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+arXiv:2301.00546v1  [cond-mat.mes-hall]  2 Jan 2023
+Tunable caging of excitation in decorated Lieb-ladder geometry with long range
+connectivity
+Atanu Nandy∗
+Department of Physics, Acharya Prafulla Chandra College,
+New Barrackpore, Kolkata West Bengal-700 131, India
+Controlled Aharonov-Bohm caging of wave train is reported in a quasi-one dimensional version
+of Lieb geometry with next nearest neighbor hopping integral within the tight-binding framework.
+This longer wavelength fluctuation is considered by incorporating periodic, quasi-periodic or fractal
+kind of geometry inside the skeleton of the original network.
+This invites exotic eigenspectrum
+displaying a distribution of flat band states. Also a subtle modulation of external magnetic flux
+leads to a comprehensive control over those non-resonant modes. Real space renormalization group
+method provides us an exact analytical prescription for the study of such tunable imprisonment of
+excitation. The non-trivial tunability of external agent is important as well as challenging in the
+context of experimental perspective.
+Keywords: Caging, flat band, interferometer, renormalization.
+I.
+INTRODUCTION
+Recent exciting headway in experimental condensed
+matter physics helps us to emulate several quantum
+mechanical phenomena in a quite tunable environment.
+This unprecedented advancement in fabrication tech-
+nique provides a scope for direct visualization of different
+theoretically proposed phenomena like localization of ex-
+citation in low dimensional networks [1, 2]. That is why
+creation of so called artificial systems for the simulation
+of complex many-body systems containing additional de-
+gree of freedom has grabbed considerable scientific im-
+pact [3]. Moreover, scientific communities have already
+addressed the celebration of sixty years of the pioneer-
+ing work of Anderson [4].
+The absence of diffusion of
+wave packet in the random disorder environment is well
+known. In fact this now becomes a general prescription
+in diverse topics of condensed matter physics starting
+from optical lattice of ultra cold atoms [5] to the acous-
+tics, wave guide arrays [6] or in micro cavities having
+exciton-polaritons [7]. Unlike the case of Anderson lo-
+calization (AL), the concept of compact localized states
+(CLS) [8]-[15] in several one or two dimensional periodic
+or non-periodic structures has attracted the spot light of
+fundamental research. The journey started nearly thirty
+years ago approximately from Sutherland [16].
+This unconventional non-diffusive progress of wave has
+generated significant attention because of its contribu-
+tion to various novel physical phenomena in strongly
+correlated system, such as unconventional Anderson
+localization [17, 18],
+Hall ferromagnetism [19, 20],
+high-temperature superconductivity [21], and superflu-
+idity [22], to name a few. Moreover, this study has kept
+scientists intrigued since it offers a suitable platform to
+investigate several phenomena that are linked with the
+information of quantum physics together with the topo-
+∗Electronic address: [email protected]
+logical effect including fractional quantum hall effect [23]
+and flat band ferromagnetism [24]. For these CLS, the
+diminishing envelope of the wave train beyond finite size
+characteristics trapping cell implies extremely low group
+velocity due to the divergent effective mass tensor. This
+means that the particle behaves like a super heavy such
+that it cannot move. The vanishing curvature of the E−k
+plot corresponding to such momentum independent self-
+localized states are generally caused by the destructive
+nature of the quantum interference occurred by multiple
+quantum dots and the local spatial symmetries involved
+with the underlying structure. Hence these are also called
+as flat band states.
+In general, occurrence of dispersionless flat band can
+be classified into two categories depending on their sta-
+bility with respect to the application of magnetic pertur-
+bation. In particular, the type of geometries discussed
+by Mielke [25] and Tasaki [19] cannot contain flat bands
+for finite magnetic flux. Whereas, the other type of lat-
+tices e.g., Lieb lattice [26], there exists macroscopically
+degenerate flat band even in the presence of flux. In fact,
+the non dispersive band is completely insensitive to the
+applied external perturbation. As it is well known that
+the inherent topology of the line-centered square lattice
+(also known as the Lieb lattice) induces interesting spec-
+tral properties such as the macroscopically degenerated
+zero-energy flat band, the Dirac cone in the low- energy
+spectrum, and the typical Hofstadter-type spectrum in
+a magnetic field. Moreover, Lieb geometry is one of the
+most prominent candidate useful for magnetism.
