ′(w∗ + ε) > 0 .
+(7)
+(b) The end point w0 (resp. wf) is a strict local minimizer of F(·) if, and only if, either
+(i) F ′(w0) > 0 (resp. F ′(wf) < 0) or
+(ii) F ′(w0) = 0 (resp. F ′(wf) = 0) and, for arbitrarily small ε > 0, F ′(w0 + ε) > 0
+(resp. F ′(wf − ε) < 0).
+Remark 3 (Three Cases for the End Points of [w0, wf]) We will apply the bisection
+method starting with the essential interval [w0, wf]. Before introducing the pertaining algo-
+rithm, we consider below the cases for the end points of this interval.
+Case I. F ′(w0)F ′(wf) < 0 : Since F ′(·) is assumed to be continuous the bisection method
+is guaranteed to find a numerical solution to (6) by the intermediate value theorem.
+Condition (7) needs to be check to see if w∗ is a strict local minimizer.
+
+Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer
+11
+Case II. F ′(w0)F ′(wf) > 0 : By the conditions in Fact 1(b)(i), at least one of w0 and wf is
+a strict local minimizer.
+Case III. F ′(w0)F ′(wf) = 0 : If one of the inequalities in Fact 1(b)(ii) is satisfied, then w0
+or wf is a strict local minimizer. It is possible that both w0 and wf are, or only one or
+neither is, a local minimizer.
+In Case I, the bisection method starts with the interval [w0, wf] and terminates with an
+approximate solution in the interior of the interval. In Case II, a local minimum is found
+immediately, and so in principle there is no need to do a further search. In Case III, however,
+the conclusion might be that neither w0 nor wf is a strict local minimizer, in which case it
+would be necessary to start the bisection method with a subinterval of [w0, wf], and consider
+Cases I–III again.
+✷
+Remark 4 In any of the scenarios elaborated in Remark 3, consideration of another subin-
+terval of [w0, wf] might as well yield a better (lower-value) solution, since the problem is
+nonconvex and we can only hope to get a locally optimal solution. In our approach here,
+however, we do not endeavour to obtain a global minimum. As a result of our discussion
+in Remark 3, we will consider only Case I, which clearly prompts us to use the bisection
+method directly. As suggested above, in the event of Case III not yielding a solution, the
+new subinterval could be chosen in such a way that one would fall into Case I.
+✷
+The derivative of F(·) is defined at the end points of the interval [w0, wf] as one-sided
+limits,
+F ′(w0) := lim
+δ→0+
+F(w0 + δ) − F(w0)
+δ
+and
+F ′(wf) := lim
+δ→0−
+F(wf + δ) − F(wf)
+δ
+,
+and in the interior, i.e., for w ∈ (w0, wf), as
+F ′(w) := lim
+δ→0
+F(w + δ) − F(w)
+δ
+,
+where F(·) is evaluated as in (4). In computations, we will use the forward, and backward,
+finite difference approximations of F ′(·). Namely, for some small δ > 0, we will set
+F ′(w) ≈
+
+
+
+
+
+
+
+F(w + δ) − F(w)
+δ
+,
+if w ∈ [w0, wf − δ) ,
+F(w) − F(w − δ)
+δ
+,
+if w ∈ [wf − δ, wf] .
+(8)
+The step δ in the difference approximation formula (8) is small for an accurate estimation of
+the derivative but not too small in order not to divide one very small number by another and
+cause numerical instabilities.
+In what follows we provide an algorithm to solve Problem (OPF). The algorithm first finds
+the essential interval [w0, wf], computes the signs of F ′(w0) and F ′(wf) and checks the cases
+I–III in Remark 3, and then if F ′(w0)F ′(wf) < 0 it uses the bisection method, to find a
+numerical solution to Problem (OPF).
+Algorithm 1
+Step 0.0 (Initialization)
+Choose utopia parameters, η1, η2 > 0, a small numerical differ-
+entiation step δ > 0, a stopping tolerance ǫ > 0, and a maximum number of iterations
+kmax . Set k := 1.
+
+Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer
+12
+Step 0.1 (Boundary points of the front) Solve (2) to get (xi, ui, ti
+f), i = 1, 2. Set
+(ϕ∗
+1, ϕ2) := (ϕ1(x1(t1
+f), t1
+f), ϕ2(x1(t1
+f), t1
+f)), (ϕ1, ϕ∗
+2) := (ϕ1(x2(t2
+f), t2
+f), ϕ2(x2(t2
+f), t2
+f)) .
+Step 0.2 (Utopia point) Set β∗ := (β∗
+1, β∗
+2) with β∗
+i := ϕ∗
+i − ηi, i = 1, 2.
+Step 0.3 (Essential interval) Determine the subinterval [w0, wf] ⊂ [0, 1] using (3).
+Step 0.4 (Signs at end points) Compute F ′(w0) and F ′(wf) using (8), with F(·) evaluated
+as in (4).
+• If Fact 1(b)(i) or (ii) is satisfied then w∗ = w0 or w∗ = wf appropriately; STOP.
+• If F ′(w0)F ′(wf) = 0 and neither of the inequalities in Fact 1(b)(ii) is satisfied then
+declare “Algorithm failed. Change the interval [w0, wf].” and STOP.
+Let a := w0 and b := wf.
+Step k.1 (Bisection) Find the midpoint c := a + (b − a)/2 of the interval [a, b].
+Step k.2 (Stopping criterion) Compute F ′(c) using (8), with F(·) evaluated as in (4).
+• If F ′(c) = 0 or (b − a)/2 < ǫ then set w∗ = c and STOP.
+• If k = kmax then declare “Maximum number of iterations exceeded.” and STOP.
+Step k.3 (New subinterval)
+Set k := k + 1 . If F ′(a)F ′(c) > 0 then update the subinterval
+as [a, b] := [c, b]; otherwise, set [a, b] := [a, c]. GO TO Step k.1.
+4
+Numerical Examples
+In this section, we illustrate the working of Algorithm 1 on two optimal control problems,
+one involving an electric circuit in Section 4.1 and the other a tuberculosis (TB) epidemic in
+Section 4.2.
+In computations, we use direct discretization of optimal control problems for which con-
+vergence theory has been an active topic of research in the literature (see for example
+[1,4,18–20,42], and see [27] for additional references and discussion).
+We employ the scalarize–discretize–then–optimize approach that was previously used in [27].
+Under this approach, one first scalarizes the multi-objective problem in the infinite-dimensional
+space, and then discretizes the scalarized problem directly and applies a usually large-scale
+finite-dimensional optimization method to find a discrete approximate solution of the scalar-
+ized problem. By the existing theory of discretization mentioned above, under certain as-
+sumptions, the discrete approximate solution converges to a solution of the continuous-time
+scalarization of the original problem, yielding a Pareto minimum of the original problem.
+When possible, we will also check a posteriori to see if the necessary optimality conditions
+are satisfied by an accurate-enough numerical solution.
+In Step 0.4 of Algorithm 1, a direct discretization of Problem (OCPw), for example em-
+ploying a Runge–Kutta scheme, such as Euler’s method or the Trapezoidal rule, is solved by
+using Ipopt, version 3.12.13, four times. In Step k.2, Problem (OCPw) is solved in a similar
+way two times. Ipopt is a popular optimization software based on an interior point method;
+see [46]. We use AMPL [23] as an optimization modelling language, which employs Ipopt as
+a solver.
+
+Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer
+13
+4.1
+Example: Tunnel-diode oscillator (Rayleigh problem)
+The tunnel-diode oscillator problem, also referred to as the Rayleigh problem in the literature,
+involves dynamics represented by the following differential equations.
+˙x1(t) = x2(t) ,
+˙x2(t) = −x1(t) + x2(t) (1.4 − 0.14 x2
+2(t)) + 4 u(t) ,
+for a.e. t ∈ [0, tf] ,
+where the state variable x1(t) denotes electric current, and the control variable u(t) stands
+for a suitable transformation of the voltage at a generator, both at time t ∈ [0, tf]—see [35]
+for a detailed exposition of the problem. In this particular instance of the problem, the initial
+and terminal values of the state variables are specified as
+(x1(0), x2(0)) = (−5, −5)
+and
+(x1(tf), x2(tf)) = (0, 0) ,
+and the dynamics are subject to constraints on the control variable such that
+−1 ≤ u(t) ≤ 1 ,
+for a.e. t ∈ [0, tf] .
+The optimal control problem is posed as a bi-objective problem with
+min
+�
+tf ,
+� tf
+0
+�
+x2
+1(t) + u2(t)
+�
+dt
+�
+,
+where the competing objectives are the minimization of the final time tf and the minimization
+of the sum of the square L2-norms, or in some sense the magnitudes, of the current and the
+generator voltage. Define a new state variable x3 such that
+˙x3(t) = x2
+1(t) + u2(t) ,
+for a.e. t ∈ [0, tf] ,
+x3(0) = 0 .
+Then the two objective functionals as in Problem (OCP), or Problem (OCPw), can be ex-
+pressed as
+ϕ1(x(tf), tf) = tf
+and
+ϕ2(x(tf), tf) = x3(tf) .
+As we have stated above, the bi-objective Rayleigh problem is in the same form as Prob-
+lem (OCP) and, in particular, Problem (OCPw). The decision maker’s objective for this
+problem will be to minimize a weighted distance to the origin of the value space. We choose
+ϕ0(xw, uw, tw
+f ) := 100 ϕ2
+1(xw(tf), tw
+f ) + ϕ2
+2(xw(tf), tw
+f ) ,
+where the scaling multiplier 100 is used to make the orders of magnitudes of ϕ1 and ϕ2 the
+same. We aim to solve Problem (OPF), to determine a scalar w ∈ (0, 1) with w1 := w and
+w2 := 1 − w that results in the best Pareto solution in the sense that ϕ0(·, ·, ·) is minimized,
+subject to the solution of Problem (OCPw).
+In [35], Maurer and Oberle numerically illustrate that an optimal solution does not exist
+for the single objective problem minimizing the quadratic functional ϕ2(x(tf), tf), in that tf
+tends to infinity. They carry out a numerical test for checking the second-order sufficient
+conditions (SSC) of optimality and show that the test fails to confirm the SSC. Therefore,
+we will impose a bound on the terminal time, namely set tf ≤ 5. On the other hand, they
+illustrate also in [35] that for certain instances of the weighted-sum problem, the SSC of
+optimality are satisfied.
+
+Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer
+14
+Problem (OCPw) can now explicitly be written for the Rayleigh problem as
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+min
+α≥0
+x(·),u(·),tf
+α
+subject to
+˙x1(t) = x2(t) ,
+x1(0) = −5 , x1(tf) = 0 ,
+˙x2(t) = −x1(t) + x2(t) (1.4 − 0.14 x2
+2(t)) + 4 u(t) , x2(0) = −5 , x2(tf) = 0 ,
+˙x3(t) = x2
+1(t) + u2(t) ,
+x3(0) = 0 ,
+−1 ≤ u(t) ≤ 1 ,
+for a.e. t ∈ [0, tf] ,
+tf ≤ 5 ,
+w (tf − β∗
+1) ≤ α ,
+(1 − w) (x3(tf) − β∗
+2) ≤ α .
+The Hamiltonian H : IR3 × IR × IR3 → IR for this problem simply is
+H(x, u, λ) := λ1x2 + λ2
+�
+(−x1 + x2 (1.4 − 0.14 x2
+2) + 4u
+�
++ λ3(x2
+1 + u2) ,
+where λ(t) := (λ1(t), λ2(t), λ3(t)) ∈ IR3 is referred to as the adjoint variable vector. Using
+the convenient notation H[t] := H(x(t), u(t), λ(t)), suppose that
+˙λ1(t) := −Hx1[t] = λ2(t) − 2λ3(t)x1(t) ,
+(9a)
+˙λ2(t) := −Hx2[t] = −λ1(t) − λ2(t)(1.4 − 0.42x2
+2(t)) ,
+(9b)
+˙λ3(t) := −Hx3[t] = 0 ,
+(9c)
+for all t ∈ [0, tf], with certain transversality conditions as required by the maximum principle.
+In (9a)–(9c), Hxi := ∂H/∂xi, i = 1, 2, 3. We will not go into the details of these (boundary)
+conditions here. However we note that λ3(t) = λ3, a constant, for all t ∈ [0, tf]. Then the
+maximum principle states that if (x, u, tf) is an optimal solution triplet then there exists
+a continuous function λ(·) satisfying (9a)–(9c), along with certain transversality conditions,
+such that λ(t) ̸= 0, for all t ∈ [0, tf], and
+u(t) = argmin
+v∈[−1,1]
+H(x(t), v, λ(t)) = argmin
+v∈[−1,1]
+�
+4λ2(t)v + λ3(t)v2�
+.
+(10)
+for a.e. t ∈ [0, tf]. If w = 1, then the problem is a single-objective one, referred to as a
+time-optimal control problem, and the condition (10) reduces to
+u(t) = argmin
+v∈[−1,1]
+λ2(t)v ,
+resulting in
+uw(t) =
+
+
+
+
+
+1 ,
+if λw
+2 (t) < 0 ,
+−1 ,
+if λw
+2 (t) > 0 ,
+undetermined ,
+if λw
+2 (t) = 0 ,
+(11)
+for a.e.
+t ∈ [0, tf]. By the discussion given in Section 3.1 (also see [27]), uw(t) given in
+(11) is the same for all w ∈ [wf, 1].
+Recall that if one does not have λw
+2 (t) = 0 for all
+[t′, t′′] ⊂ [0, tf], where t′ < t′′, then uw(t) in (11) is referred to as optimal control of bang–bang
+type. We assume (and therefore will numerically double-check) that the optimal control for
+the particular instance of the problem is of bang–bang type.
+The optimality condition (10) can be shown to yield, for any given w ∈ [w0, wf),
+uw(t) =
+
+
+
+
+
+
+
+1 ,
+if 2λw
+2 (t) < −λ
+w
+3 ,
+−2λw
+2 (t)/λ
+w
+3 ,
+if
+− λ
+w
+3 ≤ 2λw
+2 (t) ≤ λ
+w
+3 ,
+−1 ,
+if 2λw
+2 (t) > λ
+w
+3 ,
+(12)
+
+Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer
+15
+for all t ∈ [0, tf], provided λ
+w
+3 ̸= 0. Again by virtue of the discussion in Section 3.1, uw(t) in
+(12) is the same for all w ∈ [0, w0]. We define the switching function as
+σw(t) :=
+�
+2 λw
+2 (t)/λ
+w
+3 ,
+if 0 ≤ w < wf ,
+16 λw
+2 (t) ,
+if wf ≤ w ≤ 1 .
+(13)
+The constant coefficients 2 and 16 above are used for scaling purposes, so that the graphs in
+Figure 2(b) can be viewed more easily. Now, using (13), we can summarize and combine the
+expressions for the optimal control in (11) and (12) as follows.
+uw(t) =
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+1 ,
+if σw(t) < −1
+−2λw
+2 (t)/λ
+w
+3 ,
+if
+− 1 ≤ σw(t) ≤ 1 ,
+−1 ,
+if σw(t) > 1 .
+
+
+
+
+
+,
+if 0 ≤ w < wf ,
+�
+1 ,
+if σw(t) < 0 ,
+−1 ,
+if σw(t) > 0 .
+�
+,
+if wf ≤ w ≤ 1 .
+(14)
+As to why σw(·) is referred to as the switching function should now be more clear from (14):
+the value of σw(·) determines when to switch from one case of the control function uw(·) to
+another.
+For Problem (OCPw) written for the Rayleigh problem above, we have chosen the utopia
+vector as (β∗
+1, β∗
+2) = (0, 0), since ϕi(x(tf), tf) > 0, for i = 1, 2.
+Figure 2(a) depicts the
+Pareto front for the instance of the multi-objective Rayleigh problem we consider here. It
+also displays the iterations of Algorithm 1. The Rayleigh problem is discretized using the
+trapezoidal rule, the number of grid points is set to be N = 5000, and the Ipopt’s tolerance
+to 10−10, so as to get solutions for w accurate at least up to four decimal places (dp).
+The essential interval is found to be [w0, wf] = [0.8994, 0.9269], with
+(ϕw0
+1 , ϕw0
+2 ) = (5.000, 44.71)
+and
+(ϕwf
+1 , ϕwf
+2 ) = (3.668, 46.50) ,
+correct to four significant figures, where ϕw
+i := ϕi(xw(tf), tw
+f ), i = 1, 2, with w = w0 or wf, or
+as will be the case below, w = w∗. Optimization over the Pareto front results in w∗ = 0.9247,
+after 14 iterations of Algorithm 1, yielding
+ϕw∗
+0
+= 58.71
+and
+(ϕw∗
+1 , ϕw∗
+2 ) = (3.709, 45.51) .
+If there is a need to save the computational resources further, the algorithm can be asked to
+yield a less accurate result, say correct to three dp, which then yields w∗ = 0.925 in eight
+iterations with (ϕw∗
+1 , ϕw∗
+2 ) = (3.71, 45.5). In Figure 2(a) only five iterations are displayed
+(labels 1–5 appearing to the right of each iteration) for clarity in viewing.
+The Pareto
+(master) solution with w = w∗ is represented by a square.
+The numerical Pareto-optimal state and control variable solutions are presented in Fig-
+ures 2(c)–(d) for w = w0, w∗, wf. One of the boundary Pareto-optimal solutions is shown
+using solid (blue) curves for w = w0, which is the same solution for all w ∈ [0, w0], as pre-
+viously discussed in Section 3.1.
+On the other hand, the other boundary Pareto-optimal
+solution for w = wf, which holds for all w ∈ [wf, 1], is shown using dashed (green) curves.
+The latter is nothing but a time-optimal control solution for the Rayleigh problem (a solu-
+tion with the smallest tf), resulting in a bang–bang type function with the sequence of values
+{1, −1, 1}, namely with two switchings. The master Pareto solution is given for w = w∗ using
+dashed-and-dotted (red) curves.
+The switching function σw(·) plotted in Figure 2(b) by using (13) (recall that discrete
+approximations of λw
+2 (t) and λw
+3 (t) can readily be obtained from AMPL) furnishes the means
+
+Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer
+16
+3.6
+3.8
+4
+4.2
+4.4
+4.6
+4.8
+5
+45
+45.5
+46
+46.5
+   1
+   2
+   3
+   4
+   5
+(a) Pareto front, and iterations of Algorithm 1:
+Master solution is depicted by a (red) square and
+iterates by (light blue) circles.
+0
+1
+2
+3
+4
+5
+-3
+-2
+-1
+0
+1
+2
+3
+(b) Switching function as defined in (13).
+-8
+-6
+-4
+-2
+0
+2
+-6
+-4
+-2
+0
+2
+4
+6
+PSfrag replacements
+singular control switching curve
+(c) Phase plane trajectories.
+0
+1
+2
+3
+4
+5
+-1
+-0.5
+0
+0.5
+1
+(d) Control variable.
+Figure 2: Rayleigh problem—Boundary Pareto solutions, corresponding to w0 = 0.8994 and
+wf = 0.9269, are shown with (blue) solid curves and (green) dashed curves, respectively. Master
+Pareto solution, corresponding to w∗ = 0.9247, is shown with dashed-and-dotted (red) curves.
+to verify the optimality condition for uw(·) expressed in (14). It is evident from the dashed
+(green) plot of the switching function that, for w ∈ [wf, 1], when σw(·) crosses the time
+axis there is a jump (from 1 to −1 or vice versa) in the value of the corresponding uw(·)
+plot. Likewise, for w ∈ [0, w0] and for w = w∗ ∈ [w0, wf), whenever σw(·) crosses one of the
+lines σw(t) = 1 and σw(t) = −1 (shown by two black lines in Figure 2(b) for convenience)
+the expression for the control function uw(·) switches from one case in (14) to another, as
+required.
+4.2
+Example: Compartmental model for tuberculosis
+In 2020 and 2021, tuberculosis (TB) was the second leading cause of death from an infectious
+disease worldwide after COVID-19 [44]. Active TB refers to disease that occurs in someone
+infected with Mycobacterium tuberculosis. It is characterized by signs or symptoms of active
+disease, or both, and is distinct from latent tuberculosis infection, which occurs without signs
+or symptoms of active disease. Only individuals with active TB can transmit the infection.
+Many people with active TB do not experience typical TB symptoms in the early stages of the
+
+Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer
+17
+disease. These individuals are unlikely to seek care early, and may not be properly diagnosed
+when seeking care. Delays to diagnosis of active TB present a major obstacle to the control of
+a TB epidemic, it may worsen the disease, increase the risk of death and enhance tuberculosis
+transmission to the community. Both patient and the health system may be responsible for
+the diagnosis delay.
+We study the control model with control and state delays presented in Silva et al. [43]. In
+this model, reinfection and post-exposure interventions for tuberculosis are considered. The
+population is divided into five categories (compartments) (i.e., the control system has five
+state variables):
+S
+:
+susceptible individuals,
+L1
+:
+early latent individuals, recently infected (less than two years),
+I
+:
+infectious individuals, who have active TB,
+L2
+:
+persistent latent individuals,
+R
+:
+recovered individuals,
+N
+:
+total population N = S + L1 + I + L2 + R , assumed constant.
+The model has two control variables and three delays:
+u1
+:
+effort on early detection and treatment of recently infected individuals L1,
+du1
+:
+delay on the diagnosis of latent TB, and commencement of latent TB treatment,
+u2
+:
+chemotherapy or post-exposure vaccine to persistent latent individuals L2,
+du2
+:
+delay in the prophylactic treatment of persistent latent L2,
+dI
+:
+delay in I, i.e., delay in diagnosis.
+The dynamical system is given by
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+˙S(t) = µN − β
+N I(t)S(t) − µS(t),
+˙L1(t) = β
+N I(t) (S(t) + σL2(t) + σRR(t)) − (δ + τ1 + ǫ1u1(t − du1) + µ) L1(t),
+˙I(t) = φ δ L1(t) + ωL2(t) + ωRR(t) − τ0I(t − dI) + µI(t),
+˙L2(t) = (1 − φ)δL1(t) − σ β
+N I(t)L2(t) − (ω + ǫ2u2(t − du2) + τ2 + µ)L2(t).
+(15)
+The recovered population is defined by
+R(t) := N − S(t) − L1(t) − I(t) − L2(t) ,
+(16)
+with N = 30000. The system and delay parameters in the model (15) along with their values
+are listed in Table 1. In view of the delays the initial conditions and functions are:
+S(0) = 76 N/120, L1(0) = 36 N/120, L2(0) = 2 N/120, R(0) = N/120,
+I(t) = 5 N/120
+for −dI ≤ t ≤ 0,
+uk(t) = 0
+for −duk ≤ t < 0,
+(k = 1, 2).
+(17)
+The control constraints are given by
+0 ≤ uk(t) ≤ 1 ,
+∀t ∈ [0, tf] ,
+(k = 1, 2).
+(18)
+We consider the following parametric objective functional with control weights a1, a2 ≥ 0:
+tf
+�
+0
+(I(t) + L2(t) + a1u1(t) + a2u2(t)) dt .