+The
+spectral divergence of the zero-energy flat band provides
+that platform.
+In this manuscript, inspired by all the experimental re-
+alizations of Aharaonov-Bohm caging, we study a quasi-
+one dimensional Lieb-ladder network within the tight-
+binding formalism.
+The phenomenon of imprisonment
+of wave train is studied when the next nearest neighbor
+(NNN) connection term is added to the Hamiltonian. In-
+teresting modulation of self-trapping of excitation is also
+studied in details when the NNN connectivity is ‘dec-
+
+2
+orated’ by either magnetic flux or some quasi-periodic,
+fractal kind of objects.
+As a second motivation we have analyzed an Aharonov-
+Bohm interferometer model made in the form of a quasi-
+one dimensional Lieb geometry to study the flux con-
+trolled localization aspects.
+It is needless to mention
+that this flux controlled caging is a subset of widely
+used phenomena Aharonov-Bohm caging [27] and this
+has been experimentally verified in recent times [1, 2].
+However, when an electron traverses a closed loop that
+traps a finite magnetic flux Φ, its wave function picks
+up a phase factor.
+This simple sentence is at the the
+core of the pioneering Aharonov-Bohm (AB) effect [28]-
+[32] which has led to a substantial research in the stan-
+dard AB interferometry that dominated the fundamental
+physics, both theoretical and experimental perspective,
+in the mesoscopic scale over the past few decades [33]-
+[35]. It is to be noted that the current experiments by
+Yamamoto et al. [36] has stimulated more experiments on
+quantum transmission in AB interferometers [37]. Also
+the previously mentioned theoretical model studies have
+also played an important part in studying the elemen-
+tary characteristics of the electronic states and coherent
+conductance in quantum networks in the mesoscopic di-
+mensions [35]. The recent advancement in the fabrication
+and lithography processes have opened up the possibility
+to make a tailor-made geometry with the aid of quan-
+tum dots (QD) or Bose–Einstein condensates (BEC). It is
+needless to mention that this has provoked a substantial
+content of theoretical research even in model quantum
+networks with a complex topological character [38, 39].
+In this article, highly motivated by the ongoing sce-
+nario of theory and experiments in AB interferometry,
+we investigate the spectral and the transmission prop-
+erties of a model quantum network in which diamond
+shaped Aharonov-Bohm interferometers are arranged in
+the form of a quasi-one dimensional Lieb ladder geom-
+etry.
+Such diamond-based interferometer models have
+previously been analyzed as the minimal prototypes of
+bipartite networks having nodes with different coordi-
+nation numbers, and representing a family of itinerant
+geometrically frustrated electronic systems [40]. There
+are other studies which include the problem of imprison-
+ment of excitation under the influence of spin-orbit inter-
+action [41], a flux-induced semiconducting behavior [42],
+quantum level engineering for AB cages [43] or, as models
+of spin filters [44].
+In what follows we demonstrate our findings. Sec. II
+discusses the basic quasi-one dimensional Lieb ladder net-
+work in respect of energy band and transmittivity. In
+Sec. III we have incorporated a next nearest neighbor
+connectivity by inserting a rhombic loop inside the unit
+cell and discussed the flux sensitive localization.
+Af-
+ter that in Sec. IV the NNN hopping is decorated by
+a quasiperiodic Fibonacci geometry and the distribution
+of self-localized states has been studied. Sec. V demon-
+strates the self-similar pattern of compact localized states
+as a function of magnetic flux. In Sec. VI we have stud-
+ied the Lieb Aharonov-Bohm interferometer model in re-
+spect of its electronic eigenspectrum. Finally in Sec. VII
+we draw our conclusions.
+II.
+MODEL SYSTEM AND HAMILTONIAN
+We start our demonstration from the Fig. 1(a) where
+a quasi-one dimensional version of the Lieb geometry is
+shown. We make a distinction between the sites (blue col-
+ored dots marked as A site and red colored dots marked
+as B sites) based on their coordination numbers. The
+(a)
+A
+B
+x
+y
+(b)
+ε
+τ
+γ
+ξ
+FIG. 1: (Color online) (a) A quasi-one dimensional Lieb lad-
+der network with endless axial span and (b) the effective two-
+arm ladder with renormalized parameters.
+array is modeled by the standard tight-binding Hamilto-
+nian written in the Wannier basis, viz.,
+H =
+�
+j
+ǫjc†
+jcj +
+�
+⟨jk⟩
+[tjkc†
+jck + h.c.]
+(1)
+where the first term bears the potential information of
+the respective quantum dot location and the second one
+indicates the kinetic signature between two neighboring
+lattice sites. The on-site potential of the respective sites
+are marked as ǫA and ǫB and the nearest neighbor overlap
+parameter can be assigned as t.
+Without any loss of
+generality, numerically the site potentials are taken as
+uniform (equal to zero) and the nearest neighbor hopping
+is also same (equal to unity) everywhere. By virtue of real
+space renormalization group (RSRG) technique one can
+easily eliminate the amplitude of an appropriate subset
+of nodes to caste the original system into an effective
+two-strand ladder system with renormalized parameters
+as cited in the Fig. 1(b). The decimation method can be
+easily implemented with the help of difference equation,
+the discretized form of the Schr¨odinger’s equation, viz.,
+(E − ǫj)ψj =
+�
+k
+tjkψk
+(2)
+This decimation provides the renormalized uniform two-
+leg ladder network with different parameters. After this
+renormalization procedure, all the atomic sites carry
+identical on-site energy ¯ǫ and the intra-arm hopping τ.
+The inter-arm vertical connectivity is marked as γ as
+cited in the Fig. 1(b). This decimation produces a next
+nearest neighbor hopping, denoted by ξ, which generates
+overlap between the wave functions of the two diagonally
+
+3
+opposite atomic sites. The detailed forms of those pa-
+rameters are given by,
+¯ǫ = ǫ + 2t2(E − ǫ1)
+δ
+τ = t2(E − ǫ1)
+δ
+γ = 2t2t1
+δ
+ξ = t2t1
+δ
+(3)
+where ǫ1 = ǫ + t2/(E − ǫ), t1 = t2/(E − ǫ) and δ = [(E −
+ǫ1)2 − t2
+1]. With the above renormalized parameters and
+by virtue of RSRG approach, one can trivially compute
+the electronic density of states (DOS) ρ(E) for this quasi-
+one dimensional Lieb strip as a function of the energy of
+the incoming projectile by using the standard expression,
+viz.,
+ρ(E) = −
+� 1
+Nπ
+�
+Im[T rG(E)]
+(4)
+Here G(E) = [E−H +i∆]−1 is the usual green’s function
+and ∆ is the imaginary part of the energy, reasonably
+small enough, added for the numerical evaluation of DOS.
+N denotes the total number of atomic sites present in the
+system and ‘Tr’ is the trace of the green’s function.
+A.