+(19)
+
+Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer
+18
+Symbol
+Description
+Value
+β
+Transmission coefficient
+variable
+µ
+Death and birth rate
+1/70 yr−1
+δ
+Rate at which individuals leave L1
+12 yr−1
+φ
+Proportion of individuals going to I
+0.05
+ω
+Endogenous reactivation rate for persistent latent infections
+0.0002 yr−1
+ωR
+Endogenous reactivation rate for treated individuals
+0.00002 yr−1
+σ
+Factor reducing the risk of infection as a result of acquired
+immunity to a previous infection for L2
+0.25
+σR
+Rate of exogenous reinfection of treated patients
+0.25
+τ0
+Rate of recovery under treatment of active TB
+2 yr−1
+τ1
+Rate of recovery under treatment of early latent individuals L1
+2 yr−1
+τ2
+Rate of recovery under treatment of persistent latent individuals L2
+1 yr−1
+N
+Total population
+30, 000
+ǫ1
+Efficacy of treatment of early latent L1
+0.5
+ǫ2
+Efficacy of treatment of persistent latent TB L2
+0.5
+tf
+Total simulation duration
+5 years
+dI
+delay in the diagnosis of I
+0.1 years
+du1
+delay in the diagnosis of early latent individuals L1
+0.2 years
+du2
+delay in the prophylactic treatment of persistent latent individuals L2
+0.2 years
+Table 1: Parameter values for the TB control model.
+Depending on the priorities, the weights a1, a2 can be chosen in different ways (for example,
+both can be chosen to be very small or very large) giving rise to competing objectives. Namely,
+x5(tf) :=
+tf
+�
+0
+�
+I(t) + L2(t) + a11 u1(t) + a12 u2(t)
+�
+dt ,
+x6(tf) :=
+tf
+�
+0
+�
+I(t) + L2(t) + a21 u1(t) + a22 u2(t)
+�
+dt .
+(20)
+with control weights a11, a12, a21, a22 ≥ 0, constitute two competing objective functionals.
+Both functionals are given in Lagrange form. The standard method to obtain an optimal
+control problem of Bolza type is to introduce additional state variables x5 and x6 defined by
+˙x5(t) = I(t) + L2(t) + a11 u1(t) + a12 u2(t) ,
+x5(0) = 0 ,
+˙x6(t) = I(t) + L2(t) + a21 u1(t) + a22 u2(t) ,
+x6(0) = 0 .
+(21)
+Denoting the (augmented) state vector by x(t) = (S(t), L1(t), I(t), L2(t), x5(t), x6(t)) ∈ R6
+and the control vector u(t) := (u1(t), u2(t)) ∈ R2, the two competing objectives in the general
+problem (P) are given by
+ϕ1(x(tf), tf) = x5(tf) =: F1(x, u)
+and
+ϕ2(x(tf), tf) = x6(tf) =: F2(x, u) ,
+where F1(x, u) and F2(x, u) denote the two functionals in Lagrange form.
+The bi-objective TB problem is now in the same form as Problem (OCP) and, in particular,
+Problem (OCPsd). The decision maker’s objective for this problem will be to minimize the
+distance to the origin of the value space. We therefore choose
+ϕ0(xw, uw, tw
+f ) := ϕ2
+1(xw(tf), tw
+f ) + ϕ2
+2(xw(tf), tw
+f ) ,
+
+Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer
+19
+Our aim is to solve Problem (OPF), to determine a scalar w ∈ (0, 1) with w1 := w and
+w2 := 1 − w that results in the best Pareto solution in the sense that ϕ0(·, ·, ·) is minimized,
+subject to the solution of Problem (OCPw).
+Next we focus on the solution of Problem (OCPw): We aim to find a pair of functions
+(x, u) ∈ W 1,∞([0, tf], R6) × L∞([0, tf], R2) that minimizes the parameter α subject to the
+time-delayed dynamics (15) and the auxiliary dynamics (21), initial conditions (17), control
+constraints (18) and auxiliary weighted inequalities involving ϕ1 and ϕ2.
+We consider the necessary optimality conditions for the time-delayed optimal control
+problem (OCPw); see G¨ollmann and Maurer [24], Vinter [45].
+For this purpose we in-
+troduce the delayed state variable y3(t) = x3(t − dI) = I(t − dI) and delayed control
+variables vk(t) = uk(t − duk), k = 1, 2.
+Denoting the adjoint variable vector by λ(t) :=
+(λS(t), λL1(t), λI(t), λL2(t), λ5(t), λ6(t)) ∈ R6 the Hamiltonian or Pontryagin function is given
+by
+H(x, y3, λ, u1, v1, u2, v2) = λs (µN − β
+N IS − µS)
++ λL1 ( β
+N I (S + σL2 + σRR) − (δ + τ1 + ǫ1v1 + µ) L1)
++ λI ( φ δL1 + ωL2 + ωR R − τ0y3 + µI)
++ λL2 ((1 − φ)δL1 − σ β
+N IL2 − (ω + ǫ2v2 + τ2 + µ)L2)
++ λ5 (I + L2 + a11u1 + a12u2)
++ λ6 (I + L2 + a21u1 + a22u2) ,
+(22)
+where R is given as in (16). The Minimum Principle [24,45] yields the adjoint equations
+˙λS(t) = −∂H
+∂S [t],
+˙λL1(t) = − ∂H
+∂L1
+[t],
+˙λL2(t) = − ∂H
+∂L2
+[t],
+˙λx5(t) = − ∂H
+∂x5
+[t] = 0 ,
+˙λx6(t) = − ∂H
+∂x6
+[t] = 0 ,
+and the advanced adjoint equation
+˙λI(t) = −∂H
+∂I [t] − χ[0,tf −dI](t)∂H
+∂I [t + dI] ,
+where the argument [t] stands for evaluating all arguments at time t. We note that λw
+5 (t) = λ
+w
+5
+and λw
+6 (t) = λ
+w
+5 , constants, for any fixed w ∈ [0, 1]. In the last equation, the term χ[0,tf −dI](t)
+denotes the characteristic function of the interval [0, tf − dI] at time t. The minimization of
+the Hamiltonian with respect to the controls u1, u2 and delayed controls v1, v2 involves the
+switching functions σk(t) for k = 1, 2:
+σw
+k (t) = ∂H
+∂uk
+[t] + χ[0,tf −duk](t)∂H
+∂vk
+[t + duk]
+=
+�
+a1kλ
+w
+5 + a2kλ
+w
+6 − ǫkλw
+Lk(t + duk)Lw
+k (t + duk) , if 0 ≤ t ≤ tf − duk ,
+a1kλ
+w
+5 + a2kλ
+w
+6 ,
+if tf − duk ≤ t ≤ tf .
+(23)
+As in the Rayleigh problem, the superscript “w” above denotes dependence on the scalariza-
+tion parameter/weight w. Then the controls minimizing the Hamiltonian are characterized
+by the switching conditions (control law)
+uw
+k (t) =
+� 0 ,
+if σw
+k (t) > 0 ,
+1 ,
+if σw
+k (t) < 0 ,
+k = 1, 2.
+(24)
+
+Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer
+20
+Figure 3: TB problem—Pareto front, and iterations of Algorithm 1: Master solution is depicted
+by a (red) square and iterates by (light blue) circles..
+for all w ∈ [0, 1]. In particular, for positive weights a1 > 0, a2 > 0, the switching functions
+(23) and the control law (24) imply
+uw
+k (t) = 0
+∀ tf − duk ≤ t ≤ tf ,
+for all w ∈ [0, 1].
+In what follows we choose the control weights as a11 = a12 = 10 (small) and a21 = a22 =
+1000 (large) in the objective functionals ϕ1 and ϕ2.
+For Problem (OCPw) written for the TB problem, we have chosen the utopia vector as
+(β∗
+1, β∗
+2) = (0, 0). Figure 3 depicts the Pareto front for the TB problem we consider here.
+The plot also displays the iterations of Algorithm 1. The TB problem is discretized using the
+trapezoidal rule, the number of grid points is set to be N = 5000, and the Ipopt’s tolerance
+to 10−10, so as to get solutions for w accurate at least up to four decimal places (dp).
+The essential interval in this case is found to be [w0, wf] = [0.5251, 0.5709], with
+(ϕw0
+1 , ϕw0
+2 ) = (28155, 31133)
+and
+(ϕwf
+1 , ϕwf
+2 ) = (26459, 35205) ,
+where ϕw
+i := ϕi(xw(tf), tw
+f ), i = 1, 2, with w = w0 or wf, or as will be the case below, w = w∗.
+Optimization over the Pareto front results in w∗ = 0.5358, after 10 iterations of Algorithm 1,
+yielding
+ϕw∗
+0
+= 41621
+and
+(ϕw∗
+1 , ϕw∗
+2 ) = (27255, 31455) .
+In Figure 3 only five iterations are displayed (labelled 1–5) for clarity in viewing. The Pareto
+(master) solution with w = w∗ is represented by a square.
+The numerical Pareto-optimal control variable solutions uw
+1 (·) and uw
+2 (·) are presented
+in Figures 4(a)–(b) for w = w0, w∗, wf.
+As with Rayleigh, one of the boundary Pareto-
+optimal solutions is shown using solid (blue) curves for w = w0, the same solution for all
+w ∈ [0, w0]. The other boundary Pareto-optimal solution for w = wf, which holds for all
+w ∈ [wf, 1], is shown using dashed (green) curves. Both of the control solutions are of bang–
+bang type (as required by (24)), with one switching (the number of switchings not dictated
+
+X10
+0
+3.5
+3.4
+P2
+3.35
+3
+0
+4
+2.75
+2.8
+P13.2
+1
+2
+3.1
+2.65
+2.7Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer
+21
+0
+1
+2
+3
+4
+5
+-0.5
+0
+0.5
+1
+(a) Control variable uw
+1 (24) and scaled
+switching function σw
+1 (23) superposed.
+0
+1
+2
+3
+4
+5
+-0.5
+0
+0.5
+1
+(b) Control variable uw
+2 (24) and scaled
+switching function σw
+2 (23) superposed.
+Figure 4: TB problem—Boundary Pareto solutions, corresponding to w0 = 0.5251 and wf =
+0.5709, are shown with (blue) solid curves and (green) dashed curves, respectively. Master Pareto
+solution, corresponding to w∗ = 0.5358, is shown with dashed-and-dotted (red) curves.
+Scalarization
+Functional values
+Switching times
+Terminal state values
+weight w
+xw
+5 (tf)
+xw
+6 (tf)
+tw
+s1
+tw
+s2
+Sw(tf)
+Lw
+1 (tf)
+Iw(tf)
+Lw
+2 (tf)
+Rw(tf)
+w0 = 0.5251 :
+28155
+31133
+0.145
+2.864
+1193.1
+28.2
+13.3
+864.0
+27901.4
+w∗ = 0.5358:
+27255
+31455
+0.809
+3.439
+1205.8
+27.5
+13.0
+747.6
+28006.1
+wf = 0.5709 :
+26459
+35205
+4.083
+4.752
+1238.2
+23.8
+11.2
+419.3
+28307.5
+Table 2: TB problem.
+by (24) alone). The master Pareto solution is given for w = w∗ using dashed-and-dotted
+(red) curves, in which the controls are also of bang–bang type with one switching.
+The switching functions for each control and case, σw
+k (·), k = 1, 2, scaled as indicated, are
+plotted with (black) dotted curves and superposed with the control plots in Figures 4(a)–(b).
+We remind that, by using (23) (recall that discrete approximations of λw
+Lk(t), k = 1, 2, λw
+5 (t)
+and λw
+6 (t) can readily be obtained as constraint multipliers from AMPL), one verifies the
+optimality condition in (24).
+In each strategy, the two control efforts are “on” until the times tw
+sk, k = 1, 2, at which
+the respective uw
+k (·) is switched “off” (down to zero). These types of bang–bang controls
+are also referred to as on–off controls. In Table 2 the switching times for the boundary as
+well as the optimal weights are listed. Under these controls, the resulting terminal values
+of the state variables are also listed in Table 2. The plots of these variables are not pro-
+vided as they are difficult to distinguish at earlier times (as expected) and that they become
+distinguishable/comparable only near the terminal time.
+Under the controls minimizing x5(tf) (with w = wf = 0.5709 and minimum xwf
+5 (tf) =
+26459) the number of persistent latent individuals L2(tf) turns out to be about 419 (in a
+population of 30000). This number is more than doubled to 864 if x6(tf) is minimized (with
+w = w0 = 0.5709 and minimum xw0
+6 (tf) = 31133). The optimal Pareto solution minimizing
+the distance in value space to the origin yields with w = w∗ = 0.5358 the optimal L2(tf) as
+748.
+
+Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer
+22
+5
+Conclusion
+We have proposed an algorithm to solve the problem of optimization over the Pareto front.
+The algorithm employs bisection method which starts with an essential interval of weights
+of the Chebyshev scalarization. It is applicable to a wide range of optimal control problems,
+including state- and control-constrained problems with time delay. Numerical solution of two
+challenging optimal control problems has demonstrated the effectiveness of the algorithm.
+The main motive behind the algorithm we have proposed is that one can find the optimal
+solution minimizing a master objective functional without having to construct the Pareto
+front. The algorithm solves the challenging optimal control problem (OCPw) a relatively
+smaller number of times than the case of constructing the Pareto front. In the examples
+we have studied the algorithm had to solve (OCPw) 20 to 30 times. On the other hand,
+without the algorithm we propose, it is necessary to construct the Pareto front by solving
+(OCPw) thousands of times in order to obtain the same solution with the same computational
+accuracy.
+The proposed algorithm can be improved/modified in various ways. For example, scalar-
+ization techniques other than Chebyshev might be employed; see for example [8, 9] and the
+references therein. Bisection method might be replaced by methods with higher convergence
+rates, for example regula falsi and secant methods (see [10]), at the expense of approximating
+higher order derivatives of course, although the latter would make the algorithm applicable
+to problems with more than just two objective functionals.
+References
+[1] Alt, W., Baier, R. , Lempio, F., Gerdts, M.: Approximations of linear control problems
+with bang–bang solutions. Optimization, 62, 9–32 (2013)
+[2] Benson, H.P.: Optimization over the efficient set, J. Math. Anal. Appl.. 98, 562–580
+(1984)
+[3] Benson, H.P.: A finite, non-adjacent extreme point search algorithm for optimization
+over the efficient set. J. Optim. Theory Appl., 73, 47–64 (1992)
+[4] Betts, J.T.: Practical Methods for Optimal Control Using Nonlinear Programming,
+Third Edition. Advances in Design and Control, SIAM Publications, Philadelphia (2020)
+[5] Bolintin´eanu, S.: Optimality conditions for minimization over the (weakly or properly)
+efficient set. J. Math. Anal. Appl., 173(2), 523–541 (1993)
+[6] Bolintin´eanu, S.: Minimization of a quasi-concave function over an efficient set. Math.
+Prog., 61, 89–110 (1993)
+[7] Bonnel, H., Kaya, C.Y.: Optimization over the efficient set of multi-objective convex
+optimal control problems. J. Optim. Theory Appl., 147, 93–11 (2010)
+[8] Burachik, R.S., Kaya, C.Y., Rizvi, M.M.: A new scalarization technique to approximate
+Pareto fronts of problems with disconnected feasible sets. J. Optim. Theory Appl., 162,
+428–446 (2014)
+[9] Burachik, R.S., Kaya, C.Y., Rizvi, M.M.: A new scalarization technique and new algo-
+rithms to generate Pareto fronts. SIAM J. Optim., 27, 1010–1034 (2017)
+
+Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer
+23
+[10] Burden, R.L., Faires, J.D.: Numerical Analysis, 9th edition. Thompson Brooks/Cole,
+Belmont, CA, USA (2011)
+[11] B¨uskens, C.:
+Optimierungsmethoden und Sensitivit¨atsanalyse f¨ur optimale Steuer-
+prozesse mit Steuer– und Zustands–Beschr¨ankungen. PhD Thesis, Institut f¨ur Nu-
+merische Mathematik, Universit¨at M¨unster, Germany (1998)
+[12] B¨uskens, C., Maurer, H.: SQP-methods for solving optimal control problems with control
+and state constraints: Adjoint variables, sensitivity analysis and real-time control. J.
+Comp. Appl. Math., 120, 85–108 (2000)
+[13] Chorobura, A.P.: Multi-objective infinite horizon optimal control problems: character-
+ization of the Pareto fronts and Pareto solutions. Comp. Appl. Math., 40, 258 (2021)
+URL: https://doi.org/10.1007/s40314-021-01633-0.
+[14] Dauer J.P.: Optimization over the efficient set using an active constraint approach. Z.
+Oper. Res., 35, 185–195 (1991).
+[15] Dauer, J.P., Fosnaugh, T.A.: Optimization over the efficient set, J. Global Optim., 7,
+261–277 (1995).
+[16] D´esilles, A., Zidani, H.: Pareto front characterization for multiobjective optimal control
+problems using Hamilton–Jacobi approach. SIAM J. Control Optim., 57, 3884–3910
+(2019)
+[17] Dontchev, A.L., and Hager, W.W.: Lipschitz stability in in nonlinear control and opti-
+mization, SIAM J. Control Optim., 31, 569–603 (1993).
+[18] Dontchev, A.L., Hager, W.W.: The Euler approximation in state constrained optimal
+control problems. Math. Comput., 70, 173–203 (2001)
+[19] Dontchev, A.L., Hager, W.W., Malanowski, K.: Error bound for Euler approximation of
+a state and control constrained optimal control problem. Numer. Funct. Anal. Optim.,
+21(6), 653–682 (2000)
+[20] Dontchev, A.L., Hager, W.W., Veliov, V.M.: Second-order Runge-Kutta approximations
+in control constrained optimal control, SIAM J. Num. Anal., 38(1), 202—226 (2000)
+[21] Dutta, J., Kaya, C.Y.: A new scalarization and numerical method for constructing the
+weak Pareto front of multi-objective optimization problems. Optimization, 60, 1091–1104
+(2011)
+[22] Eichfelder,
+G.:
+Adaptive Scalarization Methods in Multiobjective Optimization.
+Springer, Berlin, Heidelberg (2008)
+[23] Fourer, R., Gay, D.M., Kernighan, B.W.: AMPL: A Modeling Language for Mathe-
+matical Programming, Second Edition. Brooks/Cole Publishing Company / Cengage
+Learning (2003)
+[24] G¨ollmann, L., Maurer, H.: Theory and applications of optimal control problems with
+multiple time-delays, J. of Industrial and Management Optimization, 10, 413–441 (2014)
+[25] Horst, R., Thoai, N.V.: Maximizing a concave function over the efficient or weakly-
+efficient set. European J. Oper. Res., 117, 239–252 (1999)
+[26] Horst, R., Thoai, N.V., Yamamoto, Y., Zenke, D.: On optimization over the efficient set
+in linear multicriteria programming. J. Optim. Theory Appl., 134, 433–443 (2007)
+
+Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer
+24
+[27] Kaya, C.Y., Maurer, H.: A numerical method for nonconvex multi-objective optimal
+control problems. Comp. Optim. Appl., 57, 685–702 (2014)
+[28] Logist, F., van Erdeghem, P.M.M., van Impe, J.F.: Efficient deterministic multiple
+objective optimal control of (bio)chemical processes. Chem. Eng. Sci., 64, 2527–2538
+(2009)
+[29] Logist, F., Houska, B., Diehl, M., van Impe, J.: Fast Pareto set generation for nonlinear
+optimal control problems with multiple objectives. Struct. Multidisc. Optim., 42, 591-603
+(2010)
+[30] Logist, F., Vallerio, M., Houska, B., Diehl, M., van Impe, J.: Multi-objective optimal
+control of chemical processes using ACADO toolkit. Comp. Chem. Eng., 37, 191–199
+(2012)
+[31] Liu, Z., Ehrgott, M.: Primal and dual algorithms for optimization over the efficient set.
+Optimization, 67,1661–1686 (2018)
+[32] Malanowski, K., Maurer, H.: Sensitivity analysis for parametric control problems with
+control-state constraints. Computational Optimization and Applications, 5, 253-283
+(1996)
+[33] Malanowski, K., Maurer, H.: Sensitivity analysis for state constrained optimal control
+problems. Discrete and Continuous Dynamical Systems, 4, 241-272 (1998)
+[34] Maurer, H., B¨uskens, C., Kim, J.-H.R., Kaya, C.Y.: Optimization methods for the
+verification of second-order sufficient conditions for bang–bang controls. Optim. Contr.
+Appl. Meth., 26, 129–156 (2005)
+[35] Maurer, H., Oberle, H.J.: Second order sufficient conditions for optimal control problems
+with free final time: the Riccati Approach. SIAM J. Control Optim., 41(2), 380–403
+(2002)
+[36] Maurer, H., Pesch, H.J.: Solution differentiability for nonlinear parametric control prob-
+lems. SIAM J. Control Optim., 32, 1542–1554 (1994)
+[37] Maurer, H., Pesch, H.J.: Solution differentiability for parametric nonlinear control prob-
+lems with control-state constraints. J. Optim. Theory Appl., 86, 285–309 (1995)
+[38] Miettinen, K.M.: Nonlinear Multiobjective Optimization, Kluwer (1999)
+[39] Ober-Bl¨obaum, S., Ringkamp, M., zum Felde, G.: Solving multiobjective optimal control
+problems in space mission design using discrete mechanics and reference point techniques.
+Proceedings of the 51st IEEE Conference on Decision and Control, Dec. 10-13, Maui,
+Hawaii, USA, pp. 5711–5716 (2012)
+[40] Osmolovskii, N.P., H. Maurer, H.: Applications to Regular and Bang-Bang Control:
+Second-Order Necessary and Sufficient Optimality Conditions in Calculus of Variations
+and Optimal Control. SIAM Advances in Design and Control, Vol. DC 24, SIAM Pub-
+lications, Philadelphia, 2012.
+[41] Philip, J.: Algorithms for the vector maximization problem. Math. Prog., 207–229 (1972).
+[42] Pietrus, A., Scarinci, T., Veliov, V.: High order discrete approximations to Mayer’s
+problems for linear systems SIAM J. Control Optim., 56(1), 102–119 (2018)
+
+Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer
+25
+[43] Silva, C. J., Maurer, H., Torres, D.F.M.: Optimal control of a Tuberculosis model with
+state and control delays. Math. Biosci. Eng., 14, 321–337 (2017)
+[44] Tuberculosis. World Health Organization, 14 October 2021 and 27 October 2022. URL:
+https://www.who.int/news-room/fact-sheets/detail/tuberculosis. Accessed: 15 August
+2022 and 14 January 2023.
+[45] Vinter, R. B.: State constrained optimal control problems with time delays. J. Math.
+Anal. Appl., 457(2), 1696–1712 (2018)
+[46] W¨achter, A., Biegler, L.T.: On the Implementation of a primal-dual interior point filter
+line search algorithm for large-scale nonlinear programming. Math. Progr., 106, 25–57
+(2006)
+[47] Yamamoto, Y.: Optimization over the efficient set : overview. J. Global Optim., 22(1-4),
+285–317 (2002).
+

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@@ -0,0 +1,934 @@
+OpTaS: An Optimization-based Task Specification Library for
+Trajectory Optimization and Model Predictive Control
+Christopher E. Mower, Jo˜ao Moura, Nazanin Zamani Behabadi,
+Sethu Vijayakumar, Tom Vercauteren∗, Christos Bergeles∗
+Abstract— This paper presents OpTaS, a task specification
+Python library for Trajectory Optimization (TO) and Model
+Predictive Control (MPC) in robotics. Both TO and MPC
+are increasingly receiving interest in optimal control and in
+particular handling dynamic environments. While a flurry
+of software libraries exists to handle such problems, they
+either provide interfaces that are limited to a specific problem
+formulation (e.g. TracIK, CHOMP), or are large and stati-
+cally specify the problem in configuration files (e.g. EXOTica,
+eTaSL). OpTaS, on the other hand, allows a user to specify
+custom nonlinear constrained problem formulations in a single
+Python script allowing the controller parameters to be modified
+during execution. The library provides interface to several open
+source and commercial solvers (e.g. IPOPT, SNOPT, KNITRO,
+SciPy) to facilitate integration with established workflows in
+robotics. Further benefits of OpTaS are highlighted through
+a thorough comparison with common libraries. An additional
+key advantage of OpTaS is the ability to define optimal control
+tasks in the joint space, task space, or indeed simultaneously.
+The code for OpTaS is easily installed via pip, and the source
+code with examples can be found at github.com/cmower/optas.
+I. INTRODUCTION
+High-dimensional motion planners and controllers are
+integrated in many of the approaches for solving complex
+manipulation tasks. Consider, for example, a robot operating
+in an unstructured and dynamic environment that, e.g. places
+an object onto a shelf, or drilling during pedicle screw
+fixation in surgery (see Fig. 1). In such cases, a planner
+and controller must account for objectives/constraints like
+bi-manual coordination, contact constraints between robot-
+object and object-environment, and be robust to disturbances.
+Efficient motion planning and fast controllers are an effective
+way of enabling robots to perform these tasks subject to
+C. E. Mower, C. Bergeles and T. Vercauteren are with the School of
+Biomedical Engineering & Imaging Sciences, King’s College London, UK.
+J. Moura and S. Vijaykumar are with School of Informatics, University of
+Edinburgh, UK. Correspondence: [email protected].
+This research received funding from the European Union’s Horizon 2020
+research and innovation program under grant agreement No. 101016985
+(FAROS). Further, this work was supported by core funding from the
+Wellcome/EPSRC [WT203148/Z/16/Z; NS/A000049/1]. T. Vercauteren is
+supported by a Medtronic / RAEng Research Chair [RCSRF1819\7\34],
+and C. Bergeles by an ERC Starting Grant [714562]. This work has
+received funding from the European Union’s Horizon 2020 research and
+innovation programme under grant agreement No 101017008, Enhancing
+Healthcare with Assistive Robotic Mobile Manipulation (HARMONY).
+This work was supported by core funding from the Wellcome/EPSRC
+[WT203148/Z/16/Z; NS/A000049/1]. This research is supported by Kawada
+Robotics Corporation, Japan and the Alan Turing Institute, UK.
+∗C. Bergeles and T. Vercauteren equally contributed to the work.
+For the purpose of open access, the authors have applied a CC BY public
+copyright license to any Author Accepted Manuscript version arising from
+this submission.
+(a)
+(b)
+Fig. 1: Examples of contact-rich manipulation showing (a)
+a robot placing an item on a shelf, (b) a human interacting
+with a robot performing a drilling task during pedicle screw
+fixation. Image credit: University Hospital Balgrist, Daniel
+Hager Photography & Film GmbH.
+motion constraints, system dynamics, and changing task
+objectives.
+Sampling-based planners [1] are effective, however, they
+typically require considerable post-processing (e.g. trajectory
+smoothing). Optimal planners (i.e. that are provably asymp-
+totically optimal, e.g. RRT∗) are promising but inefficient (in
+terms of computation duration) for solving high-dimensional
+problems [2].
+Gradient-based trajectory optimization (TO) is a key ap-
+proach in optimal control, and has also been utilized for mo-
+tion planning. This approach underpins many recent works
+in robotics for planning and control, e.g. [3], [4], [5], [6],
+[7], [8], [9], [10]. Given an initialization, optimization finds a
+locally optimal trajectory, comprised of a stream of state and
+control commands subject to motion constraints and system
+dynamics (i.e. equations of motion).
+Several reliable open-source and commercial optimization
+solvers exist for solving TO problems, e.g. IPOPT [11], KNI-
+TRO [12], and SNOPT [13]. However, despite the success
+of the optimization approaches proposed in the literature
+and motion planning frameworks such as MoveIt [14], there
+is a lack of libraries enabling fast development/prototyping
+of optimization-based approaches for multi-robot setups that
+easily interfaces with these efficient solvers.
+To fill this gap, this paper proposes OpTaS, a user-friendly
+task-specification library for rapid development and deploy-
+ment of nonlinear optimization-based planning and control
+approaches such as Model Predictive Control (MPC). The
+library leverages the symbolic framework of CasADi [15],
+enabling function derivatives to arbitrary order via automatic
+differentiation. This is important since some solvers (e.g.
+SNOPT) utilize the Jacobian and Hessian.
+arXiv:2301.13512v1  [cs.RO]  31 Jan 2023
+
+ACEFig. 2: System overview for the proposed OpTaS library. Red highlights the main features of the proposed library. Green
+shows configuration parameter input. Grey shows third-party frameworks/libraries. Finally, the image in the top-right corner
+shows integration with the ROS-PyBullet Interface [16].
+A. Related work
+In this section, we review popular optimization solvers
+and their interfaces. Next, we describe works similar (in
+formulation) to our proposed library. Finally, we summarize
+the key differences and highlight our contributions. Table I
+summarizes alternatives and how they compare to OpTaS.
+There are several capable open-source and commercial
+optimization solvers. First considering quadratic program-
+ming, the OSQP method provides a general purpose solver
+based on the alternating direction method of multipliers [17].
+Alternatively, CVXOPT implements a custom interior-point
+solver [18]. IPOPT implements an interior-point solver for
+constrained nonlinear optimization. SNOPT provides an in-
+terface to an SQP algorithm [13]. KNITRO also solves gen-
+eral mixed-integer programs [12]. Please note that SNOPT
+and KNITRO are proprietary.
+These solvers are often implemented in low-level program-
+ming languages such as C, C++, or FORTRAN. However,
+there are also many interfaces to these methods via higher
+level languages, such as Python, to make implementation and
+adoption easier. The SciPy library contains the optimize
+module [19] to interface with low-level routines, e.g. conju-
+gate gradient and BFGS algorithm [20], the Simplex method
+[21], COBYLA [22], and SLSQP [23]. A requirement when
+using optimization-based methods is the need for function
+gradients. Several popular software packages implement
+automatic differentiation [24], [15], [25]. We leverage the
+CasADi framework [15] for deriving gradients. Our choice
+for CasADI is based on the fact that it comes readily
+integrated with common solvers for optimal control. To the
+best of our knowledge, JAX and PyTorch are not currently
+integrated with constrained nonlinear optimization solvers.
+Similar to our proposed library are the following pack-
+ages. The MoveIt package provides the user with specific
+TABLE I: Comparison between OpTaS and common alter-
+natives in literature.
+Languages
+End-pose Traj. MPC Solver
+AutoDiff ROS Re-form
+OpTaS
+Python
+
+
+
+QP/NLP 
+
+
+EXOTica
+Python/C++ 
+
+
+QP/NLP 
+
+
+MoveIt
+Python/C++ 
+
+
+QP
+
+
+
+TracIK
+Python/C++ 
+
+
+QP
+
+
+
+RBDL
+Python/C++ 
+
+
+QP
+
+
+
+eTaSL
+C++
+
+
+
+QP
+
+1
+
+OpenRAVE Python
+
+
+
+QP
+
+
+
+IK/planning formulations and provides interfaces to solvers
+for the particular problem [14]. The eTaSL library [26]
+allows the user to specify custom tasks specifications, but
+only supports problems formulated as quadratic programs.
+The CASCLIK library uses CasADi and provides support
+for constraint-based inverse kinematic controllers [27], to the
+best of our knowledge they allow optimization in the joint
+space. We provide joint space, task space optimization and
+also the ability to simultaneously optimize in the joint/task
+space. Furthermore, our framework supports optimization of
+several robots in a single formulation. The EXOTica library
+allows the user to specify a problem formulation from an
+XML file [28]. The package, however, requires the user to
+supply analytic gradients for additional sub-task models.
+B. Contributions
+This paper makes the following contributions:
+• A task-specification library, in Python, for rapid devel-
+opment/deployment of TO approaches for multi-robot
+setups.
+• Modeling of the robot kinematics (forward kinematics,
+geometric Jacobian, etc.), to arbitrary derivative order,
+given a URDF specification.
+1Enabled with external pluggins.
+
+Task specification
+Goals
+Obstacles
+Regularization
+Optimization builder
+URDF
+ROS-PyBullet
+Robot model 1
+Interface
+ Model options
+Decision
+Linear (In)equality
+..
+variables
+constraints
+Cost
+URDF
+terms
+":ROS
+Robot model N
+(In)equality
+Parameters
+Model options
+constraints
+Solver
+Solver options
+Optimization problem type identifie
+Deployment
+Feedback
+Joint
+ configuration
+Solver interface
+Optimization problem
+Obstacle
+tracking
+Teleoperator
+Controller/Planner
+input
+Controller options• An interface that allows a user to easily reformulate
+an optimal control problem, and define parameterized
+constraints for online modification of the optimization
+problem.
+• Analysis comparing the performance of the library (i.e.
+solver convergence, solution quality) versus existing
+software packages. Further demonstrations highlight the
+ease in which nonlinear constrained optimization prob-
+lems can be set up and deployed in realistic settings.
+II. PROBLEM FORMULATION
+We can write an optimal control formulation of a TO or
+planning problems as
+min
+x,u cost(x, u; T)
+subject to
+�
+�
+�
+�
+�
+˙x = f(x, u)
+x ∈ X
+u ∈ U
+(1)
+where t denotes time, and x = x(t) ∈ Rnx and u =
+u(t) ∈ Rnu denote the states and controls, with T being the
+time-horizon for the planned trajectory. The scalar function
+cost : Rnx ×Rnu → R represents the cost function (typically
+a weighted sum of terms each modeling a certain sub-task),
+the dot notation denotes a derivative with respect to time
+(i.e. ˙x ≡ dx
+dt ), f represents the system dynamics (equations
+of motion), and X ⊆ Rnx and U ⊆ Rnu are feasible
+regions for the states and controls respectively (modeled by
+a set of equality and inequality constraints). Direct optimal
+control, optimizes for the controls u for a discrete set of time
+instances, using numerical methods (e.g. Euler or Runge-
+Kutta), to integrate the system dynamics over the time
+horizon T [29]. Given an initialization xinit, uinit, a locally
+optimal trajectory x∗, u∗ is found by solving (1).
+As discussed in Sec. I, many works propose optimization-
+based approaches for planning and control. These can all be
+formulated under the same framework, i.e. a TO problem as
+in (1). The goal of our work is to deliver a library that allows
+a user to quickly develop and prototype constrained nonlinear
+TO for multi-robot problems, and deploy them for motion
+generation. The library includes two types of problems, IK
+and task-sace TO, and indeed both simultaneously. Common
+steps, such as transcription that transforms the problem’s
+task-level description into a form accepted by numerical
+optimization solver routines, should be automated and thus
+not burden the user. Furthermore, many works in practice
+require the ability to adapt constraints dynamically to handle
+changes in the environment (e.g. MPC). This motivates a
+constraint parameterization feature.
+III. PROPOSED FRAMEWORK
+In this section, we describe the main features of the
+proposed library shown in Fig. 2. The library is completely
+implemented in the Python programming language. We chose
+Python because it is simple for beginners but also versatile
+with many well-developed libraries, and it easily facilitates
+fast prototyping.
+A. Robot model
+The robot model (RobotModel) provides the kinematic
+modeling and specifies the time derivative orders required for
+the optimization problem. The only requirement is a URDF
+to instantiate the object2. A key feature is that we can include
+several robots in the TO, which is useful for dual arm and
+whole-body optimization. Additional base frames and end-
+effector links can be added programatically (for example,
+when several robots are included the optimization their
+base frames should be registered within a global coordinate
+frame).
+The RobotModel class allows access to data such as:
+the number of degrees of freedom, the names of the actuated
+joints, the upper and lower actuated joint limits, and the kine-
+matics model. Furthermore, we provide methods to compute
+the forward kinematics and geometric Jacobian in any given
+reference frame. Several methods modeling the kinematics
+are supplied, given a specification from the user for the base
+frame and end-effector frame. These methods include: the 4×
+4 homogeneous transformation matrix, translation position,
+rotational representations (e.g. Euler angles, quaternions),
+the geometric and analytical Jacobian. Each of the methods
+above depend on a joint state (supplied as either a Python
+list, NumPy array, or CasADi symbolic array).
+B. Task model
+Several works optimize robot motion in the task space
+and then compute the IK as a secondary step, e.g. [8], [9].
+The task model (TaskModel) provides a representation for
+any arbitrary trajectory. For example, the three dimensional
+position trajectory of an end-effector. In the same way as
+the robot model, the time derivatives can be specified in the
+interface an arbitrary order.
+C. Optimization builder
+This section introduces and describes the optimization
+builder class (OptimizationBuilder). The purpose of
+this class is to aid the user to easily setup a TO problem,
+and then automatically build an optimization problem model
+(Sec. III-D) that interfaces with a solver interface (Sec.
+III-E). The development cycle consists in specifying the
+task (i.e. decision variables, parameters, cost function, and
+constraints) using intuitive syntax and symbolic variables.
+Then, the builder creates an optimization problem class,
+which interfaces with several solvers.
+D. Optimization problem model
+The standard TO is stated in (1). This task/problem is
+specified by the optimization builder class in intuitive syntax
+for the user. Transcribing the problem to a form that can be
+solved by off-the-shelf solvers is non-trivial. The output of
+the optimization builder method build is an optimization
+problem model that allows us to interface with several
+solvers.
+2http://wiki.ros.org/urdf
+
+The most general optimization problem that is modeled
+by OpTaS is given by
+X∗ = arg min
+X
+f(X; P)
+(2a)
+subject to
+k(X; P) = M(P)X + c(P) ≥ 0
+(2b)
+a(X; P) = A(P)X + b(P) = 0
+(2c)
+g(X; P) ≥ 0
+(2d)
+h(X; P) = 0
+(2e)
+where X = [vec(x)T , vec(u)T ]T ∈ RnX is the decision
+variable array such that x, u are as defined in (1) and vec(·)
+is a function that returns its input as a 1-dimensional vector,
+P ∈ RnP is the vectorized parameters, f : RnX → R
+denotes the objective function, k : RnX → Rnk denotes
+the linear inequality constraints, a : RnX → Rna denotes
+the linear equality constraints, g : RnX → Rng denotes
+the nonlinear inequality constraints, and h : RnX → Rnh
+denotes the nonlinear equality constraints. The decision vari-
+ables X are all the joint states and other variables specified
+by the user stacked into a single vector. Similarly for the
+parameters, cost terms, and constraints. Vectorization is made
+possible by the SXContainer data structure implemented
+in the sx container module. This data structure enables
+automatic transcription of the TO problem specified in (1)
+into the form (2).
+Of course, not all task specifications will require defini-
+tions for each of the functions in (2). Depending on the struc-
+ture of the objective function and constraints, the required
+time budget, and accuracy, some solvers will be more appro-
+priate for solving (2). For example, a quadratic programming
+solver that only handles linear constraints (e.g. OSQP [17])
+is unsuitable for solving a problem with nonlinear objective
+function and nonlinear constraints. The build process auto-
+matically identifies the optimization problem type, exposing
+only the relevant solvers. Several problem types are available
+to the user: unconstrained quadratic cost, linearly constrained
+with quadratic cost, nonlinear constrained with quadratic
+cost, unconstrained with nonlinear cost, linearly constrained
+with nonlinear cost, nonlinear cost and constraints.
+1) Initialization: Upon initialization of the optimization
+builder class we can specify (i) the number of time steps
+in the trajectory, (ii) several robot and task models (given a
+unique name for each), (iii) the joint states (positions and
+required time-derivatives) that integrate the decision variable
+array, (iv) task space labels, dimensions, and derivatives
+to also integrate the decision variable array, (v) a Boolean
+describing the alignment of the derivatives (Fig. 3), and (vi)
+a Boolean indicating whether to optimize time steps.
+The alignment of time-derivatives can be specified in
+two ways. Each derivative is aligned with its corresponding
+state (alignement), or otherwise. This is specified by the
+derivs align flag in the optimization builder interface
+and shown diagramatically in Fig. 3.
+In addition, the user can also optimize the time-steps
+between each state. The time derivatives can be integrated
+Fig. 3: Joint state alignment with time. User supplies
+derivs align that specifies how joint state time deriva-
+tives should be aligned.
+over time, e.g. qt+1 = qt + δτt ˙qt, where δτt is an increment
+in time. When optimize time=True, then each δτt is
+included as decision variables in the optimal control problem.
+2) Decision variables and parameters: Decision variables
+are specified in the optimization builder class interface for
+the joint space, task space, and time steps. Each group
+of variables is given a unique label and can be retrieved
+using the get model state method. States are retrieved
+by specifying a robot name or task name, the required time
+index, and the time derivative order required. Additional
+decision variables can be included in the problem by using
+the add decision variables method given a unique
+name and dimension.
+Parameters for the problem (e.g. safe distances) can be
+specified using the add parameter method. To specify a
+new parameter, a unique name and dimension is required.
+3) Cost and constraint functions: The cost function in (1)
+is assumed to be made up of several cost terms, i.e.
+cost(x, u; T) =
+�
+i
+ci(x, u; T)
+(3)
+where ci : Rnx × Rnu → R is an individual cost term
+modeling a specific sub-task. For example, let us define the
+cost terms c0 = ∥ψ(xT ) − ψ∗∥2 and c1 = λ
+� T
+0
+∥u∥2 dt
+(note, discretization is implicit in this formulation) where
+ψ : Rnx → R3 is a function for the forward kinematics
+position (note, this can be provided by the robot model
+class as described in Sec. III-A), ψ∗ ∈ R3 is a goal task
+space position, and 0 < λ ∈ R is a scaling term used
+to weight the relative importance of one constraint against
+the other. Thus, c0 describes an ideal state for the final
+state, and c1 encourages trajectories with minimal control
+signals (e.g. minimize joint velocities). Each cost term is
+added to the problem using the add cost term method;
+the build sequence ensures each term is added to the
+objective function.
+Several constraints can be added to the optimization
+problem by using the add equality constraint and
+add leq inequality constraint methods that add
+equality and inequality constraints respectively. When the
+constraints are added to the problem, they are first checked to
+see if they are linear constraints with respect to the decision
+variables. This functionality allows the library to differentiate
+between linear and nonlinear constraints.
+Additionally, OpTaS offers several methods that provide
+an implementation for common constraints, as, for example,
+
+derivs_align=True
+1-
+-1-
+—...
+q0
+q1
+q2
+q3
+q4
+QT-1
+qT
+:b
+:pb
+q0
+q1
+q2
+q3
+q4
+QT-1
+qT
+derivs_align=False
+1
+..·
+-1
+qo
+q1
+q2
+q3
+q4
+: b
+qT-1
+qT
+qo
+q1
+q2
+q3
+: pb
+qT-1joint position/velocity limits and time-integration for the
+system dynamics f (e.g joint velocities can be integrated
+to positions).
+E. Solver interface
+OpTaS provides interfaces to solvers (open-source and
+commercial) that interface with CasADi [15] (such as
+IPOPT [11]), SNOPT [13], KNITRO [12], and Gurobi [30]),
+the Scipy minimize method [19], OSQP [17], and CVX-
+OPT [18].
+1) Initialization of solver: When the solver is initialized,
+several variables are setup and the optimization problem
+object is set as a class attribute. The user must then call
+the setup method - that itself is an interface to the solver
+initialization that the user has chosen. The requirement of
+this method is to setup the interface for the specific solver;
+relevant solver parameters are passed to the interface at this
+stage.
+2) Resetting the interface: When using the solver as a
+controller, it is expected that the solver should be called more
+than once. In the case for feedback controllers or controllers
+with parameterized constraints (e.g. obstacles), this requires
+a way to reset the problem parameters. Furthermore, the
+initial seed for the optimizer is often required to be reset
+at each control loop cycle. To reset the initial seed and
+problem parameters the user calls reset initial seed,
+and reset parameters, respectively. Both the initial
+seed and parameters are initialized by giving the name of the
+variables. The required vectorization is internally performed
+by the solver utilizing features of the SXContainer data
+structure. Note, if any decision variables or parameters are
+not specified in the reset methods then they automatically
+default to zero. This enables warm-starting the optimization
+routine, e.g. with the solution of the previous time-step
+problem.
+3) Solving an optimization problem: The optimization
+problem is solved by calling the solve method. This
+method passes the optimization problem to the desired
+solver. The resulting data from the solver is collected and
+transformed back into the state trajectory for each robot. A
+method is provided, named interpolate, is used to in-
+terpolate the computed trajectories across time. Additionally,
+the method stats retrieves available optimization statistics
+(e.g. number of iterations).
+4) Extensible solver interface: The solver interface has
+been implemented to allow for extensibility, i.e. additional
+optimization solvers can be easily integrated into the frame-
+work. When a user would like to include a new solver
+interface, they must create a new class that inherits from
+the Solver class. In their sub-class definition they must
+implement three methods: (i) setup which (as described
+above) initializes the solver interface, (ii) solve that calls
+the solver and returns the optimized variable X∗, and (iii)
+stats that returns any statistics from the solver.
+F. Additional features
+Support for integration with ROS [31] is provided out-
+of-the-box. The ROS node provided is integrated with the
+import optas
+# Setup robot and optimization builder
+T = 100 # number of time steps in trajectory
+urdf = ’/path/to/robot.urdf’
+r = optas.RobotModel(urdf, time_deriv=[0, 1])
+n = r.get_name()
+b = optas.OptimizationBuilder(T=T, robots=[r])
+# Retrieve variables and setup parameters
+q0 = b.get_model_state(n, t=0)
+qT = b.get_model_state(n, t=-1) # final state
+pg = b.add_parameter(’pg’, 3) # goal pos.
+qc = b.add_parameter(’qc’, r.ndof) # init q
+o = b.add_parameter(’o’, 3) # obstacle pos.
+r = b.add_parameter(’r’)
+# obstacle radius
+dt = b.add_parameter(’dt’) # time step
+# Forward kinematics
+p = r.get_global_link_position(tip, qT)
+# Cost and constraints
+b.add_cost_term(’c’, optas.sumsqr(p - pg))
+b.integrate_model_states(
+n, time_deriv=1, dt=dt)
+b.add_equality_constraint(’init’, q0, qc)
+for t in range(T):
+b.add_leq_inequality_constraint(
+optas.sumsqr(p - o), r**2)
+# Build optimization problem and setup solver
+solver = optas.CasADiSolver(
+b.build()).setup(’ipopt’)
+Fig. 4: Example code for TO described in Section IV.
+ROS-PyBullet Interface [16] so the publishers/subscribers
+can connect a robot in the optimization problem with a robot
+simulated in PyBullet.
+In addition, we provide a port of the spatialmath
+library by Corke [32] that supports CasADi variables. This
+library defines methods for manipulating homogeneous trans-
+formation matrices, quaternions, Euler angles, etc. using
+CasADi symbolic variables.
+IV. CODE EXAMPLE
+In this section, we describe a common TO problem and
+give the code that models the problem. We aim to highlight
+how straightforward it is to setup a problem.
+Consider a serial link manipulator, and goal to find a
+collision-free plan over time horizon T to a goal end-
+effector position pg given a starting configuration qc. A single
+spherical collision is represented by a position o and radius r.
+The robot configuration qt represent states, and the velocities
+˙qt are controls.
+The cost function is given by ∥p(qT ) − pg∥2 where p
+is the position of the end-effector given by the forward
+kinematics. We solve the problem by minimizing the cost
+function subject to the constraints: (i) initial configuration,
+q0 = qc, (ii) joint limits q− ≤ qt ≤ q+, and (iii) obstacle
+avoidance, ∥p(qt) − o∥2 ≥ r2. The system dynamics is
+represented by several equality constraints qt+1 = qt + δt ˙qt
+that can be specified by methods already in-built into OpTaS.
+The code for the TO problem above, is shown in Fig. 4.
+
+(a)
+(b)
+Fig. 5: Comparison of end-effector task space trajectories
+computed using two different formulations. (a) Shows the
+start (left), and final configurations (right) for the robot under
+each approach. (b) Plots the end-effector position trajectory
+two dimensions.
+V. EXPERIMENTS
+A. Optimization along custom dimensions
+Popular solvers, such as TracIK [33], require the user to
+provide a 6D pose as the task space goal. Whilst this is ap-
+plicable to several robotics problems (e.g. pick-and-place) it
+may not be necessary to optimize each task space dimension
+(e.g. spraying applications does not require optimization in
+the roll angular direction). Furthermore, optimizing in more
+dimensions than necessary may be disadvantageous.
+OpTaS can optimize or neglect any desired task space
+dimension. This can have certain advantages, for example
+increasing the robot workspace. Consider a non-prehensile
+pushing task along the plane, optimizing the full 6D pose
+may not be ideal since the task is two dimensional. By
+optimizing in the two dimensional plane and specifying
+boundary constraints on the third linear spatial dimension,
+increases the robots workspace.
+We setup a tracking experiment in OpTaS using a sim-
+ulated Kuka LWR robot arm to compare the two cases:
+(i) optimize the full 6D pose, and (ii) optimize 2D linear
+position. The robot is given an initial configuration (Fig. 5a
+left) and the task is to move the end-effector with velocity of
+constant magnitude and direction in the 2D plane. The end
+configuration for each approach is shown in Fig. 5a right and
+the end-effector trajectories are shown in Fig. 5b. We see
+that the 2D optimization problem is able to reach a greater
+distance, highlighting that the robot workspace is increased.
+B. Performance comparison
+In this section, we demonstrate that OpTaS can formulate
+similar problems and compare its performance to alterna-
+tives. First, we model, with OpTaS, the same problem as
+used in TracIK [33] and in addition we also model the
+problem using EXOTica [28]. The Scipy SLSQP solver [23]
+was used for OpTaS and EXOTica. With same Kuka LWR
+Fig. 6: Figure-of-eight trajectory tracked by the Kuka LWR.
+(a)
+(b)
+Fig. 7: Solver duration comparisons for figure of eight
+motion. (a) Compares an IK tracking approach described
+in Section V, (b) is a similar comparison that includes a
+maximization term for manipulability. Green is OpTaS, red
+is TracIK, and blue is EXOTica.
+robot arm in the previous experiment, we setup a task where
+the robot must track a figure-of-eight motion in task space
+(Fig. 6) and record the CPU time for the solver duration at
+each control loop cycle. The results are shown in Fig. 7a.
+TracIK is the fastest (0.049 ± 0.035ms), which is expected
+since it is optimized for a specific problem formulation. We
+see that OpTaS (2.608 ± 0.239ms) is faster than EXOTica
+(3.694 ± 0.300ms)
+A second experiment, using the same setup as be-
+fore, was performed comparing the performance of OpTaS
+against EXOTica with an additional cost term to maxi-
+mize manipulability [34]. The results are shown in Fig.
+7b. Despite using the same formulation and solver, OpTaS
+(2.650 ± 0.270ms) achieved better performance than EXOT-
+ica (7.640±1.404ms). Without extensive profiling it is diffi-
+cult to precisely explain this difference. However, EXOTica
+requires the user to supply analytical gradients for sub-tasks
+(called task maps in the EXOTica documentation). EXOTica
+does not provide the gradients for the manipulability task,
+and thus falls-back to using the finite difference method to
+estimate the gradient - this can can be slow to compute.
+VI. CONCLUSIONS
+In this paper, we have proposed OpTaS: an optimization-
+based task tpecification Python library for TO and MPC.
+OpTaS allows a user to setup a constrained nonlinear pro-
+grams for custom problem formulations and has been shown
+to perform well against alternatives. Parameterization enables
+programs to act as feedback controllers, motion planners, and
+benchmark problem formulations and solvers.
+We hope OpTaS will be used by researchers, students, and
+industry to facilitate the development of control and motion
+planning algorithms. The code base is easily installed via
+pip and has been made open-source under the Apache 2
+license: https://github.com/cmower/optas.
+
+0.1
+Optimize6D
+Optimize2D
+Ideal path
+Start
+(m)
+0.0
+-0.1
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+X (m)CPU Time (ms)
+OpTaS
+TraclK
+EXOTica
+7
+0
+0.0
+2.5
+5.0
+7.5
+5 10.0 12.5 15.0 17.5 20.0
+Time (s)CPU Time (ms)
+10
+OpTaS
+EXOTica
+8
+6
+4
+2
+0.0
+2.5
+5.0
+7.5
+10.0 12.5 15.0 17.5 20.0
+Time (s)REFERENCES
+[1] S. M. LaValle, Planning algorithms.
+Cambridge university press,
+2006.
+[2] S. Karaman and E. Frazzoli, “Sampling-based algorithms for optimal
+motion planning,” The international journal of robotics research,
+vol. 30, no. 7, pp. 846–894, 2011.
+[3] N. Ratliff, M. Zucker, J. A. Bagnell, and S. Srinivasa, “Chomp:
+Gradient optimization techniques for efficient motion planning,” in
+2009 IEEE International Conference on Robotics and Automation,
+2009, pp. 489–494.
+[4] J. Schulman, Y. Duan, J. Ho, A. Lee, I. Awwal, H. Bradlow, J. Pan,
+S. Patil, K. Goldberg, and P. Abbeel, “Motion planning with sequential
+convex optimization and convex collision checking,” The International
+Journal of Robotics Research, vol. 33, no. 9, pp. 1251–1270, 2014.
+[5] M. Posa, C. Cantu, and R. Tedrake, “A direct method for trajectory op-
+timization of rigid bodies through contact,” The International Journal
+of Robotics Research, vol. 33, no. 1, pp. 69–81, 2014.
+[6] S. Kuindersma, R. Deits, M. Fallon, A. Valenzuela, H. Dai,
+F. Permenter, T. Koolen, P. Marion, and R. Tedrake, “Optimization-
+based locomotion planning, estimation, and control design for the
+atlas humanoid robot,” Autonomous Robots, vol. 40, no. 3, pp.
+429–455, Mar 2016. [Online]. Available: https://doi.org/10.1007/
+s10514-015-9479-3
+[7] T. Stouraitis, I. Chatzinikolaidis, M. Gienger, and S. Vijayakumar,
+“Online hybrid motion planning for dyadic collaborative manipulation
+via bilevel optimization,” IEEE Transactions on Robotics, vol. 36,
+no. 5, pp. 1452–1471, 2020.
+[8] C. E. Mower, J. Moura, and S. Vijayakumar, “Skill-based shared
+control,” in Robotics: Science and Systems, 2021.
+[9] J. Moura, T. Stouraitis, and S. Vijayakumar, “Non-prehensile planar
+manipulation via trajectory optimization with complementarity con-
+straints,” in 2022 International Conference on Robotics and Automa-
+tion (ICRA).
+IEEE, 2022, pp. 970–976.
+[10] M. Toussaint, J. Harris, J.-S. Ha, D. Driess, and W. H¨onig, “Sequence-
+of-constraints mpc: Reactive timing-optimal control of sequential
+manipulation,” 2022.
+[11] A. W¨achter and L. T. Biegler, “On the implementation of an
+interior-point filter line-search algorithm for large-scale nonlinear
+programming,” Mathematical Programming, vol. 106, no. 1, pp.
+25–57,
+Mar
+2006.
+[Online].
+Available:
+https://doi.org/10.1007/
+s10107-004-0559-y
+[12] R. H. Byrd, J. Nocedal, and R. A. Waltz, “KNITRO: An integrated
+package for nonlinear optimization,” in Large-scale nonlinear opti-
+mization.
+Springer, 2006, pp. 35–59.
+[13] P.
+E.
+Gill,
+W.
+Murray,
+and
+M.
+A.
+Saunders,
+“Snopt:
+An
+sqp
+algorithm
+for
+large-scale
+constrained
+optimization,”
+SIAM
+Rev., vol. 47, no. 1, p. 99–131, jan 2005. [Online]. Available:
+https://doi.org/10.1137/S0036144504446096
+[14] D. Coleman, I. Sucan, S. Chitta, and N. Correll, “Reducing the barrier
+to entry of complex robotic software: a MoveIt! case study,” 2014.
+[15] J. A. E. Andersson, J. Gillis, G. Horn, J. B. Rawlings, and M. Diehl,
+“CasADi – A software framework for nonlinear optimization and
+optimal control,” Mathematical Programming Computation, vol. 11,
+no. 1, pp. 1–36, 2019.
+[16] C. E. Mower, T. Stouraitis, J. Moura, C. Rauch, L. Yan, N. Z. Be-
+habadi, M. Gienger, T. Vercauteren, C. Bergeles, and S. Vijayakumar,
+“ROS-PyBullet Interface: A framework for reliable contact simulation
+and human-robot interaction,” in [to appear] Proceedings of the
+Conference on Robot Learning, ser. Proceedings of Machine Learning
+Research.
+PMLR, 2022.
+[17] B. Stellato, G. Banjac, P. Goulart, A. Bemporad, and S. Boyd, “OSQP:
+an operator splitting solver for quadratic programs,” Mathematical
+Programming Computation, vol. 12, no. 4, pp. 637–672, 2020.
+[Online]. Available: https://doi.org/10.1007/s12532-020-00179-2
+[18] M. Andersen, J. Dahl, and L. Vandenberghe, “Cvxopt: Convex opti-
+mization,” Astrophysics Source Code Library, pp. ascl–2008, 2020.
+[19] P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy,
+D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J.
+van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J.
+Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, ˙I. Polat, Y. Feng,
+E. W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman,
+I. Henriksen, E. A. Quintero, C. R. Harris, A. M. Archibald, A. H.
+Ribeiro, F. Pedregosa, P. van Mulbregt, and SciPy 1.0 Contributors,
+“SciPy 1.0: Fundamental Algorithms for Scientific Computing in
+Python,” Nature Methods, vol. 17, pp. 261–272, 2020.
+[20] J. Nocedal and S. J. Wright, Numerical optimization. Springer, 1999.
+[21] J. A. Nelder and R. Mead, “A simplex method for function minimiza-
+tion,” The computer journal, vol. 7, no. 4, pp. 308–313, 1965.
+[22] M. J. D. Powell, A Direct Search Optimization Method That Models
+the Objective and Constraint Functions by Linear Interpolation.
+Dordrecht:
+Springer
+Netherlands,
+1994,
+pp.
+51–67.
+[Online].
+Available: https://doi.org/10.1007/978-94-015-8330-5 4
+[23] D. Kraft, “A software package for sequential quadratic programming,”
+Tech. Rep. DFVLR-FB 88-28, DLR German Aerospace Center –
+Institute for Flight Mechanics, Koln, Germany, 1988.
+[24] J. Bradbury, R. Frostig, P. Hawkins, M. J. Johnson, C. Leary,
+D. Maclaurin, G. Necula, A. Paszke, J. VanderPlas, S. Wanderman-
+Milne,
+and
+Q.
+Zhang,
+“JAX:
+composable
+transformations
+of
+Python+NumPy programs,” 2018. [Online]. Available: http://github.
+com/google/jax
+[25] A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan,
+T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, A. Desmaison, A. Kopf,
+E. Yang, Z. DeVito, M. Raison, A. Tejani, S. Chilamkurthy, B. Steiner,
+L. Fang, J. Bai, and S. Chintala, “Pytorch: An imperative style, high-
+performance deep learning library,” in Advances in Neural Information
+Processing Systems 32, H. Wallach, H. Larochelle, A. Beygelzimer,
+F. d'Alch´e-Buc, E. Fox, and R. Garnett, Eds. Curran Associates, Inc.,
+2019, pp. 8024–8035.
+[26] E. Aertbeli¨en and J. De Schutter, “etasl/etc: A constraint-based task
+specification language and robot controller using expression graphs,”
+in 2014 IEEE/RSJ International Conference on Intelligent Robots and
+Systems, 2014, pp. 1540–1546.
+[27] M. H. Arbo, E. I. Grøtli, and J. T. Gravdahl, “Casclik: Casadi-
+based closed-loop inverse kinematics,” 2019. [Online]. Available:
+https://arxiv.org/abs/1901.06713
+[28] V. Ivan, Y. Yang, W. Merkt, M. P. Camilleri, and S. Vijayakumar,
+EXOTica: An Extensible Optimization Toolset for Prototyping and
+Benchmarking Motion Planning and Control.
+Cham: Springer
+International Publishing, 2019, pp. 211–240. [Online]. Available:
+https://doi.org/10.1007/978-3-319-91590-6 7
+[29] M. Kelly, “An introduction to trajectory optimization: How to do your
+own direct collocation,” SIAM Review, vol. 59, no. 4, pp. 849–904,
+2017.
+[30] Gurobi Optimization, LLC, “Gurobi Optimizer Reference Manual,”
+2022. [Online]. Available: https://www.gurobi.com
+[31] M. Quigley, B. Gerkey, K. Conley, J. Faust, T. Foote, J. Leibs,
+E. Berger, R. Wheeler, and A. Ng, “ROS: an open-source robot
+operating system,” in Proc. of the IEEE Intl. Conf. on Robotics and
+Automation (ICRA) Workshop on Open Source Robotics, Kobe, Japan,
+2009.
+[32] P. I. Corke, Robotics, Vision & Control: Fundamental Algorithms in
+MATLAB, 2nd ed.
+Springer, 2017, iSBN 978-3-319-54413-7.
+[33] P. Beeson and B. Ames, “Trac-ik: An open-source library for improved
+solving of generic inverse kinematics,” in 2015 IEEE-RAS 15th
+International Conference on Humanoid Robots (Humanoids), 2015,
+pp. 928–935.
+[34] T. Yoshikawa, “Manipulability of robotic mechanisms,” The Interna-
+tional Journal of Robotics Research, vol. 4, no. 2, pp. 3–9, 1985.
+[Online]. Available: https://doi.org/10.1177/027836498500400201
+

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+TUM-HEP 1447/22, IPMU22-0070
+Enhanced prospects for direct detection of
+inelastic dark matter from a non-galactic diffuse
+component
+Gonzalo Herrera1,2, Alejandro Ibarra1, and Satoshi Shirai3
+1Physik-Department, Technische Universität München, James-Franck-Straße, 85748
+Garching, Germany
+2Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6,80805
+München, Germany
+3Kavli Institute for the Physics and Mathematics of the Universe (WPI),The University of
+Tokyo Institutes for Advanced Study, The University of Tokyo, Kashiwa 277-8583, Japan
+Abstract
+In some scenarios, the dark matter particle predominantly scatters inelastically with
+the target, producing a heavier neutral particle in the final state. In this class of scenarios,
+the reach in parameter space of direct detection experiments is limited by the velocity of
+the dark matter particle, usually taken as the escape velocity from the Milky Way. On the
+other hand, it has been argued that a fraction of the dark matter particles in the Solar
+System could be bound to the envelope of the Local Group or to the Virgo Supercluster,
+and not to our Galaxy, and therefore could carry velocities larger than the escape velocity
+from the Milky Way. In this paper we estimate the enhancement in sensitivity of current
+direct detection experiments to inelastic dark matter scatterings with nucleons or electrons
+due to the non-galactic diffuse components, and we discuss the implications for some well
+motivated models.
+1
+Introduction
+The existence of dark matter in galaxies, clusters of galaxies and the Universe at large scale is
+by now established by their gravitational effects on ordinary matter (for reviews, see e.g. [1–
+4]). If the dark matter is constituted by new particles, it is plausible that they could interact
+with the ordinary matter through other interactions aside from gravity. A promising avenue to
+probe these putative interactions consists in the search for nuclear or electron recoils induced
+by dark matter particles entering a dedicated detector at the Earth [5, 6] (for reviews, see e.g.
+[7–9]). This search strategy, denominated direct detection, has seen an impressive increase in
+sensitivity since it was first proposed more than three decades ago. Yet, no conclusive dark
+matter signal has been found to date.
+Assuming that the dark matter scatters elastically with the nucleus, current direct detection
+experiments restrict the spin-independent interaction cross-section to be smaller than ∼ 1
+1
+arXiv:2301.00870v1  [hep-ph]  2 Jan 2023
+
+zeptobarn in the mass range ∼ 10 GeV - 1 TeV [10]. These stringent constraints put pressure on
+several well motivated dark matter scenarios, especially those for which the dark matter particle
+couples at tree level with the valence quarks in models addressing the electroweak hierarchy
+problem [1]. On the other hand, there are many other dark matter scenarios, arguably also
+well motivated theoretically, which are largely unconstrained by current searches.
+In this paper we will focus on scenarios where the dark matter cannot scatter elastically with
+a nucleus (or an electron), so that the stringent limits on the elastic scattering cross-section do
+not necessarily hold. This seemingly strong assumption naturally arises in some models. For
+instance, the elastic scattering mediated by vector current is forbidden for Majorana dark matter
+χ, due to the Majorana nature of fermion: ¯χγµχ = 0 [11]. However, Majorana dark matter
+particles may leave an imprint in direct search experiments if they could scatter inelastically
+producing a heavier Majorana fermion χ′ in the final state, since there is an off-diagonal fermion
+current ¯χ′γµχ ̸= 0. This scenario is approximately realized in the Minimal Supersymmetric
+Standard Model, when the lightest supersymmetric particle is almost a pure Higgsino state,
+and the other supersymmetric particles are very heavy. In this case, the elastic scattering of
+the Higgsino dark matter is suppressed by the large sfermion and gaugino masses, while it has
+a large inelastic scattering cross section by the electroweak gauge interactions [12]. Scenarios
+of inelastic dark matter have also been motivated phenomenologically, e.g. in [12–22].
+The kinematics of the inelastic scattering differs from the one in the elastic scenario. In
+order to allow the production of a heavier neutral particle in the final state, the velocity of the
+incoming dark matter particle must be larger than a certain threshold. Therefore, as the mass
+difference between the initial and final neutral particles increases, faster and faster dark matter
+particles are necessary in order to open kinematically the inelastic process. For dark matter
+particles bound to our galaxy, and which have speeds smaller than the escape velocity from
+the Milky Way, vesc = 544 km/s [23, 24], the inelastic scattering off a nucleus is kinematically
+allowed when the mass difference between the two states is δm < 1/2µv2
+esc, with µ the reduced
+mass of the DM-nucleus system; for the scattering off an electron, the inelastic channel is open
+when δm < 1/2µev2
+esc − |Enl|, where µe is the reduced mass of the DM-electron system, and
+|Enl| is the binding energy of an electron in the (n, l) shell of the target nucleus. In practice,
+experiments can only detect recoiling nuclei/ionized electrons within a given energy range,
+therefore the mass difference that can be probed in direct searches is smaller than this value.
+In this letter we argue that the parameter space of inelastic dark matter scenarios that
+can be probed in direct search experiments is larger than the one previously considered in
+the literature, that implicitly assumes that the Milky Way is an isolated galaxy. Instead, the
+Milky Way is one among the various members of the Local Group, which include M31, M33
+and several dwarf galaxies. It has been argued that the Local Group contains a diffuse dark
+matter component, which is not bound to any individual galaxy, and which is distributed
+roughly homogeneously over the Local Group [25–27]. Notably, a non-negligible fraction of
+the dark matter particles in the Solar System is expected to be associated to this non-galactic
+diffuse component, rather than to the Milky Way halo, and could have velocities larger than
+the escape velocity from the Milky Way. Consequently, the mass splitting that could be probed
+in experiments correspondingly increases. Likewise, the Local Group is one among the many
+groups of galaxies embedded in the Virgo Supercluster, which could also contain a diffuse
+component [28]. Although the fraction of dark matter particles in the Solar System associated
+to the Virgo Supercluster is fairly small, they have very large velocities, and could be pivotal in
+generating a signal in direct search experiments when the inelastic scattering is kinematically
+2
+
+inaccessible for the dark matter bound to the Milky Way and to the Local Group.
+The paper is organized as follows. In section 2, we present the non-galactic dark matter
+flux at Earth. In section 3, we derive constraints on inelastic dark matter from nuclear recoil
+searches, and in section 4, we derive constraints from electron recoil searches. Finally, in section
+5, we present our conclusions.
+2
+Dark matter flux at Earth
+A correct description of the dark matter flux at Earth is crucial for assessing the prospects for
+detection of a given dark matter model. The largest contribution to the flux is expected to arise
+from dark matter particles in the Milky Way halo. The local density of dark matter particles
+and their velocity distribution is unknown. However, it is common in the literature to adopt
+the Standard Halo Model (SHM), characterized by a local density ρloc
+SHM = 0.3 GeV/cm3 and
+an isotropic velocity distribution described by a Maxwell-Boltzmann distribution truncated at
+the escape velocity of the Milky Way [29, 30]. In the galactic frame, the velocity distribution
+reads:
+fSHM(⃗v) =
+1
+(2πσ2
+v)3/2Nesc
+exp
+�
+− v2
+2σ2
+v
+�
+for v ≤ vesc ,
+(1)
+where v = |⃗v|, σv ≈ 156 km/s is the velocity dispersion [30, 31], and vesc = 544 km/s is the
+escape velocity from our Galaxy [23, 24]. Further, Nesc is a normalization constant, given by:
+Nesc = erf
+� vesc
+√
+2σv
+�
+−
+�
+2
+π
+vesc
+σv
+exp
+�
+−v2
+esc
+2σ2
+v
+�
+.
+(2)
+For our chosen parameters, Nesc ≃ 0.993. The contribution to the local dark matter flux from
+the Milky Way halo then reads:
+FSHM(⃗v) = ρloc
+SHM
+mDM
+vfSHM(⃗v) .
+(3)
+It is plausible that the dark matter flux at Earth also contains a contribution from dark mat-
+ter particles not bound to the Milky Way. Astronomical observations indicate the presence of
+diffuse dark matter components homogeneously distributed between clusters and Superclusters
+of galaxies [32]. Since these dark matter particles are not gravitationally bound to the Milky
+Way, they carry larger velocities than the escape velocity of the Milky Way. In this work, we
+consider the contribution to the dark matter flux from the Local Group and from the Virgo
+Supercluster. The dark matter particles from the Local Group contribute at the Solar System
+with a local density of ρLG ∼ 10−2 GeV/cm3, and are expected to move isotropically with a
+narrow velocity distribution, σv.LG ∼ 20 km/s, and with mean velocity vLG ∼ 600 km/s [33].
+The contribution from the Local Group to the dark matter flux at the location of the Solar
+System then reads:
+FLG(⃗v) = ρloc
+LG
+mDM
+δ(v − vLG)
+4πv
+.
+(4)
+Dark matter particles bound to the Virgo Supercluster give a small contribution to the local
+dark matter density. Observations indicate that the average density in the diffuse component
+3
+
+of the Virgo Supercluster is close to the cosmological value ∼ 10−6 GeV/cm3 [28]. However, the
+gravitational focusing due to the Local Group leads to an increase in the density at the location
+of the Sun by a factor ∼ 1 + v2
+esc/v2
+σVS, where vσVS is the velocity dispersion of the dark matter
+particles from the Virgo Supercluster [33]. This value is highly uncertain, but it is expected
+to be comparable to that of the observable members of the Supercluster, which ranges from
+vσVS ∼ 50 km/s to vσVS ∼ 500 km/s [28, 34]. We consider for concreteness an enhancement on
+the local density of dark matter particles from the Virgo Supercluster of ∼ 10, consistent with
+the value of the velocity dispersion of the observable members of the Supercluster, which leads
+to ρloc
+VG ∼ 10−5 GeV/cm3. Current knowledge on the dark matter velocity distribution in the
+Virgo Supercluster is much poorer. Following [33], we assume that the dark matter particles
+have the typical velocities of the members of the Virgo Supercluster, corresponding to (at least)
+vVS ∼ 1000 km/s. The contribution to the dark matter flux at the location of the Solar System
+from the Virgo Supercluster can then be written as:
+FVS(⃗v) = ρloc
+VS
+mDM
+δ(v − vVS)
+4πv
+.
+(5)
+The total (galactic plus non-galactic) dark matter flux at the Solar System is therefore
+approximately given by:
+F(⃗v) = FSHM(⃗v) + FLG(⃗v) + FVS(⃗v).
+(6)
+Following [33], we adopt values for the local density of each component such that the total sum
+yields the canonical value of the local density used by direct detection experiments ρloc = 0.3
+GeV/cm3, namely ρloc
+SHM = 0.26 GeV/cm3 (∼ 88%), ρloc
+LG = 0.037 GeV/cm3 (∼ 12%), and
+ρloc
+VS = 10−5 GeV/cm3 (∼ 0.00003%).
+3
+Impact on nuclear recoils
+The differential rate of nuclear recoils induced by inelastic up-scatterings of dark matter parti-
+cles traversing a detector at the Earth is given by:
+dR
+dER
+=
+�
+i
+ξi
+mAi
+�
+v≥vi
+min(ER)
+d3vF(⃗v + ⃗v⊙) dσi
+dER
+(v, ER) .
+(7)
+Here, ⃗v is the dark matter velocity in the rest frame of the detector, F(⃗v + ⃗v⊙) is the dark
+matter flux in the detector frame, and ⃗v⊙ is the velocity of the Sun with respect to the Galactic
+frame with |⃗v⊙| ≈ 232 km/s [35]. For the inelastic scattering with mass splitting between two
+dark matter states, δDM, the minimum velocity necessary to induce a recoil with energy ER of
+the nucleus i with mass mAi and mass fraction ξi in the detector reads
+vi
+min(ER) =
+1
+�
+2ERmAi
+�ERmAi
+µAi
++ δDM
+�
+.
+(8)
+Further, for spin-independent interactions, the differential dark matter-nucleus cross section
+reads,
+dσSI
+i
+dER
+(v, ER) =
+mAi
+2µ2
+Aiv2σSI
+0,iF 2
+i (ER) .
+(9)
+4
+
+Here mAi is mass of the nucleus i, µAi is the reduced mass of the dark matter-nucleus i system
+and F 2
+i (ER) is the nuclear form-factor, for which we adopt the Helm prescription. Besides,
+σSI
+0,i is the spin-independent dark matter-nucleus scattering cross section at zero momentum
+transfer, which depends on the details of the dark matter model and the target nucleus. From
+the differential rate, one can calculate the total recoil rate using:
+R =
+� ∞
+0
+dER ϵi(ER) dR
+dER
+,
+(10)
+where ϵi(ER) is the efficiency of that experiment. Finally, the total number of expected recoil
+events is N = R · E, with E the exposure (i.e. mass multiplied by live-time).
+In our analysis, we will consider two scenarios for the coupling of dark matter to nucleons.
+First, we will consider a Majorana dark matter candidate. In this case
+σSI
+0,i = 4µ2
+Ai
+π
+�
+Zif p
+S + (Ai − Zi)f n
+S
+�2
+,
+(11)
+where f p
+S and f n
+S parametrize the strength of the scalar interactions to the proton and the
+neutron (see e.g. [7, 36]). It is common to write Eq. (11) as
+σSI
+0,i = µ2
+Ai
+µ2
+p
+�
+Zi + (Ai − Zi)f n
+S
+f p
+S
+�2
+σDM,p ,
+(12)
+with µp the reduced mass of the DM-proton system and σDM,p an effective DM-proton inter-
+action cross-section. Within the Majorana dark matter scenario, we will consider in particular
+the widely adopted benchmark case where the interaction is “isoscalar”, i.e. when the dark
+matter couples with equal strength to protons and neutrons, for which
+σSI
+0,i = µ2
+Ai
+µ2
+p
+A2
+i σDM,p .
+(13)
+We will also consider a scenario where the dark matter has hypercharge Y , and interacts
+with the quarks via the exchange of a Z boson.
+In this case, σSI
+0,i has the same form as
+Eq. (11), replacing the scalar couplings by the corresponding vector couplings, f p,n
+S
+→ f p,n
+V .
+For interactions with the Z boson, f p
+V and f n
+V are explicitly given by:
+f p
+V = GFζY
+2
+√
+2 (1 − 4 sin2 θW) ,
+f n
+V = −GFζY
+2
+√
+2 ,
+(14)
+with ζ = 1 (ζ = 2) for fermionic (bosonic) dark matter [5, 21, 37]. In this scenario, the dark
+matter-nucleus cross section can be related to the dark matter-proton cross-section through:
+σSI
+0,i = µ2
+Ai
+µ2
+p
+�
+Zi −
+(Ai − Zi)
+(1 − 4 sin2 θW)
+�2
+σDM,p ,
+(15)
+which is independent of the dark matter hypercharge and spin.
+To assess the impact of the non-galactic diffuse components for direct detection experiments,
+we plot in Figure. 1 the differential rate of inelastic scatterings in the LUX-ZEPLIN experiment
+5
+
+10
+20
+30
+40
+50
+60
+70
+80
+ER [keV]
+10−6
+10−4
+10−2
+100
+102
+104
+106
+dR
+dER [keV−1]
+mDM = 1 TeV
+σDM−p = 10−38cm2
+LUX-ZEPLIN (SHM, δDM = 100 keV)
+LUX-ZEPLIN (SHM, δDM = 200 keV)
+LUX-ZEPLIN (SHM+Non-galactic, δDM = 100 keV)
+LUX-ZEPLIN (SHM+Non-galactic, δDM = 200 keV)
+CEνNS (Solar neutrinos)
+Figure 1: Differential rate for the inelastic scattering of a Majorana dark matter candidate in
+the “isoscalar” scenario with mass mDM = 1 TeV, for δDM = 100 keV (light blue) and 200 keV
+(dark blue), for a dark matter flux at Earth as modelled by the Standard Halo Model (dotted
+line) or including also the contribution from the non-galactic diffuse dark matter component
+(solid line). For the plots it was assumed σDM,p = 10−38 cm2.
+for the “isoscalar” scenario, assuming mDM = 1 TeV and σDM,p = 10−38 cm2, for δDM = 100
+keV (light blue) and 200 keV (dark blue), including in the flux only the contribution from dark
+matter bound to the Milky Way (dotted lines), as commonly assumed in the literature, and
+including the contribution from the non-galactic diffuse component (solid lines). The impact
+of the non-galactic component in the differential rate is apparent from the figure, and increases
+the number of events at all recoil energies, especially in the region with low ER which is not
+kinematically accessible to the galactic dark matter. The non-galactic dark matter, therefore,
+has implications not only for enhancing the sensitivity of the experiment, but also for the
+interpretation of a putative dark matter signal.
+Current direct search experiments have not observed a significant excess of nuclear recoils,
+which allows to derive upper limits on the dark matter nucleon cross section for given com-
+binations of the dark matter mass and mass splitting between the dark matter particle and
+the neutral particle in the final state. In Figure 2, we show upper limits on the dark matter-
+proton spin-independent scattering cross section versus mass splitting for mDM = 1 TeV from
+LUX-ZEPLIN (blue) [10], PICO60 (green) [38], CRESST-II (red) [39], and from a radiopurity
+measurement in a CaWO4 crystal (orange) [40, 41]. The dotted lines represent the limits ob-
+tained considering the galactic dark matter (described by the SHM) as the only contribution
+to the dark matter flux, while the solid lines were obtained including also the contributions to
+the flux from the non-galactic diffuse component in the Solar System. In the upper left plot,
+we show the limits for a Majorana dark matter candidate in the “isoscalar” scenario, and in the
+upper right plot, the most conservative limit for the Majorana dark matter, without making
+assumptions on the coupling strengths, derived following the approach of [42]. Lastly, in the
+lower plot we show the limits for a scenario where the dark matter interacts with the nucleus via
+the exchange of a Z-boson. In the latter plot we also show the dark matter-proton scattering
+cross-section for scenarios of a fermionic dark matter, and Y = 1/2 (corresponding to the well
+motivated scenario of the Higgsino dark matter in the limit of high scale supersymmetry [12]),
+6
+
+0
+200
+400
+600
+800
+1000
+1200
+δDM [keV]
+10−48
+10−46
+10−44
+10−42
+10−40
+10−38
+10−36
+10−34
+10−32
+10−30
+σSI
+DM−p[cm2]
+mDM = 1 TeV
+Majorana DM, f n = f p
+LUX-ZEPLIN (SHM)
+LUX-ZEPLIN (SHM + Non-galactic)
+PICO60 (SHM)
+PICO60 (SHM + Non-galactic)
+CRESST II (SHM)
+CRESST II (SHM + Non-galactic)
+CaWO4 (SHM)
+CaWO4 (SHM + Non-galactic)
+0
+200
+400
+600
+800
+1000
+1200
+δDM [keV]
+10−48
+10−46
+10−44
+10−42
+10−40
+10−38
+10−36
+10−34
+10−32
+10−30
+σSI
+DM−p[cm2]
+mDM = 1 TeV
+Majorana DM, f n, f p free
+LUX-ZEPLIN (SHM)
+LUX-ZEPLIN (SHM + Non-galactic)
+PICO60 (SHM)
+PICO60 (SHM + Non-galactic)
+CRESST II (SHM)
+CRESST II (SHM + Non-galactic)
+CaWO4 (SHM)
+CaWO4 (SHM + Non-galactic)
+0
+200
+400
+600
+800
+1000
+1200
+δDM [keV]
+10−48
+10−46
+10−44
+10−42
+10−40
+10−38
+10−36
+10−34
+10−32
+σSI
+DM−p[cm2]
+mDM = 1 TeV
+Y=1/2
+Y=1
+Y=3/2
+Z-boson mediation
+LUX-ZEPLIN (SHM)
+LUX-ZEPLIN (SHM + Non-galactic)
+PICO60 (SHM)
+PICO60 (SHM + Non-galactic)
+CRESST II (SHM)
+CRESST II (SHM + Non-galactic)
+CaWO4 (SHM)
+CaWO4 (SHM + Non-galactic)
+Figure 2: 90% C.L upper limits on the spin-independent dark matter-proton inelastic cross
+section for a dark matter mass of 1 TeV as a function of the mass splitting, from LUX-ZEPLIN
+(blue), PICO60 (green), CRESST-II (red and orange) and from a CaWO4 detector radiopurity
+measurement (orange). We show the limits for three different scenarios: Majorana dark matter
+with scalar interactions f p = f n (upper left plot), arbitrary f p and f n (upper right plot),
+and dark matter interacting via the Z-boson (lower plot). In the lower plot, we also show for
+reference the predicted value of the cross-section with a xenon target for scenarios of fermionic
+dark matter with hypercharge Y = 1/2, 1, 3/2.
+Y = 1 and Y = 3/2 (which correspond to different scenarios of minimal dark matter [37]), for a
+xenon target. For other targets, the expected cross section for mDM = 1 TeV scales as ∼ Ai/Zi,
+being indistinguishable in the Figure.
+As seen in the plots, for all the scenarios the non-galactic diffuse component enhances the
+sensitivity of experiments to inelastic dark matter, allowing to probe larger mass splittings.
+For instance, for our representative dark matter mass of 1 TeV, the LUX-ZEPLIN experiment
+is insensitive to dark matter particles of the Milky Way scattering inelastically if the mass
+difference with the neutral particle in the final state is δDM ≳ 300 keV. However, the presence
+of dark matter in the Solar System from the envelope of the Local Group extends the reach
+up to δDM ≃ 330 keV and allows to probe uncharted parameter space for large mass splittings.
+7
+
+Concretely, the LUX-ZEPLIN experiment sets for the isoscalar scenario the limit σSI
+DM−p ≲
+10−44 cm2 for δDM = 250 keV, which is about three orders of magnitude stronger than the limit
+obtained assuming that all dark matter is bound to the Milky Way, and only a factor of 100
+weaker than the limit on the elastic scattering cross-section i.e. for δDM = 0. For the interaction
+mediated by the Z-boson the upper limit is σSI
+DM−p ≲ 10−44 cm2, and the most conservative limit
+without making assumptions on the form of the interaction is σSI
+DM−p ≲ 10−40 cm2, obviously
+much weaker than for concrete scenarios. The dark matter particles from the Virgo Supercluster
+extend the reach to even larger mass differences, up to δDM ≃ 450 keV and sets for the isoscalar
+scenario the limit σSI
+DM−p ≲ 5 × 10−40 cm2 for δDM = 450 keV; for the interaction mediated
+by the Z-boson the upper limit is σSI
+DM−p ≲ 10−41 cm2, while the model independent limit is
+σSI
+DM−p ≲ 5 × 10−36 cm2. Similar conclusions apply for the PICO and CRESST experiments,
+and from the radiopurity measurements on a CaWO4 target.
+It is interesting to note the complementarity of the different experiments in probing the
+parameter space of inelastic dark matter scenarios. Both in the scenario of a Majorana dark
+matter with f n = f p and for the scenario with Z-boson mediation, LUX-ZEPLIN is the most
+sensitive probe for small δDM, whereas the radiopurity measurements on a CaWO4 is the most
+sensitive probe for large δDM.
+PICO-60 is relevant for intermediate values of δDM, and is
+in fact the most sensitive current probe of some well motivated dark matter scenarios, as
+suggested by the gray lines in the Figure, which correspond to the expected cross-section for
+different scenarios of electroweakly interacting fermionic dark matter. The complementarity
+of experiments in probing these scenarios is investigated in Figure 3. The dotted lines show
+the upper limit on the mass splitting as a function of the dark matter mass assuming the
+Standard Halo Model. Under this common assumption, LUX-ZEPLIN is the most constraining
+experiment over the whole parameter space considered. However, when including the non-
+galactic components, different experiments contribute to set the upper limit, as reflected by the
+breaks in the solid lines in the Figure: LUX-ZEPLIN remains as the most sensitive experiment
+for small dark matter masses, while PICO-60 is the best experiment for larger masses. Further,
+the dark matter mass at which PICO-60 becomes the leading experiment becomes larger and
+larger as the dark matter hypercharge increases. As seen in the Figure, for this class of scenarios
+the non-galactic components in the dark matter flux enhance the sensitivity of experiments to
+the mass splitting by a factor ∼ 2 for mDM = 100 GeV - 1 TeV.
+It is noteworthy the pivotal role of the radiopurity measurements on a CaWO4 target to
+probe large mass splittings in inelastic dark matter scenarios. This can be understood from the
+expression for the minimum DM velocity required to induced a recoil with energy ER, Eq. (8).
+Let us consider a velocity distribution where the maximum speed is v∗. Then, for an experiment
+capable of detecting a recoil of a nucleus Ai with energy ER, the maximum mass splitting that
+can be probed is:
+δDM ≤
+�
+2ERmAiv∗ − ERmAi
+µAi
+≤ 1
+2µAiv2
+∗ ,
+(16)
+where the absolute maximum is reached when ER = µ2
+Aiv2
+∗/(2mAi). This is shown in Figure 4,
+for v∗ = 764 km/s, v∗ = 820 km/s, v∗ = 1220 km/s (solid lines), corresponding respectively to
+the maximal velocity at the Earth of dark matter particles bound to the Milky Way (described
+by the Standard Halo Model), from the Local Group envelope and from the Virgo Supercluster.
+The plot also shows the range of recoil energies that can be detected by the CRESST-II ex-
+periment and by the radiopurity measurements in CaWO4 crystals. As seen in the plot, while
+8
+
+102
+103
+104
+mDM [GeV]
+100
+200
+300
+400
+500
+δDM [keV]
+Upper limits at 90% CL from LZ+PICO60+CaWO4, Dirac dark matter
+Y = 1/2, SHM
+Y = 1/2, SHM + Non-galactic
+Y = 1, SHM
+Y = 1, SHM + Non-galactic
+Y = 3/2, SHM
+Y = 3/2, SHM + Non-galactic
+Figure 3: Upper limits on the mass splitting for electroweakly charged (pseudo-)dirac dark
+matter as a function of the dark matter mass, for different choices of the hypercharge, and
+including in the flux only the Standard Halo Model component (dotted lines) or also the non-
+galactic diffuse components (solid lines).
+CRESST-II can only probe up to δDM ∼ 700 keV, the radiopurity measurements allow to probe
+up to δDM ∼ 1200 keV, when including the flux component from the dark matter bound to the
+Virgo Supercluster (however with a lower sensitivity due to the smaller exposure). From this
+plot it follows that the CRESST experiment would have an enhanced sensitivity to inelastic
+dark matter scenarios if the window of recoil energies used in the analysis were extended to
+larger values. Let us note that for low dark matter masses, extending the search window to
+higher recoil energies would not help in probing larger values of the mass splitting. This is
+illustrated in the Figure for mDM = 100 GeV, from where it is apparent that to increase the
+reach in mass splittings it is necessary to extend the search to lower recoil energies.
+Finally, we show in Figure 5 the isocontours with the 90% C.L. upper limits on the cross-
+section for different dark matter masses and mass splittings, from LUX-ZEPLIN (top panels),
+PICO60 (middle panels) and from radiopurity measurements on a CaWO4 target (bottom
+panels), considering that all dark matter in the Solar System is bound to the Milky Way, as
+commonly assumed (left panels), and including the non-galactic components (right panels).
+The enhancement in sensitivity is clear from the plots.
+4
+Impact on electron recoils
+The differential ionization rate induced by dark matter-electron inelastic scattering in liquid
+xenon, with mass splitting between the two dark matter states given by δDM, reads:
+dRion
+dlnEer
+= NT
+�
+n,l
+�
+v≥vnl
+min(Eer)
+d3vF(⃗v + ⃗v⊙) dσnl
+ion
+dlnEer
+(v, Eer) ,
+(17)
+where NT is the number of target nuclei and
+vnl
+min(Eer) =
+�
+2
+mDM
+(Eer + |Enl| + δDM)
+(18)
+9
+
+100
+101
+102
+103
+104
+ER [keV]
+200
+400
+600
+800
+1000
+1200
+1400
+δDM [keV]
+mDM = 100 GeV
+CaWO4
+CRESST-II
+SHM
+SHM+LG
+SHM+LG+VS
+100
+101
+102
+103
+104
+ER [keV]
+200
+400
+600
+800
+1000
+1200
+1400
+δDM [keV]
+mDM = 1 TeV
+CaWO4
+CRESST-II
+SHM
+SHM+LG
+SHM+LG+VS
+Figure 4: Values of the mass splitting δDM that can produce a recoil energy in a 184W target
+for mDM = 100 GeV (left plot) and mDM = 1 TeV (right plot) when the maximal velocity of
+the dark matter particles at Earth is v∗ = 764 km/s (dotted lines), v∗ = 820 km/s (dashed
+lines) and v∗ = 1220 km/s (solid lines), corresponding respectively to dark matter bound to
+the Milky Way (described by the Standard Halo Model), bound to the Local Group and bound
+to the Virgo Supercluster.
+For comparison, we also show the range of recoil energies that
+can be detected by the CRESST-II experiment (red band) and by the CaWO4 radiopurity
+measurement (yellow band).
+is the minimum dark matter velocity necessary to ionize a bound electron in the (n, l) shell of
+a xenon atom (with energy Enl), giving a free electron with energy Eer. Further, dσnl
+ion/dlnEer
+is the differential ionization cross section, given by:
+dσnl
+ion
+dlnEer
+(v, Eer) =
+¯σDM−e
+8µ2
+DM,ev2
+� qnl
+max
+qnl
+min
+dqq
+��f nl
+ion(k′, q)
+��2 |FDM(q)|2 .
+(19)
+Here, µDM,e is the reduced mass of the dark matter-electron system, ¯σDM−e is the dark matter-
+free electron scattering cross section at fixed momentum transfer q = αme,
+��f nl
+ion(k′, q)
+��2 is the
+ionization form factor of an electron in the (n, l) shell with final momentum k′ = √2meEer
+and momentum transfer q, and FDM(q) is a form factor that encodes the q-dependence of the
+squared matrix element for dark matter-electron scattering and depends on the mediator under
+consideration. The maximum and minimum values of the momentum transfer needed to ionize
+a bound electron in the (n, l) shell recoil with energy Eer from the interaction of a dark matter
+particle with speed v are:
+qnl
+max
+min(Eer) = mDMv
+�
+�1 ±
+�
+1 −
+�vnl
+min(Eer)
+v
+�2�
+� ,
+(20)
+with vnl
+min(Eer) defined in Eq. (18). Finally, the total number of expected ionization events reads
+N = Rion · E, with Rion the total ionization rate, calculated from integrating Eq.(17) over the
+experimentally measured recoil energies, and E the exposure (i.e. mass multiplied by live-time)
+of the experiment.
+10
+
+102
+103
+104
+mDM [GeV]
+100
+200
+300
+400
+500
+600
+δDM [keV]
+Upper limits at 90% C.L from LUX-ZEPLIN, SHM, Isoscalar
+10−47
+10−45
+10−43
+10−41
+10−39
+10−37
+σDM−p
+102
+103
+104
+mDM [GeV]
+100
+200
+300
+400
+500
+600
+δDM [keV]
+Upper limits at 90% C.L from LUX-ZEPLIN, Non-galactic, Isoscalar
+10−47
+10−45
+10−43
+10−41
+10−39
+10−37
+σDM−p
+102
+103
+104
+mDM [GeV]
+100
+200
+300
+400
+500
+600
+δDM [keV]
+Upper limits at 90% C.L from PICO60, SHM, Isoscalar
+10−45
+10−43
+10−41
+10���39
+10−37
+10−35
+σDM−p
+102
+103
+104
+mDM [GeV]
+100
+200
+300
+400
+500
+600
+δDM [keV]
+Upper limits at 90% C.L from PICO60, Non-galactic, Isoscalar
+10−45
+10−43
+10−41
+10−39
+10−37
+10−35
+σDM−p
+102
+103
+104
+mDM [GeV]
+200
+400
+600
+800
+1000
+1200
+1400
+δDM [keV]
+Upper limits at 90% C.L from CaWO4, SHM, Isoscalar
+10−41
+10−39
+10−37
+10−35
+10−33
+10−31
+10−29
+10−27
+σDM−p
+102
+103
+104
+mDM [GeV]
+200
+400
+600
+800
+1000
+1200
+1400
+δDM [keV]
+Upper limits at 90% C.L from CaWO4, Non-galactic, Isoscalar
+10−41
+10−39
+10−37
+10−35
+10−33
+10−31
+10−29
+10−27
+σDM−p
+Figure 5: Isocontours of the 90% C.L. upper limits on the spin-independent dark matter-proton
+inelastic cross-section for the isoscalar scenario (f p = f n) in the parameter space spanned by
+the dark matter mass and mass splitting, from LUX-ZEPLIN (top panels), PICO60 (middle
+panels) and radiopurity measurements in a CaWO4 target (lower panels), assuming that all
+dark matter in the Solar System is bound to the Milky Way (left panels) or including the
+non-galactic diffuse component (right panels).
+11
+
+In semiconductor detectors, the electron excitation rate induced by dark matter-electron
+inelastic scatterings, with a mass splitting δDM, reads [43, 44]
+R = 1
+ρT
+¯σDM−e
+µ2
+DM,e
+π
+α
+�
+d3vF(⃗v + ⃗v⊙)
+v
+�
+d3q
+(2π)3q2 |FDM(q)|2
+� dω
+2π
+1
+1 − e−βω Im
+�
+−1
+ϵ(ω, ⃗q)
+�
+δ
+�
+ω + δDM +
+q2
+2mχ
+− ⃗q · ⃗v
+�
+,
+(21)
+where w is the energy deposited in the material, ⃗q is the momentum transfer of the process,
+and ρT is the target density. The rate involves an integration of the Electronic Loss Function
+(ELF) of the target material, which we calculate with DarkELF [44]. For the dielectric function
+ϵ(ω, q), we use the Lindhard method, which treats the target as a non-interacting Fermi liquid.
+Finally, the total number of events reads N = R · E, with E the exposure (i.e. mass multiplied
+by live-time) of the experiment.
+The non-observation of a significant excess of electron recoils in a given experiment allows
+to set upper limits on the dark matter-electron scattering cross section, for a given dark matter
+mass and a given mass splitting between the dark matter particle and the heavier neutral state.
+We show in Figure 6, upper limits on the inelastic dark matter-electron cross section versus mass
+splitting for a fixed dark matter mass of mDM = 1 GeV from XENON1T [45](blue lines), and
+from the semiconductor experiment SENSEI [46](purple lines), both when considering the SHM
+flux only (solid lines), and when including the non-galactic components to the dark matter flux
+(dotted lines). In the upper plots, we take the form factor FDM = α2m2
+e/q2, corresponding to
+an ultralight or massless mediator. In the middle plots, we take the form factor FDM = αme/q,
+corresponding to an electric dipole interaction, and in the lower plots we take the form factor
+FDM = 1, corresponding to a heavy mediator [47, 48].
+As can be seen in the Figure, the non-galactic components enhance the sensitivity to the
+mass splitting of both XENON1T and SENSEI by a factor of ∼ 2, compared to the sensitivity
+estimated from considering just the galactic component. This conclusion holds independently
+of the choice of the dark matter form factor. Further, the reach in cross-section is enhanced due
+to the non-galactic components, especially at low mass splittings, being the effect stronger for
+XENON1T than for SENSEI. For comparison, we also show as a grey band the cross section
+for which the observed dark matter abundance is reproduced via freeze-in in the case of an
+ultralight mediator [49], or via freeze-out in the case of a heavy mediator [50]. Clearly, the
+non-galactic dark matter components allow to probe larger values of the mass splitting.
+5
+Conclusions
+We have investigated the impact of a non-galactic diffuse dark matter component inside the
+Solar System for the detection of the inelastic scattering of a dark matter particle in direct
+search experiments. Concretely, we have considered the contribution to the dark matter flux
+from dark matter particles in the envelope of the Local Group and from the Virgo Supercluster.
+Their speeds in the galactic frame are ∼ 600 km/s and ∼ 1000 km/s, respectively, which are
+larger than the maximal speed of dark matter particles bound to the Milky Way, ∼ 540 km/s.
+As a result, the region of parameter space that can be probed with current experiments is larger
+than reported in previous works, that implicitly assumed that the Milky Way is an isolated
+galaxy in the Universe.
+12
+
+100
+101
+102
+δDM [eV]
+10−47
+10−44
+10−41
+10−38
+10−35
+10−32
+10−29
+10−26
+¯σe[cm2]
+FDM = α2m2
+e/q2
+mDM = 1 GeV
+Freeze-in
+Ultralight mediator
+SENSEI (SHM)
+SENSEI (SHM+Non-galactic)
+XENON1T (SHM)
+XENON1T (SHM+Non-galactic)
+100
+101
+102
+δDM [eV]
+10−47
+10−44
+10−41
+10−38
+10−35
+10−32
+10−29
+10−26
+¯σe[cm2]
+FDM = αme/q
+mDM = 1 GeV
+Dipole interaction
+SENSEI (SHM)
+SENSEI (SHM+Non-galactic)
+XENON1T (SHM)
+XENON1T (SHM+Non-galactic)
+100
+101
+102
+δDM [eV]
+10−47
+10−44
+10−41
+10−38
+10−35
+10−32
+10−29
+10−26
+¯σe[cm2]
+FDM = 1
+mDM = 1 GeV
+Freeze-out (Pseudo-Dirac fermion)
+Massive mediator
+SENSEI (SHM)
+SENSEI (SHM+Non-galactic)
+XENON1T (SHM)
+XENON1T (SHM+Non-galactic)
+Figure 6: 90% C.L upper limits on the spin-independent dark matter-electron inelastic cross
+section for a dark matter mass of 1 GeV, as a function of the mass splitting, from XENON1T
+(blue) and SENSEI (purple), when the dark matter-electron interaction is mediated by an
+ultralight dark photon (upper left plot), by a dipole operator (upper right plot), or by a heavy
+mediator (lower plot).
+For nuclear recoils, the non-galactic component expands the reach in mass splitting at
+the LUX-ZEPLIN, PICO60, and CRESST-II experiments by a factor ∼ 2 in the mass range
+mDM = 10 GeV- 10 TeV, and enhances significantly the reach in cross-section, especially close
+to the kinematic threshold for the galactic dark matter. For instance, for mDM = 1 TeV and
+δDM = 250 keV, the sensitivity to the cross-section improves by about three orders of magnitude.
+We have also stressed the relevance of experiments capable of detecting high recoil energies
+for probing the parameter space of inelastic dark matter scenarios. We have illustrated this
+capability with the radiopurity measurements in CaWO4 crystals performed by the CRESST
+collaboration, and which allows to probe up to δDM ∼ 1.2 MeV (1.4 MeV) for mDM = 1 TeV
+(10 TeV). For electron recoils, the conclusions are analogous, allowing to increase reach in mass
+splitting of the XENON1T and SENSEI experiments also by a factor ∼ 2 for dark matter
+13
+
+10−2
+10−1
+100
+101
+mDM [GeV]
+5
+10
+15
+20
+25
+30
+δDM [eV]
+Upper limits at 90% C.L from SENSEI, SHM, Massive mediator
+10−37
+10−35
+10−33
+10−31
+10−29
+10−27
+10−25
+¯σDM−e
+10−2
+10−1
+100
+101
+mDM [GeV]
+5
+10
+15
+20
+25
+30
+δDM [eV]
+Upper limits at 90% C.L from SENSEI, Non-galactic, Massive mediator
+10−37
+10−35
+10−33
+10−31
+10−29
+10−27
+10−25
+¯σDM−e
+10−2
+10−1
+100
+101
+mDM [GeV]
+100
+200
+300
+400
+500
+600
+δDM [eV]
+Upper limits at 90% C.L from XENON1T, SHM, Massive mediator
+10−41
+10−40
+10−39
+10−38
+10−37
+10−36
+10−35
+10−34
+¯σDM−e
+10−2
+10−1
+100
+101
+mDM [GeV]
+100
+200
+300
+400
+500
+600
+δDM [eV]
+Upper limits at 90% C.L from XENON1T, Non-galactic, Massive mediator
+10−41
+10−40
+10−39
+10−38
+10−37
+10−36
+10−35
+10−34
+¯σDM−e
+Figure 7: Isocontours of the 90% C.L. upper limits on the dark matter-electron inelastic scat-
+tering cross-section for the heavy mediator scenario (FDM = 1) in the parameter space spanned
+by the dark matter mass and mass splitting, from SENSEI (top panels), and XENON1T (lower
+panels), assuming that all dark matter in the Solar System is bound to the Milky Way (left
+panels) or including the non-galactic component diffuse (right panels).
+masses in the range mDM = 0.01 GeV-10 GeV,
+Acknowledgments
+The work of GH and AI was supported by the Collaborative Research Center SFB1258 and by
+the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
+Excellence Strategy - EXC-2094 - 390783311. The work of SS is supported by Grant-in-Aid
+for Scientific Research from the Ministry of Education, Culture, Sports, Science, and Technol-
+ogy (MEXT), Japan, 18K13535, 20H01895, 20H05860 and 21H00067, and by World Premier
+International Research Center Initiative (WPI), MEXT, Japan.
+14
+
+A
+Derivation of upper limits from direct detection exper-
+iments
+To derive upper limits on the inelastic dark matter-nucleon scattering cross section, as a function
+of the dark matter mass and/or the dark matter mass splitting, we follow a poissonian-likelihood
+approach, and we calculate the rates for the different experiments/detectors independently. For
+the LUX-ZEPLIN experiment, we use the data from [10], with an exposure of 0.904 tonne×year,
+a region of interest extending from 2 keV to 70 keV, and the efficiency function reported by
+the collaboration. Given the agreement of the number of signal events with the background
+prediction reported by the collaboration, we take a 90% C.L. upper limit on the number of
+signal events of 2.71. For the PICO-60 experiment, we use the results from [38], corresponding
+to an exposure of 9.356 kg×year, a region of interest extending from 13.5 keV to 100 keV, and
+the efficiency function reported by the collaboration. Since PICO-60 observed no signal events,
+we take a 90% C.L. upper limit on the number of signal events of 2.71. For CRESST-II, we use
+the published data [39], corresponding to an exposure of 52 kg×days. We do not consider as
+signal events those belonging to the acceptance region of the experiment at low recoil energies,
+but instead, we consider the recoil energy region extending from 30 keV to 120 keV, which gives
+an upper limit of 4 signal events. Finally, for the CaWO4 radiopurity measurement from [40],
+we take an exposure of 90.10 kg×days, with a recoil energy region extending from 300 keV to
+2000 keV, and a number of 3 signal events.
+For the inelastic dark matter-electron scattering cross-section, we derive upper limits at 90%
+C.L at fixed momentum transfer q = αme using data from XENON1T [45] and SENSEI [46].
+We consider the observed event rate XENON1T between 150-3000 photoelectrons (PE), which
+corresponds to the range 0.18 keVee to 3.5 keVee (kiloelectronvolt electron equivalent). We take
+the efficiency function from [45], an exposure of 22 ± 3 tonne-days and an upper limit on the
+number of events of 39.2. For SENSEI, we sum-up the observed events in the energy bins
+ranging from 4.91 eV to 16.31 eV, resulting in an upper limit of 4.957 events per gram day of
+exposure. Further, we use the efficiency reported by the collaboration in every energy bin [46].
+References
+[1]
+Gerard Jungman, Marc Kamionkowski, and Kim Griest. “Supersymmetric dark matter”.
+In: Phys. Rept. 267 (1996), pp. 195–373. doi: 10.1016/0370-1573(95)00058-5. arXiv:
+hep-ph/9506380.
+[2]
+Gianfranco Bertone, Dan Hooper, and Joseph Silk. “Particle dark matter: Evidence, can-
+didates and constraints”. In: Phys. Rept. 405 (2005), pp. 279–390. doi: 10.1016/j.
+physrep.2004.08.031. arXiv: hep-ph/0404175.
+[3]
+Lars Bergström. “Nonbaryonic dark matter: Observational evidence and detection meth-
+ods”. In: Rept. Prog. Phys. 63 (2000), p. 793. doi: 10.1088/0034- 4885/63/5/2r3.
+arXiv: hep-ph/0002126.
+[4]
+Jonathan L. Feng. “Dark Matter Candidates from Particle Physics and Methods of Detec-
+tion”. In: Ann. Rev. Astron. Astrophys. 48 (2010), pp. 495–545. doi: 10.1146/annurev-
+astro-082708-101659. arXiv: 1003.0904 [astro-ph.CO].
+15
+
+[5]
+Mark W. Goodman and Edward Witten. “Detectability of Certain Dark Matter Can-
+didates”. In: Phys. Rev. D 31 (1985). Ed. by M. A. Srednicki, p. 3059. doi: 10.1103/
+PhysRevD.31.3059.
+[6]
+R. Bernabei et al. “Investigating electron interacting dark matter”. In: Phys. Rev. D 77
+(2008), p. 023506. doi: 10.1103/PhysRevD.77.023506. arXiv: 0712.0562 [astro-ph].
+[7]
+David G. Cerdeno and Anne M. Green. “Direct detection of WIMPs”. In: (Feb. 2010),
+pp. 347–369. doi: 10.1017/CBO9780511770739.018. arXiv: 1002.1912 [astro-ph.CO].
+[8]
+Teresa Marrodán Undagoitia and Ludwig Rauch. “Dark matter direct-detection experi-
+ments”. In: J. Phys. G 43.1 (2016), p. 013001. doi: 10.1088/0954-3899/43/1/013001.
+arXiv: 1509.08767 [physics.ins-det].
+[9]
+J. D. Lewin and P. F. Smith. “Review of mathematics, numerical factors, and corrections
+for dark matter experiments based on elastic nuclear recoil”. In: Astropart. Phys. 6 (1996),
+pp. 87–112. doi: 10.1016/S0927-6505(96)00047-3.
+[10]
+J. Aalbers et al. “First Dark Matter Search Results from the LUX-ZEPLIN (LZ) Exper-
+iment”. In: (July 2022). arXiv: 2207.03764 [hep-ex].
+[11]
+Jose F. Nieves. “Electromagnetic Properties of Majorana Neutrinos”. In: Phys. Rev. D 26
+(1982), p. 3152. doi: 10.1103/PhysRevD.26.3152.
+[12]
+Natsumi Nagata and Satoshi Shirai. “Higgsino dark matter in high-scale supersymmetry”.
+In: Journal of High Energy Physics 2015.1 (Jan. 2015). doi: 10.1007/jhep01(2015)029.
+url: https://doi.org/10.1007%2Fjhep01%282015%29029.
+[13]
+Lawrence J. Hall, Takeo Moroi, and Hitoshi Murayama. “Sneutrino cold dark matter
+with lepton-number violation”. In: Physics Letters B 424.3-4 (Apr. 1998), pp. 305–312.
+doi: 10.1016/s0370-2693(98)00196-8. url: https://doi.org/10.1016%2Fs0370-
+2693%2898%2900196-8.
+[14]
+Nicole F. Bell, Giorgio Busoni, and Sandra Robles. “Heating up neutron stars with in-
+elastic dark matter”. In: Journal of Cosmology and Astroparticle Physics 2018.09 (Sept.
+2018), pp. 018–018. doi: 10.1088/1475-7516/2018/09/018. url: https://doi.org/
+10.1088%2F1475-7516%2F2018%2F09%2F018.
+[15]
+Anirban Biswas et al. “Improved White Dwarves Constraints on Inelastic Dark Matter
+and Left-Right Symmetric Models”. In: (June 2022). arXiv: 2206.06667 [hep-ph].
+[16]
+Daniele S.M. Alves et al. “Composite inelastic dark matter”. In: Physics Letters B 692.5
+(Sept. 2010), pp. 323–326. doi: 10.1016/j.physletb.2010.08.006. url: https:
+//doi.org/10.1016%2Fj.physletb.2010.08.006.
+[17]
+Thomas Schwetz and Jure Zupan. “Dark matter attempts for CoGeNT and DAMA”. In:
+Journal of Cosmology and Astroparticle Physics 2011.08 (Aug. 2011), pp. 008–008. doi:
+10.1088/1475- 7516/2011/08/008. url: https://doi.org/10.1088%2F1475-
+7516%2F2011%2F08%2F008.
+[18]
+Nima Arkani-Hamed et al. “A theory of dark matter”. In: Physical Review D 79.1 (Jan.
+2009). doi: 10 . 1103 / physrevd . 79 . 015014. url: https : / / doi . org / 10 . 1103 %
+2Fphysrevd.79.015014.
+16
+
+[19]
+Spencer Chang, Neal Weiner, and Itay Yavin. “Magnetic inelastic dark matter”. In:
+Physical Review D 82.12 (Dec. 2010). doi: 10.1103/physrevd.82.125011. url: https:
+//doi.org/10.1103%2Fphysrevd.82.125011.
+[20]
+G. Barello, Spencer Chang, and Christopher A. Newby. “A model independent approach
+to inelastic dark matter scattering”. In: Physical Review D 90.9 (Nov. 2014). doi: 10.
+1103/physrevd.90.094027. url: https://doi.org/10.1103%2Fphysrevd.90.094027.
+[21]
+Natsumi Nagata and Satoshi Shirai. “Electroweakly interacting Dirac dark matter”. In:
+Physical Review D 91.5 (Mar. 2015). doi: 10.1103/physrevd.91.055035. url: https:
+//doi.org/10.1103%2Fphysrevd.91.055035.
+[22]
+Timon Emken et al. “Electron recoils from terrestrial upscattering of inelastic dark mat-
+ter”. In: Physical Review D 105.5 (Mar. 2022). doi: 10.1103/physrevd.105.055023.
+url: https://doi.org/10.1103%2Fphysrevd.105.055023.
+[23]
+Martin C. Smith et al. “The RAVE Survey: Constraining the Local Galactic Escape
+Speed”. In: Mon. Not. Roy. Astron. Soc. 379 (2007), pp. 755–772. doi: 10.1111/j.1365-
+2966.2007.11964.x. arXiv: astro-ph/0611671.
+[24]
+Til Piffl et al. “The RAVE survey: the Galactic escape speed and the mass of the Milky
+Way”. In: Astron. Astrophys. 562 (2014), A91. doi: 10.1051/0004-6361/201322531.
+arXiv: 1309.4293 [astro-ph.GA].
+[25]
+F. D. Kahn and L. Woltjer. “Intergalactic Matter and the Galaxy.” In: Astrophys. J. 130
+(1959), p. 705. doi: 10.1086/146762.
+[26]
+James Binney and Scott Tremaine. Galactic Dynamics: Second Edition. 2008.
+[27]
+T. J. Cox and Abraham Loeb. “The Collision Between The Milky Way And Andromeda”.
+In: Mon. Not. Roy. Astron. Soc. 386 (2008), p. 461. doi: 10.1111/j.1365-2966.2008.
+13048.x. arXiv: 0705.1170 [astro-ph].
+[28]
+Dmitry Makarov and Igor Karachentsev. “Galaxy groups and clouds in the Local (z ∼
+0.01) universe”. In: Mon. Not. Roy. Astron. Soc. 412 (2011), p. 2498. doi: 10.1111/j.
+1365-2966.2010.18071.x. arXiv: 1011.6277 [astro-ph.CO].
+[29]
+J. I. Read. “The Local Dark Matter Density”. In: J. Phys. G 41 (2014), p. 063101. doi:
+10.1088/0954-3899/41/6/063101. arXiv: 1404.1938 [astro-ph.GA].
+[30]
+Anne M. Green. “Astrophysical uncertainties on direct detection experiments”. In: Mod. Phys. Lett. A
+27 (2012), p. 1230004. doi: 10.1142/S0217732312300042. arXiv: 1112.0524 [astro-ph.CO].
+[31]
+F. J. Kerr and Donald Lynden-Bell. “Review of galactic constants”. In: Mon. Not. Roy. Astron. Soc.
+221 (1986), p. 1023.
+[32]
+I. D. Karachentsev. “Missing dark matter in the local universe”. In: Astrophysical Bulletin
+67.2 (Apr. 2012), pp. 123–134. issn: 1990-3421. doi: 10.1134/s1990341312020010. url:
+http://dx.doi.org/10.1134/S1990341312020010.
+[33]
+A.N. Baushev. “Extragalactic dark matter and direct detection experiments”. In: Astrophys. J.
+771 (2013), p. 117. doi: 10.1088/0004-637X/771/2/117. arXiv: 1208.0392 [astro-ph.CO].
+[34]
+I. D. Karachentsev et al. “Local galaxy flows within 5 mpc”. In: Astron. Astrophys. 398
+(2003), pp. 479–492. doi: 10.1051/0004-6361:20021566. arXiv: astro-ph/0211011.
+17
+
+[35]
+Christopher McCabe. “The Earth’s velocity for direct detection experiments”. In: JCAP 02
+(2014), p. 027. doi: 10.1088/1475-7516/2014/02/027. arXiv: 1312.1355 [astro-ph.CO].
+[36]
+G. Belanger et al. “Dark matter direct detection rate in a generic model with micrOMEGAs
+2.2”. In: Comput. Phys. Commun. 180 (2009), pp. 747–767. doi: 10.1016/j.cpc.2008.
+11.019. arXiv: 0803.2360 [hep-ph].
+[37]
+Marco Cirelli, Nicolao Fornengo, and Alessandro Strumia. “Minimal dark matter”. In:
+Nucl. Phys. B 753 (2006), pp. 178–194. doi: 10.1016/j.nuclphysb.2006.07.012.
+arXiv: hep-ph/0512090.
+[38]
+C. Amole et al. “Dark matter search results from the PICO-60 CF3I bubble chamber”.
+In: Phys. Rev. D 93.5 (2016), p. 052014. doi: 10.1103/PhysRevD.93.052014. arXiv:
+1510.07754 [hep-ex]. url: https://doi.org/10.1103%2Fphysrevd.93.052014.
+[39]
+G. Angloher et al. “Results on light dark matter particles with a low-threshold CRESST-
+II detector”. In: The European Physical Journal C 76.1 (Jan. 2016). doi: 10.1140/epjc/
+s10052-016-3877-3. url: https://doi.org/10.1140%2Fepjc%2Fs10052-016-3877-3.
+[40]
+A Münster et al. “Radiopurity of CaWO4 crystals for direct dark matter search with
+CRESST and EURECA”. In: 2014.05 (May 2014), pp. 018–018. doi: 10.1088/1475-
+7516/2014/05/018. url: https://doi.org/10.1088%2F1475-7516%2F2014%2F05%
+2F018.
+[41]
+Ningqiang Song, Serge Nagorny, and Aaron C. Vincent. “Pushing the frontier of WIMPy
+inelastic dark matter: Journey to the end of the periodic table”. In: Phys. Rev. D 104.10
+(2021), p. 103032. doi: 10.1103/PhysRevD.104.103032. arXiv: 2104.09517 [hep-ph].
+[42]
+Anja Brenner et al. “Complementarity of experiments in probing the non-relativistic
+effective theory of dark matter-nucleon interactions”. In: JCAP 06.06 (2022), p. 026. doi:
+10.1088/1475-7516/2022/06/026. arXiv: 2203.04210 [hep-ph].
+[43]
+Simon Knapen, Tongyan Lin, and Kathryn M. Zurek. “Light dark matter: Models and
+constraints”. In: Physical Review D 96.11 (Dec. 2017). doi: 10.1103/physrevd.96.
+115021. url: https://doi.org/10.1103%2Fphysrevd.96.115021.
+[44]
+Simon Knapen, Jonathan Kozaczuk, and Tongyan Lin. “python package for dark matter
+scattering in dielectric targets”. In: Physical Review D 105.1 (Jan. 2022). doi: 10.1103/
+physrevd.105.015014. url: https://doi.org/10.1103%2Fphysrevd.105.015014.
+[45]
+E. Aprile et al. “Dark Matter Search Results from a One Ton-Year Exposure of XENON1T”.
+In: Physical Review Letters 121.11 (Sept. 2018). doi: 10 . 1103 / physrevlett . 121 .
+111302. url: https://doi.org/10.1103%2Fphysrevlett.121.111302.
+[46]
+Liron Barak et al. “SENSEI: Direct-Detection Results on sub-GeV Dark Matter from
+a New Skipper CCD”. In: Physical Review Letters 125.17 (Oct. 2020). doi: 10.1103/
+physrevlett.125.171802. url: https://doi.org/10.1103%2Fphysrevlett.125.
+171802.
+[47]
+Rouven Essig et al. “Snowmass2021 Cosmic Frontier: The landscape of low-threshold
+dark matter direct detection in the next decade”. In: 2022 Snowmass Summer Study.
+Mar. 2022. arXiv: 2203.08297 [hep-ph].
+[48]
+Riccardo Catena et al. “Atomic responses to general dark matter-electron interactions”. In:
+Physical Review Research 2.3 (Aug. 2020). doi: 10.1103/physrevresearch.2.033195.
+url: https://doi.org/10.1103%2Fphysrevresearch.2.033195.
+18
+
+[49]
+Rouven Essig, Tomer Volansky, and Tien-Tien Yu. “New Constraints and Prospects for
+sub-GeV Dark Matter Scattering off Electrons in Xenon”. In: Phys. Rev. D 96.4 (2017),
+p. 043017. doi: 10.1103/PhysRevD.96.043017. arXiv: 1703.00910 [hep-ph].
+[50]
+Mariana Carrillo González and Natalia Toro. “Cosmology and signals of light pseudo-
+Dirac dark matter”. In: JHEP 04 (2022), p. 060. doi: 10.1007/JHEP04(2022)060. arXiv:
+2108.13422 [hep-ph].
+19
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+2CIM: Area-Efficient 2-Cycle Integer Multipliers
+Ahmad Houraniah
+Department of Computer Science
+¨Ozye˘gin University
+Istanbul, Turkey
+[email protected]
+H. Fatih Ugurdag
+Department of Electrical
+and Electronics Engineering
+¨Ozye˘gin University
+Istanbul, Turkey
+[email protected]
+Cengiz Emre Dedeagac
+Department of Computer Science
+¨Ozye˘gin University
+Istanbul, Turkey
+[email protected]
+Abstract—Fast multipliers with large bit widths can occupy
+significant silicon area, which, in turn, can be minimized by
+employing multi-cycle multipliers. This paper introduces archi-
+tectures and parameterized Verilog circuit generators for 2-
+cycle integer multipliers. When implementing an algorithm in
+hardware, it is common that less than 1 multiplication needs
+to be performed per clock cycle. It is also possible that the
+multiplications per cycle is a fractional number, e.g., 3.5. In such
+case, we can surely use 4 multipliers, each with a throughput of 1
+result per cycle. However, we can instead use 3 such multipliers
+plus a multiplier with a throughput of 1/2. Resource sharing
+allows a multiplier with a lower throughput to be smaller, hence
+area savings. These multipliers offer customization in regards
+to the latency and clock frequency. All proposed designs were
+automatically synthesized and tested for various bit widths. Two
+main architectures are presented in this work, and each has
+several variants. Our 2-cycle multipliers offer up to 21%, 42%,
+32%, 41%, and 48% of area savings for bit widths of 8, 16, 32,
+64, and 128, with respect to synthesizing the “*” operator with
+throughput of 1. Furthermore, some of the proposed designs also
+offer power savings under certain conditions.
+Index Terms—computer arithmetic, multi-cycle multiplier,
+resource-sharing, pipelining
+I. INTRODUCTION
+An integer multiplier is an essential building block for
+various ASICs and CPUs. Integer multipliers can get quite
+expensive in terms of area as bit width increases. Large
+integers are used in a wide range of applications that require a
+high degree of precision. Floating-point operations are unsuit-
+able for some applications because of so-called “catastrophic
+cancellations” [1]. The addition and subtraction of floating-
+point numbers that greatly vary in magnitude can produce
+rounding errors, resulting in significant data loss. Floating-
+point operations are significantly more expensive than fixed-
+point operations regarding area complexity and latency. For
+these reasons, fixed point representation is preferred for many
+applications. The CUDA [2] programming model and software
+environment, which NVIDIA develops, recently introduced a
+128-bit integer data type, which is intended for applications
+that require a higher degree of precision within a predeter-
+mined range. This signifies the importance of large fixed-
+point integers. Multiplying such large integers can require
+significant hardware resources. For this reason, decreasing the
+area complexity for integer multipliers can be very beneficial.
+There exists a large number of applications containing data
+flow paths that require less than one multiplication every
+two or more clock cycles. Such a case is found in RSIC-V
+softcore implementations, where area efficiency is vital, even
+if it comes at the expense of increasing the latency of ALU
+operations such as multiplication.
+Applications can contain numerous multiplications in their
+data flow paths. Due to the area complexity of these multipli-
+ers, the same multiplication circuits are used multiple times to
+minimize the area complexity of the system. These multiplica-
+tions typically need to be computed within a predefined period.
+Since a conventional multiplier can only be used once in every
+clock cycle, using a single multiplier can severely limit the
+system’s throughput. The throughput can be maintained by
+using several multiplication circuits operating in parallel. The
+number of multipliers necessary can be calculated by dividing
+the number of multiplications by the period they need to be
+computed within (in clock cycles). Using this formula, the
+number of multipliers required is not always an integer. The
+conventional approach is to round up this value, which causes
+one of these multipliers to be underutilized. A fully-pipelined
+multiplier that can accept new inputs in consecutive clock
+cycles would remain underutilized in such applications.
+In this work, various 2-cycle unsigned integer multiplier
+(2CIM) designs are proposed to decrease the area complexity
+for these underutilized multipliers. 2CIM architectures can
+be extended for signed integer multiplication as well. 2CIM
+designs offer partial multipliers that can offer area-efficient
+designs with a throughput of 1/2. The conventional approach
+of multiplying unsigned integers comprises two steps: partial
+product generation (PPG) and partial production summation.
+The PPG stage is typically done using AND gates, where one
+integer is shifted and multiplied with each bit of the other
+integer. The PPG stage produces multiple variables that must
+be summed to produce the multiplication result. This is done
+using a tree of full adder (FA) and half adder (HA) cells.
+Using a standard ripple carry adder structure to handle the
+partial product summation can result in a long critical path
+and an increased area complexity. The optimization of integer
+multiplications has been a thoroughly studied topic, and the
+conventional approach is to use a carry-save adder for the
+summation stage.
+Multiplication can be solved using a divide and conquer
+arXiv:2301.13332v1  [cs.AR]  30 Jan 2023
+
+strategy, where any multiplication can be divided into multiple
+smaller ones, this is often seen as an algorithmic problem, and
+a great deal of research was done to reduce the algorithmic
+complexity of multiplications. The same concepts used for
+reducing the algorithmic complexity can be applied to hard-
+ware for area reductions since multiplication can be spread
+across multiple clock cycles. The conventional schoolbook
+approach is to divide multiplication using the distributive
+property. One multiplication can be split into various smaller
+multiplications. Equation 1 shows how a single multiplication
+can be divided into two smaller multiplications using the
+schoolbook approach, where N is the bit width of the second
+multiplicand B.
+Y = A ∗ B = A ∗ {B1, B0} = A ∗ B0 + A ∗ B1 ∗ 2N/2 (1)
+Such an approach can be applied for area reductions, where a
+smaller multiplication circuit can be used repetitively to com-
+pute several smaller multiplications. Although multiplication
+can be implemented using a recursive approach, implementing
+several division levels requires more control logic. Since the
+smaller multiplier needed for a single division level is expected
+to be used twice for each multiplication, new inputs can only
+arrive once every two clock cycles (CCs), which will be called
+the initiation interval (II).
+The rest of the paper is organized as follows: Section
+II presents the previous work. Section III presents the ar-
+chitectures proposed in this work. Section IV describes the
+methodology used for the design generation, synthesis, and
+verification. Section V presents the implementation results of
+all the proposed architectures and explains the implications of
+these results. Section VI concludes the paper.
+II. PREVIOUS WORK
+A great degree of research has been done to improve the
+efficiency of integer multipliers. This is due to the large area
+complexities that can require. In [3], Wallace et al. proposed
+an efficient approach to dealing with integer multiplications
+by using carry-save trees for row and column compression.
+Their approach significantly decreased the critical path for the
+summation of more than two numbers. A carry-save adder
+structure uses FA and HA cells in a parallel fashion, reducing
+the number of rows to two before the ripple carry adder,
+thus significantly reducing the critical path. The architecture
+proposed by Wallace [3] heavily impacted the industry and
+continues to be applied in modern research. In [4], Dadda
+presented an improvement over the previously proposed carry
+save adder tree structure proposed by [3]. This reduced the
+number of FA and HA cells required for the column and row
+compression. Ugurdag et al. in [5] proposed a new and faster
+carry-save tree structure called row and column compression
+trees (RoCoCo). Their structure allowed for smaller and faster
+final adders for integer multiplication circuits. Using these
+structures, they presented faster multiplication circuits than
+much of the literature on FPGAs, outperforming Dadda mul-
+tipliers [4] and the built-in multiplication circuits by Xilinx.
+Considering how RoCoCo trees can offer an improvement
+over Wallace [3] and Dadda [4] trees, their architectures were
+utilized in this work.
+Multi-cycle (MC) multipliers have also been studied in the
+past, both for FPGA and ASIC applications. The focus of
+these studies was to decrease the area complexity for integer
+multiplication (similar to ours). However, recent work only
+target FPGA applications due to the limited resources avail-
+able. The authors in [6] extensively studied the topic of large
+integer multiplication, with a focus on FPGA implementation.
+They proposed several MC architectures involving Schoolbook
+multiplication, Comba multiplication, Karatsuba multiplica-
+tion, and Number Theoretic Transforms. These architectures
+are each suited for different applications, each having different
+characteristics. In their architectures, the latency depends on
+the bit widths of the multiplicands, allowing for a higher
+degree of resource sharing depending on the multiplication
+size, minimizing the number of required DSP slices. DSP
+slices are an FPGA-specific type of resource. Thus, more op-
+timized architectures can be presented for ASIC applications.
+In [7], a design implementing the Karatsuba algorithm was
+implemented. Due to the recursive nature of the Karatsuba
+algorithm, they proposed using a “Coprocessor,” which was
+responsible for the sub-functions of the Karatsuba algorithm,
+such as multiplication, addition, and shifting operations. The
+“Coprocessor” has a fixed size, and a Block RAM was used
+to store intermediate results. They were able to produce a
+circuit that could handle different bit widths using the same
+area resources. The only varying part is the latency required.
+This approach means the latency can increase rapidly as the
+multiplication size increases, requiring 120 clock cycles to im-
+plement a 128×128 multiplication. This approach also implies
+a long II since the “Coprocessor” is expected to be reused sev-
+eral times for a single computation. This architecture provides
+an area-efficient approach to computing large multiplications
+on FPGAs, yet, it is only suitable for very low-bandwidth
+applications. This architecture made further improvements to
+other FPGA-based MC multiplication circuits proposed in
+[8], [9], and [10]. Such architectures heavily rely on the
+usage of DSP slices and Block RAMs, which are expected to
+increase the speed of an FPGA application while decreasing
+the slice usage of the design. Since ASICs do not have these
+built-in hardware resources, more optimized solutions can be
+proposed. The authors in [11] proposed a design based on an
+MC Karatsuba multiplication. Their design achieves significant
+area reduction for FPGAs. Their design required 1/9th of the
+DSP resources for a conventional 2048×2048 multiplication.
+For a 2048×2048 multiplication, they achieve a latency of
+118 cycles and an II of 9 cycles. Although the design has
+a relatively low throughput of 1/9, they achieve significant
+area savings for large multiplications, which would otherwise
+be too costly in terms of area requirements. This design is
+only feasible for very large multiplications with a more limited
+number of applications.
+A few work have evaluated MC multiplications for ASIC,
+offering significant area reductions at the cost of speed/latency.
+Li et al. presented an area-efficient MC multiplier in [12].
+
+Their architecture heavily relies on resource sharing. Their
+designs require an II of N for N×N multiplication and have
+a latency of N+1. Such a high II and latency can severely
+limit the design’s applications and scalability. In [13], an
+iterative multiplication circuit for 64×64 was proposed, having
+an II of 4 CCs and a latency of 10 CCs. They used an
+internally generated clock using NOT gates. Therefore, the
+clock generation circuitry requires manual modification to
+work with different system clocks. Such an approach can limit
+the design’s capability of being pipelined. Another design was
+proposed in [14], implementing an MC multiplication circuit
+based on a modified-Booth encoding, similar to [13]. They use
+self-timed clocks that do not require manual modification to
+change the operating frequency, allowing them to work with
+any system clock. Their architecture offered an area reduction
+of 86.6% in comparison with an array implementation, coming
+at the cost of an 18.8% reduction in speed. Although this
+architecture presents significant area savings, a speed reduction
+can be a major limitation for high-speed applications. Further-
+more, using a self-timed clock can limit the design’s ability to
+be pipelined. This architecture is considered a combinational
+circuit; hence, the design can accept inputs in consecutive
+clock cycles. The system clock, however, would be limited
+by the maximum frequency of the design.
+III. PROPOSED ARCHITECTURES
+Integer multiplication can be represented by three stages,
+partial product generation (PPG), partial product reduction
+(PPR), and the final addition. Since multiplication can be
+described as the sum of several smaller multiplications as
+shown in equation 1. These smaller multiplications can be
+computed using the same hardware resource when applying
+resource-sharing. Thus, the circuit responsible for the smaller
+multiplications will be used multiple times for each multipli-
+cation, reducing the throughput to less than one. This approach
+reduces the area requirements and the critical path of the
+circuit. Both the critical path and the area requirements for the
+PPG stage for a conventional integer multiplier can be reduced
+by 50% when implementing a 2-cycle approach (implementing
+equation 1). For any multiplication greater than 4×4, more
+than two rows are generated after the PPG stage. This means
+that the PPR stage is expected to handle large reductions.
+Furthermore, all the intermediate values must be stored in
+registers until they reach the PPR stage. Such an approach
+can limit the area reduction that can be achieved. A more
+optimized approach is to separate the PPR stage into two parts.
+The first part would be connected to the outputs of the PPG
+stage. This allows for applying resource sharing on a large
+part of the PPR stage since this circuit is expected to be used
+twice for each multiplication. This combination of a PPG and a
+smaller PPR is called a partial product multiplier (PPM). PPMs
+have a shorter critical path than regular multipliers since they
+do not require the final addition stage. The PPM will be used
+twice for each multiplication, producing four results. These
+results are then reduced using the second part of the PPR,
+which will be called a compressor.
+A. Sub-module Architectures
+Our architectures consist of three core sub-modules: PPM,
+compressor, and final adder. Several architectures for these
+sub-modules were used in this work, each having certain
+advantages. DW02 multp is a synthesis-based PPM offered by
+Synopsys [15] which can produce fast and efficient PPMs. The
+outputs’ sizes for an M × N multiplication is M+N+2 due to
+the nature of the design. DW02 multp produces signed results
+that require sign extension. Since the sign of both outputs can
+vary, the sign extension was implemented using the following
+three steps.
+1) Applying a NOT gate to the most significant bits of all
+PPMs’ outputs (bit position M+N+2, where M and N
+are the sizes of the multiplicands).
+2) Pad the outputs with 1’s after the most significant bit
+(bit positions greater than M+N+2).
+3) Sum all constants to reduce the compression size (before
+synthesis).
+Steps 1 and 2 allow DW02 multp to be used for unsigned
+multiplications, even though DW02 multp produces signed
+results. Step 3 sums up all the constants, including the sign
+extension padding bits, reducing the size of the numbers that
+should be compressed in the next stage, and providing further
+area reductions. Like DW02 mult, DW02 tree is a synthesis-
+based compression tree offered by Synopsys [15], which can
+provide fast and efficient compressors.
+RoCoCo, proposed in [5], presents a row and column com-
+pression tree that can be used to create fast and efficient integer
+multipliers and compressors. RoCoCo aims to maximize the
+reduction in the row and column compression tree, allowing
+for a smaller final addition. RoCoCo multipliers can be used
+as PPMs by omitting the final addition stage. RoCoCo also
+presents RTL generators that can produce compressors. Their
+architecture can reduce the area complexity and the critical
+path of the overall design. The compressor used for the Ro-
+CoCo multiplier maximizes the reduction made, where several
+of the least significant bits of the second output are reduced to
+0. This reduces required computations in the following stages,
+reducing the overall area complexity and the critical path.
+A custom area-efficient compression tree is proposed in this
+work as well. This compression tree aims at reducing the num-
+ber of rows into two while minimizing the resources required.
+This compressor is tailored for a feed-forward architecture
+(which will be discussed in III-C) utilizing a DW02 multp
+PPM. Since this compressor is designed to minimize the
+required hardware resources, it does not achieve identical bit
+reductions as RoCoCo. This approach can be helpful since the
+final addition can be spread into multiple cycles using resource
+sharing or pipelining.
+The final addition stage, which comes after the partial
+product reduction stage, typically consists of ripple carry
+adders, which can often be the critical path of a design.
+This can increase the area complexity since the synthesis
+tool is forced to use larger library cells with greater driving
+strength and a shorter propagation delay. Since the designs
+
+target MC multiplications, the final adder will remain idle
+50% of the time for 2-cycle multipliers. This presents another
+opportunity to implement resource sharing. The size of the
+adder can be reduced by 50% by applying resource-sharing
+and using it in two consecutive cycles. This approach creates
+a loop around the adders, which makes pipelining the design
+complex, requiring additional control logic. Such a 2-cycle
+resource shared adder (2CA) architecture is unsuitable for
+strict timing targets. The feedback loop presents a limitation
+since the designs cannot always meet the timing target. For
+more relaxed timing targets, creating a feedback loop around
+a smaller adder reduces the area complexity because it reduces
+the size of the final adder. Pipelining the 2CA architecture can
+allow resource sharing to be implemented without limiting the
+maximum frequency. A pipelined 2-cycle adder (2CPA) can
+be achieved by placing registers in the path of the final adders.
+Such an approach would not work since variables would start
+to overlap. This was solved by adding a delay register in the
+path, making the total latency for the final adder five clock
+cycles. However, when inputs arrive at odd intervals with
+respect to the previous inputs, this again causes an overlap
+of variables. Thus, an internal state machine is required for
+such cases, which requires more complicated control logic and
+memory elements.
+MC multiplication circuits offer various opportunities for
+resource sharing since any multiplication stage can be reused
+for II times. However, resource sharing can also create feed-
+back loops in the design. Feedback loops limit a design’s
+ability to be pipelined. And thus, there exists a trade-off
+between the design’s ability to be pipelined (which determines
+the maximum frequency) and the degree of resource sharing
+to be implemented. A design containing no feedback loops
+can easily be pipelined by placing registers in the path. This
+allows a design to meet very strict timing targets at the
+cost of increased latency. Designs containing feedback loops
+have a fixed critical path since the feedback loop cannot
+be pipelined. This means that the maximum frequency can
+become a limitation for high-speed applications. Furthermore,
+since the area of a standard library cell depends on its driving
+strength and speed, pipelining allows for the reduction of area
+usage since the synthesis tool can use smaller cells that have
+longer propagation delays. Due to this, a feed-forward design
+can significantly outperform a design with a feedback loop
+in its data flow path when the clock target is strict enough,
+even if the designs implementing a feedback loop can meet
+timing. Nevertheless, feedback loops enable a greater degree of
+resource sharing, maximizing the area reductions when dealing
+with more relaxed timing targets. As a result, both approaches
+have their benefits. Depending on the target frequency, either
+approach could outperform the other.
+Two designs are proposed in this work using the previously
+discussed concepts, each having different variations in the
+sub-modules used. These designs are optimized for specific
+applications in terms of operating frequency, maximizing the
+area reductions that can be achieved.
+B. Feedback Design
+Feedback loops allow for a greater degree of resource
+sharing. As a result, we present an architecture that uses feed-
+back loops to reduce the area complexity. In this architecture,
+all three multiplier stages are resource-shared: the PPM, the
+compressor, and the final adder. These stages are fully utilized
+for 100% of clock cycles under regular operation (when
+inputs arrive back to back). This is achieved by creating a
+feedback loop around the compressor and final adder, reducing
+the size of both. The feedback loop in this design contains
+a 3:2 compressor and a ripple carry adder. For an M×N
+multiplication, this architecture uses an M × (⌈N/2⌉) PPM,
+a final adder, and a 3:2 compressor of width M+⌈N/2⌉. This
+design is represented by figure 1. The number of FA cells in
+the feedback loop, which determines the critical path of the
+design, can be calculated using equation 2, where M and N
+are the bit widths of the multiplicands.
+CriticalPath = 1 + (⌈M/2⌉ + N),
+(2)
+This approach uses the least amount of resources out of our
+designs. However, due to the loop, it can be outperformed
+by feed-forward designs for strict timing targets, where the
+feedback loop can be a limiting factor. This design is based on
+the schoolbook approach and can be represented by figure 1 for
+2C multiplication. This architecture can be extended for any
+3:2 
+Compressor 
+PPM 
++
+REG
+MUX
+0
+Fig. 1.
+2-cycle feedback design
+II, decreasing the size of the core sub-modules. The II and the
+area complexity of the three stages (PPM, compressor, and the
+final adder) have an exponential decay relationship. However,
+the number of registers storing intermediate results increases
+linearly. Due to this relationship, diminishing returns will be
+seen as the II increases. Furthermore, continuously increasing
+the II after some point would also increase the area complexity
+of the design.
+
+C. Feed-forward Design
+Multiplication circuits are frequently used with high-
+frequency applications; ergo, architectures with short critical
+paths can be very beneficial. The critical path directly affects
+both area and speed. The critical path limits the maximum
+operating frequency, and the area complexity is indirectly
+affected by the critical path. A circuit synthesized using its
+maximum frequency consumes significantly more area than
+one synthesized for a relaxed frequency. When operating under
+strict timing conditions, the synthesis tool instantiates larger
+library cells with shorter delays to meet timing. In contrast,
+the synthesis tool can use small library cells with longer prop-
+agation delays when dealing with relaxed timing targets. The
+critical path is caused by the combinational circuit that has to
+be executed within the same clock cycle. If this combinational
+logic were to be divided across multiple clock cycles, the
+critical path would be decreased, and thus, both speed and area
+would improve. As previously discussed, having a feedback
+loop in the design limits the design’s ability to be pipelined.
+For this reason, we propose a design that contains no feedback
+loops. This architecture has three main steps. Firstly, the partial
+product multiplications are computed using one module, thus
+requiring two clock cycles. In the final clock cycle, four results
+need to be added, these are first sent to a compressor, and then
+the result of the compression is sent to a ripple carry adder.
+Only the PPM is used twice in this architecture which does not
+create any feedback loops. This design can easily be pipelined
+to meet very strict timing targets. The area savings of this
+design come from the fact that it requires smaller PPG and
+PPR stages, which contribute a large portion of the overall
+complexity for an integer multiplier. All the stages in this
+design can be efficiently pipelined, which allows the critical
+path to be continuously decreased at the cost of longer latency.
+The PPM’s size equals M + ⌈(N/2)⌉. The compressor’s size
+depends on the bit widths of the inputs, as well as the type of
+PPM used. The compression tree has at most four rows to be
+reduced, and several bit-positions contain fewer rows due to
+the shifting operations. DW02 multp produces signed outputs
+that require sign extension, while RoCoCo produces unsigned
+results; ergo, the compression trees needed for these two PPMs
+are not identical. This architecture is ideal to be used with an
+II of 2. In such a case, the stage can be reduced by around
+50% while keeping the control logic simple and not requiring a
+significant number of registers for storing intermediate results.
+Since the PPM has two outputs and it is used twice, a 4:2
+compressor is required, where the inputs would be the first
+2 PPM results and the shifted version of the second 2 PPM
+results. This design is represented by figure 2.
+This architecture can also be extended for different IIs;
+however, maintaining a feed-forward approach becomes a
+limitation. A feed-forward design with an II of 3 would
+require a 6:2 compressor. This increase in the compressor’s
+size undermines the area reductions gained by the smaller
+PPM. Furthermore, such a design requires significantly more
+registers to store the intermediate results until all PPMs are
+REG
+REG
+4:2 
+Compressor 
+Final
+Adder
+PPM
+Fig. 2. 2 cycle feed-forward with a 4:2 comp
+computed. Such an architecture is not expected to provide
+any area savings. A better approach would be to add a loop
+around the 4:2 compressor. Such a design allows the II to
+be increased without significantly affecting the architecture.
+However, The feedback loop limits the design’s ability to be
+pipelined. This loop has a short critical path of only 2 FAs,
+but it also requires more registers to store the intermediate
+results. Moreover, this approach also requires a significantly
+larger compressor, requiring 96% more FAs and HAs for the
+case of 3C versus 2C II. Although a 4:2 compressor is used
+in both cases, the 4:2 compressor required for a 2C design
+contains more columns with only 2 bits to be reduced. All this
+results in a higher area requirement when compared to the 2C
+version. Therefore, a feed-forward design with this approach
+is only viable for an II of 2.
+IV. IMPLEMENTATION
+The proposed designs were tested thoroughly, using differ-
+ent bit widths, latencies, and timing targets. This required sev-
+eral steps, including design generation, synthesis, pipelining,
+simulation, and reporting power. These steps were automated
+using a series of scripts that handle these tasks accordingly.
+This automation allows for a greater degree of testing required
+for a complete evaluation of all these designs. The designs
+are generated using RTL generation scripts written in Python,
+synthesized using the Synopsys Design Compiler and a TSMC
+40 nm technology, and simulated using Icarus Verilog.
+In this work, two main architectures are proposed, which
+can have variations in the type of compressors, PPMs, and
+final adders. Due to this, design generation scripts were used
+to accommodate these variations easily. All designs are instan-
+tiated with a wrapper, automatically setting the input/output
+delays and loads. The wrapper applies a register to each of
+the inputs and outputs of the design except the clock signal.
+
+There are two different generators for 2CIM designs, one for
+each architecture. All the generators create both a wrapper
+and a testbench, thus allowing for easy and accurate testing
+for each design. The feedback architecture’s generator takes
+the size of the multiplicands, the type of PPM (DW02 multp
+or RoCoCo), and the added pipeline stages as inputs and
+then generates the design. The feed-forward design has some
+differences in the design generation parameters since it has
+multiple options for the compressor and final adder to be
+used. This generator takes the multiplicands’ size, the PPM
+and compressor’s type, and the number of pipeline stages
+as the input parameters. These variations in sub-modules are
+required to achieve optimal performance since each offers
+some advantages.
+Retiming is a technique used to optimize digital circuits
+by moving flip-flops. It can significantly reduce the critical
+path and improve the area. Since strict timing targets require
+larger library cells, reducing the critical path can also decrease
+the area complexity. The Synopsys Design Compiler offers
+the retiming feature, which was utilized to achieve optimized
+pipelining for any design. To increase the depth of the pipeline,
+registers are added at the end of the design. The synthesis tool
+can freely move these registers as long as they do not affect
+a feedback loop in the design. Increasing the pipeline’s depth
+also increases the latency, but it can decrease both the area
+complexity and the critical path while maintaining the same
+throughput.
+Synthesis is not a linear computation. It has to handle var-
+ious constraints to meet timing, optimize area, and minimize
+power consumption. Over-constraining is another technique to
+meet strict timing targets when the synthesis tool fails. This
+is done by further restricting the timing target, allowing the
+synthesis tool to make decisions that expect a more strict
+timing target. This can often allow our designs to meet the
+required timing target even if the initial synthesis attempt
+was unsuccessfully in meeting timing. The automation scripts
+attempt over-constraining whenever a design does not meet
+timing; this is done by reducing the timing target by 5%
+and re-attempting synthesis. Over-constraining is attempted
+three times before increasing the depth of the pipeline (adding
+registers to the design and applying retiming).
+In this work, area complexities of all 2CIM designs and the
+standard multiplication circuit generated by Synopsys using
+the “*” operator (which will be referred to as Star) are
+compared. The same target frequency should be used for a fair
+comparison of results between any two designs. The area com-
+plexity, power consumption, and operating frequency are all
+interdependent. However, in most applications, the operating
+frequency is a design decision that needs to be preserved.The
+initial timing target is set as the maximum frequency achieved
+by the Star multiplier without adding any additional pipeline
+stages. In addition, over-constraining is used to test if a higher
+maximum frequency can be achieved. Timing is then increased
+by 15% and 30% to see how the designs perform in more
+relaxed timing targets. Furthermore, the target is set to 0.31,
+representing the clock-to-q + setup + hold delay for 1 FA
+placed between registers, using the smallest library cell for
+a FA. However, such a strict timing target is not suitable
+for multiplications larger than 32×32 since large multipliers
+require many pipeline stages to meet the target timing. Larger
+multipliers used less strict timing targets according to the
+actual multiplication size. Using such strict timing targets is
+useful to check how effectively the designs can be pipelined
+to meet very strict timing targets. The designs are also
+synthesized with additional pipeline stages until the latency
+reaches the maximum latency of other designs. This is because
+increasing the pipeline depth can decrease the area complexity
+as well. All this was accomplished by using automation scripts.
+The scripts first synthesize the Star multiplier to get the three
+timing targets. Two more timing targets are used, representing
+strict and relaxed timing conditions. For each timing target, it
+follows a series of steps to produce the complete results table.
+The first step is the design generation, which is done using
+the RTL generators of each architecture. It then synthesizes
+each design using retiming and over-constraining to meet the
+timing target. After that, it simulates the generated netlist and
+estimates the power consumption. This is repetitively done for
+all designs, variations, and timing targets. Since the designs are
+tested using a strict timing target, the target is not always met
+from the initial synthesis attempt. The script will first attempt
+to meet timing using over-constraining. If the design is still
+unable to meet timing, a pipeline stage is added. This iteration
+is repetitively done until either timing is met or the number of
+pipeline stages exceeds 8. This usually means that the target
+timing cannot be met even when pipelined. All architectures
+are simulated using a set of 200 randomly generated inputs.
+Self-checking test benches were used, where the output is
+sampled depending on the latency of the design. Furthermore,
+both design and netlist simulations were performed.
+Power consumption can be an important aspect to consider
+when designing a digital circuit. All the proposed designs were
+analyzed for power consumption. Randomly generated inputs
+were used, being set every 2 clock cycles. This represents a
+worst-case scenario, where the power consumption is analyzed
+under heavy loads. Post-synthesis simulation was performed
+on all the designs, generating a file that contains the switching
+activities of all nets and ports. This file is then used by
+the synthesis tool (Synopsys Design Compiler) to estimate
+the power consumption of a design, including both dynamic
+and static power consumption in the estimation. Analyzing
+power using the post-synthesis netlist simulation can generate
+more accurate results when compared to the RTL simulation
+since it contains more accurate switching activities. Power
+consumption can be affected by several factors. However, there
+are three main factors to consider, the critical path of the
+circuit, the length of the MC path (which includes the parts to
+be resource shared), and the area of the design. Under strict
+timing conditions, the synthesis tool uses large library cells to
+meet timing, which consumes more power than smaller library
+cells having longer delays. Longer MC paths produce more
+glitches in the design, and these glitches (part of the switching
+activity) increase power consumption. And lastly, a larger area
+
+implies that there are more gates in the design, thus consuming
+more power since even idle gates contribute to static power
+consumption.
+V. RESULTS
+All the proposed 2CIM designs were tested thoroughly in
+this work. Each design has advantages and disadvantages,
+offering significant area saving under the right conditions. All
+the proposed 2CIM designs and variants should be synthesized
+under the same timing conditions and compared to perform a
+thorough evaluation. All 2CIM designs were synthesized under
+a wide range of multiplication sizes and various timing targets.
+All the designs were tested using a wide range of operating
+frequencies since each 2CIM architecture targets a specific
+type of application with respect to the operating frequency.
+Moreover, the scalability of these designs is an important
+feature. The designs were tested for various multiplication
+sizes to show that these area savings are consistent for various
+multiplication sizes. This section will present two tables that
+can represent strict and relaxed timing conditions. Since our
+designs usually achieve a longer latency, they should be
+compared with a Star multiplier design using the same number
+of pipeline stages. However, the Star multiplier is relatively
+faster, so it will reach its minimum area using a smaller
+number of pipeline stages than that of 2CIM designs. The
+area results that are reported for the Star multiplier are the
+best area results that can be achieved using any number of
+pipeline stages.
+A. Synthesis Under Relaxed Timing Conditions
+A relaxed timing target can show the actual area require-
+ments of each design since a strict timing target would affect
+the results depending on the critical path. The relaxed timing
+conditions case is represented by a timing target of 10 ns.
+Such a target allowed all the designs to meet timing without
+requiring additional pipeline stages while being able to use
+small library cells. Table I presents the synthesis results for a
+16×16 multiplication. Since the timing target is very relaxed,
+retiming and over-constraining were not required. Thus, the
+latency presented in table I represents the minimum latency
+(L) for these designs.
+The feedback (FB) designs best suit applications that do
+not require very strict timing targets. The area complexity of
+these designs is always the lowest under such circumstances.
+For 16×16 multiplications, the feedback designs can offer
+around 30% area savings. But it comes at the cost of an
+increase in power consumption. And so, a trade-off exists
+between the area complexity and power consumption for such
+multiplications with relaxed timing targets. It can be seen that
+the 2CA final adder cannot offer any benefits in terms of
+area savings compared to either the feed-forward (FF) design
+implementing no spread (SCA) or the feedback design. This
+can be explained by the fact that under very relaxed timing
+conditions, the area complexity coming from the final adder is
+lower than the required logic to implement resource sharing.
+Thus, 2CA is not expected to offer additional area savings
+TABLE I
+SYNTHESIS RESULTS FOR 16×16 MULTIPLICATIONS UNDER RELAXED
+TIMING CONDITIONS (TARGET = 10 ns)
+Design
+Final
+PPM
+Comp.
+L
+Area
+Power
+Adder
+(uW)
+Star
+N/A
+N/A
+N/A
+1
+1348
+79
+FB
+N/A
+DW02
+FAs
+2
+942
+100
+N/A
+RoCoCo
+FAs
+2
+960
+108
+FF
+RCA
+DW02
+DW02
+2
+1096
+124
+RCA
+DW02
+RoCoCo
+2
+1145
+127
+RCA
+DW02
+Custom
+2
+1105
+124
+RCA
+RoCoCo
+DW02
+2
+1051
+120
+RCA
+RoCoCo
+RoCoCo
+2
+1122
+122
+2CA
+DW02
+DW02
+3
+1352
+159
+2CA
+DW02
+RoCoCo
+3
+1181
+148
+2CA
+DW02
+Custom
+3
+1420
+168
+2CA
+RoCoCo
+DW02
+3
+1115
+138
+2CA
+RoCoCo
+RoCoCo
+3
+1155
+140
+2CPA
+DW02
+DW02
+6
+2102
+250
+2CPA
+DW02
+RoCoCo
+6
+2053
+251
+2CPA
+DW02
+Custom
+6
+2086
+248
+2CPA
+RoCoCo
+DW02
+6
+2120
+244
+2CPA
+RoCoCo
+RoCoCo
+6
+2129
+243
+under these conditions. 2CPA is designed to provide area
+savings for strict timing targets. Therefore, it cannot offer any
+area savings under such relaxed timing conditions. This is due
+to the added complexity in the control logic, which is required
+for scheduling inputs based on their arrival time, and the
+increased number of registers needed for storing intermediate
+results. The proposed designs consume more power than
+the Star multiplier since they have a longer MC path. A
+longer MC path increases the circuit’s glitches, increasing the
+dynamic power consumption. The Star multiplier employs no
+resource sharing. Therefore, only a part of the circuit is active
+simultaneously when a throughput of less than one is needed.
+In our testing, inputs are received every two cycles, making
+the Star multiplier remain idle 50% of the time.
+B. Synthesis Under Strict Timing Conditions
+An essential aspect of ASICs is their ability to operate at
+high frequencies. For this reason, the ability of any multi-
+plication circuit to operate at high frequencies is vital. All
+2CIM designs were tested under high frequencies as well.
+They were synthesized using a very strict timing target of
+0.31 ns, equal to the clock-to-q + setup + hold delay for 1
+FA placed between registers. Such a strict target shows how
+well these designs can be pipelined. Table II contains the
+synthesis results of the proposed designs, both retiming and
+over-constraining were used to meet timing. The feed-forward
+designs that use a 2CA final adder have a longer critical path
+than the feedback designs; ergo, they are unsuitable for high-
+frequency applications. Designs using the 2CPA final adders
+could achieve high frequencies as intended. However, the
+added complexity from the more complicated control logic
+resulted in a greater area complexity. Therefore, 2CPA is
+consistently outperformed by a pipelined RCA using the same
+latency. For such high-frequency applications, the only viable
+options are fully feed-forward designs because of their ability
+
+to be efficiently pipelined without requiring additional control
+logic. As seen in table II, the feed-forward (FF) designs can
+TABLE II
+SYNTHESIS RESULTS FOR 16×16 MULTIPLICATIONS UNDER STRICT
+TIMING CONDITIONS (TARGET = 0.31 ns)
+Design
+FA
+PPM
+Comp.
+L
+Area
+Timing
+Power
+(ns)
+(mW)
+Star
+N/A
+N/A
+N/A
+7
+5178
+0.31
+7.52
+FF
+RCA
+RoCoCo
+DW02
+9
+3963
+0.31
+7.13
+RCA
+DW02
+DW02
+9
+3984
+0.31
+7.92
+RCA
+RoCoCo
+RoCoCo
+7
+4065
+0.31
+6.57
+RCA
+DW02
+Custom
+9
+4065
+0.31
+7.83
+RCA
+DW02
+RoCoCo
+9
+4200
+0.31
+7.98
+2CPA
+DW02
+DW02
+11
+4971
+0.31
+10.17
+2CPA
+DW02
+Custom
+10
+5115
+0.31
+9.72
+2CPA
+RoCoCo
+RoCoCo
+12
+5192
+0.31
+9.58
+2CPA
+RoCoCo
+DW02
+12
+5202
+0.31
+10.11
+2CPA
+DW02
+RoCoCo
+11
+5307
+0.31
+10.06
+2CA
+DW02
+DW02
+5
+4394
+0.46
+4.43
+2CA
+DW02
+RoCoCo
+7
+4208
+0.49
+4.18
+2CA
+DW02
+Custom
+5
+4600
+0.48
+4.46
+2CA
+RoCoCo
+DW02
+5
+3255
+0.55
+2.96
+2CA
+RoCoCo
+RoCoCo
+6
+3434
+0.58
+3.05
+FB
+N/A
+DW02
+FAs
+4
+3712
+0.46
+4.47
+N/A
+RoCoCo
+FAs
+6
+3554
+0.49
+4.34
+offer up to 23% area savings and 13% power reduction. The
+Star multiplier is not able to meet timing without pipelining.
+It requires a minimum pipeline depth of 6 for the designs to
+meet timing and a depth of 7 to achieve the optimal area.
+The Star multiplier achieves a similar latency to the proposed
+designs while having a greater area complexity. Therefore, the
+MC path for 2CIM designs is shorter or equivalent to that of
+Star, explaining the power reduction 2CIM designs offer for
+such strict timing targets.
+C. Discussions
+The proposed 2CIM designs can offer significant area
+savings for various applications. Table III contains the area
+savings provided by the best-performing design under different
+bit widths and timing targets. This table presents the optimal
+design under either very strict or semi-relaxed timing condi-
+tions. The feed-forward (FF) design is best suited for strict
+timing targets, and the feedback (FB) design is best suited for
+more relaxed timing targets. The 2CA and 2CPA final adders
+do not offer any area savings when compared to the other
+designs. The 2CA creates a feedback loop in the feed-forward
+design. This means that a feed-forward design implementing
+a 2CA final adder will consistently be outperformed by either
+the feedback designs or a feed-forward design that uses an
+RCA final adder.
+2CIM designs have a throughput of 1/2, i.e., they can
+compute one multiplication every two clock cycles. 2CIM
+designs can be used when i multiplications are required within
+j clock cycles, and (i mod j)/j is less than or equal to 1/2.
+Another case in which 2CIM designs can be used is when
+there is a latency constraint. For 128×128 multipliers with
+a clock target of 0.8, both the feed-forward design and the
+Star multiplier achieve the same latency. This is because the
+feed-forward architecture can be pipelined very efficiently.
+TABLE III
+AREA SAVINGS FOR DIFFERENT BIT WIDTHS
+Design
+PPM
+Comp.
+Timing
+L
+Area
+Savings
+(ns)
+8×8
+Star
+N/A
+N/A
+0.31
+4
+1377
+-
+FF
+RoCoCo
+RoCoCo
+0.31
+5
+1088
+21%
+Star
+N/A
+N/A
+0.5705
+2
+738
+-
+FB
+DW02
+FAs
+0.5705
+4
+600
+19%
+16×16
+Star
+N/A
+N/A
+0.31
+7
+5179
+-
+FF
+RoCoCo
+DW02
+0.31
+9
+3964
+23%
+Star
+N/A
+N/A
+1.001
+1
+2160
+-
+FB
+DW02
+FAs
+1.001
+3
+1255
+42%
+32×32
+Star
+N/A
+N/A
+0.31
+10
+17790
+-
+FF
+DW02
+Custom
+0.31
+9
+13653
+23%
+Star
+N/A
+N/A
+1.287
+2
+6057
+-
+FB
+DW02
+FAs
+1.287
+3
+4093
+32%
+64×64
+Star
+N/A
+N/A
+0.4
+7
+51638
+-
+FF
+DW02
+Custom
+0.4
+7
+47496
+8%
+Star
+N/A
+N/A
+1.3915
+2
+22841
+-
+FB
+DW02
+FAs
+1.3915
+3
+13389
+41%
+128×128
+Star
+N/A
+N/A
+0.8
+4
+121634
+-
+FF
+DW02
+Custom
+0.8
+4
+63777
+48%
+Star
+N/A
+N/A
+1.457
+2
+89165
+-
+FB
+DW02
+FAs
+1.457
+3
+48911
+45%
+Suppose there is a latency constraint of 4 cycles, and the
+throughput of these multipliers does not exceed 1/2. In such
+a case, multiple instances of the feed-forward design can be
+used to calculate any number of multiplications, providing
+a great degree of area savings. Such cases are only seen
+for large multiplications operating under high frequencies. In
+most cases, however, a 2CIM design can slightly increase the
+latency. This is because conventional multipliers are somewhat
+faster than the proposed 2CIM designs. Thus, 2CIM designs
+might require more pipeline stages to meet a strict timing
+target. This is usually around one or two additional clock
+cycles. For large multipliers with very strict timing targets,
+however, the feed-forward architecture can be pipelined more
+efficiently, achieving identical latencies as those of the Star
+multiplier.
+VI. CONCLUSION
+The number of multiplications required in a clock cycle is
+not always an integer, which is the case when an odd number
+of multiplications is required within two clock cycles, e.g.,
+three multiplications are required within two clock cycles, i.e.,
+1.5 multiplications per clock cycle. Such cases can be found
+in a variety of applications across any domain since multipli-
+cation circuits are essential building blocks. The conventional
+way of dealing with such cases is to utilize complete multi-
+pliers for these fractional multiplications. Or in other words,
+using a multiplier for only 50% of the time. This approach
+is not optimal since it requires more area than necessary
+and does not take full advantage of the allocated resources.
+This work presents a range of MC unsigned integer multi-
+
+pliers that offer fractional multipliers. These designs provide
+significant area savings for a variety of applications. 2CIM
+designs are designed to replace fully-pipelined multipliers,
+i.e., multipliers that can accept inputs in a consecutive clock
+cycle, for applications where they remain underutilized. 2CIM
+designs have a throughput of 1/2, i.e., they can compute one
+multiplication every two clock cycles. This work presents two
+main architectures, each having multiple possible variations.
+The feed-forward architecture has the advantage of speed. It
+can run at very high frequencies due to the feed-forward design
+aspect, which allows it to be continuously pipelined until the
+timing is met. The feedback design implements a higher degree
+of resource sharing, enabling it to require significantly fewer
+hardware resources. However, it also requires a feedback loop.
+Feedback loops can limit a design’s ability to operate at high
+frequencies since feedback loops are not easily pipelined. Both
+architectures have advantages, and depending on the target
+application, either can be the optimal choice. 2CIM designs
+can be used for any application that requires i multiplications
+within j clock cycles, and (i mod j)/j is less than or equal
+to 1/2. This can be especially useful for area-efficient low-
+bandwidth applications. 2CIM designs can provide up to 21%,
+42%, 32%, 41%, and 48% area savings for multiplications
+bit widths of 8, 16, 32, 64, and 128, respectively. 2CIM
+designs were tested thoroughly through automation scripts
+using various multiplication sizes and timing targets. 2CIM
+designs consistently offered significant area savings through-
+out our testing. Moreover, 2CIM designs can provide power
+savings when dealing with strict timing targets. All designs
+were synthesized using the Synopsys Design Compiler and a
+40 nm TSMC technology.
+REFERENCES
+[1] D. Goldberg, “What every computer scientist should know about
+floating-point arithmetic,” ACM Computing Surveys, vol. 23, pp. 5–48,
+1991.
+[2] C.
+Hoekstra,
+K.
+Shukla,
+and
+M.
+Harris,
+“Im-
+plementing
+high-precision
+decimal
+arithmetic
+with
+CUDA
+int128,”
+https://developer.nvidia.com/blog/
+implementing-high-precision-decimal-arithmetic-with-cuda-int128
+(accessed 14 July 2022).
+[3] C. S. Wallace, “A suggestion for a fast multiplier,” IEEE Transactions
+on Electronic Computers, vol. EC-13, pp. 14–17, 1964.
+[4] L. Dadda, “Some schemes for parallel multipliers,” Alta Frequenza,
+vol. 34, pp. 349–356, 1965.
+[5] F. Ugurdag, O. Keskin, C. Tunc, F. Temizkan, G. Fici, and S. Dedeoglu,
+“RoCoCo: Row and column compression for high-performance multipli-
+cation on FPGAs,” in Proc. East-West Design & Test Symp. (EWDTS),
+2011, pp. 98–101.
+[6] C. Rafferty, M. O’Neill, and N. Hanley, “Evaluation of large integer
+multiplication methods on hardware,” IEEE Transactions on Computers,
+vol. 66, pp. 1369–1382, 2017.
+[7] I. San and N. At, “On increasing the computational efficiency of long
+integer multiplication on FPGA,” in Proc. IEEE Int. Conf. on Trust,
+Security and Privacy in Computing and Communications (TrustCom),
+2012, pp. 1149–1154.
+[8] J. Von Zur Gathen and J. Shokrollahi, “Efficient FPGA-based karat-
+suba multipliers for polynomials over F2,” in Proc. Selected Areas in
+Cryptography (SAC).
+Springer, 2005, pp. 359–369.
+[9] S. Gao, D. Al-Khalili, and N. Chabini, “Efficient scheme for implement-
+ing large size signed multipliers using multigranular embedded DSP
+blocks in FPGAs,” Int. Journal of Reconfigurable Computing, 2009.
+[10] F. de Dinechin and B. Pasca, “Large multipliers with fewer DSP blocks,”
+in Proc. IEEE Int. Conf. on Field Programmable Logic and Applications
+(FPL), 2009, pp. 250–255.
+[11] M. Langhammer and B. Pasca, “Folded integer multiplication for
+FPGAs,” in Proc. ACM/SIGDA Int. Symp. on Field-Programmable Gate
+Arrays (FPGA), 2021, pp. 160–170.
+[12] J. Li, Y. Du, and J. Wang, “Design a pocket multi-bit multiplier in
+FPGA,” in Proc. IEEE Int. Conf. on ASIC (ASICON), 1996, pp. 275–
+279.
+[13] M. R. Santoro and M. A. Horowitz, “SPIM: a pipelined 64*64-bit
+iterative multiplier,” IEEE Journal of Solid-state Circuits, vol. 24, pp.
+487–493, 1989.
+[14] M.-C. Shin, S.-H. Kang, and I.-C. Park, “An area-efficient iterative
+modified-booth multiplier based on self-timed clocking,” in Proc. IEEE
+Int. Conf. on Computer Design: VLSI in Computers and Processors
+(ICCD), 2001, pp. 511–512.
+[15] Synopsys,
+“DesignWare
+Library,”
+https://www.synopsys.com/dw/
+buildingblock.php (accessed: 2021-03-01).
+