+Density of eigenstates and transport
+In Fig. 2(a) the variation of DOS is presented as a func-
+tion of energy where we see the presence of the absolutely
+continuous Bloch bands populated by extended eigen-
+functions. We have checked that for any energy belong-
+ing to the resonant band, the overlap parameter keeps
+on non-decaying behavior and that is a signature of the
+state being delocalized. At the band center (E = 0), the
+central spike confirms the existence of momentum inde-
+pendent flat band state which is an inherent signature of
+the Lieb geometry. The spectral divergence correspond-
+ing to the zero energy mode comes from the vanishing
+group velocity of the wave packet as ρ ∝
+�
+v−1
+g dk. With
+the aid of difference equation one can obtain the distribu-
+tion of amplitude for such self-localized eigenstate. The
+non-vanishing amplitudes are pinned at the intermediate
+sites as shown in Fig. 2(b) and one such characteristic
+trapping island is isolated from the other by a distinct
+physical boundary formed by the sites with zero ampli-
+tude as a result of destructive quantum interference. The
+dispersionless nature of the central band is responsible for
+anomalous behavior in the transport and optical prop-
+erties. The construction of this state definitely resem-
+bles the essence of a molecular state which is spatially
+quenched within a finite size cluster of atomic sites. The
+analogous wave function does not present any evolution
+(a)
+(b)
+0
+0
+0
+0
++1
+−1
+0
+0
+0
+0
+−1
++1
+−1
++1
+0
+0
+0
+0
+(c)
+�4
+�2
+0
+2
+4
+0.0
+0.1
+0.2
+0.3
+0.4
+E
+T �E�
+FIG. 2: (Color online) (a) Plot of density of eigenstates as
+a function of energy E for quasi-one dimensional Lieb-ladder
+geometry, (b) denotes the amplitude distribution profile for
+E = 0 and (c)variation of transmittance with energy.
+dynamics beyond the trapping cell. Extremely low mo-
+bility of the wave train is the key factor for the disper-
+sionless signature of the state. But here we should point
+out that since the compact localized state, thus formed,
+lies inside the continuum zone of extended states, here
+the hopping integral never dies out for E = 0. Hence,
+one should observe non-zero transport for that particu-
+lar mode. The localization character can be prominently
+viewed in presence of any perturbation when the spec-
+trum shows central gap around E = 0, if any.
+To corroborate the above findings related to the spec-
+tral landscape we now present a precise discussion to
+elucidate the electronic transmission characteristics for
+this quasi-one dimensional system. For this analysis we
+have considered a finite-sized underlying network. Now
+the ladder-like system needs to be clamped in between
+two pairs of semi-infinite periodic leads with the corre-
+sponding parameters. One can then adopt the standard
+green’s function approach [45, 46] and compute the same
+for the composite system (lead-system-lead). The trans-
+mission probability [47]-[51] can be written in terms of
+this green’s function including the self-energy term as,
+τij = T r[ΓiGr
+i ΓjGa
+i ]
+(5)
+
+1.0
+0.8
+0.6
+Q
+0.4
+0.2
+0.0
+-2
+-3
+-1
+0
+2
+3
+1
+E4
+Here the terms Γi and Γj respectively denote the con-
+nection of the network with the i-th and j-th leads and
+G’s are the retarded and advanced Green’s functions of
+the system. The result is demonstrated in the Fig. 2(c).
+It describes a wide resonant window for which we have
+obtained ballistic transport.
+The existence of Bloch-
+like eigenfunctions for this wide range of Fermi energy
+is solely responsible for this high transmission behavior.
+The conducting nature of the spectral density is basically
+reflected in this transmission plot.
+B.
+Band dispersion
+To study the energy-momentum relation of this peri-
+odic system we will cast the original Hamiltonian in terms
+of wave vector k by virtue of the following expression,
+H =
+�
+k
+ψ†
+kH(k)ψk
+(6)
+Using this relation, the Hamiltonian matrix in k-space
+reads as,
+H(k) =
+
+
+ǫ
+t
+0
+t(1 + e−ika)
+0
+t
+ǫ
+t
+0
+0
+0
+t
+ǫ
+0
+t(1 + e−ika)
+t(1 + eika) 0
+0
+ǫ
+0
+0
+0 t(1 + eika)
+0
+ǫ
+
+
+(7)
+The straightforward diagonalization of the above matrix
+�Π
+� Π
+2
+0
+Π
+2
+Π
+�2
+�3
+�1
+0
+1
+2
+3
+ka
+E
+FIG. 3: (Color online) Band dispersion diagram of a quasi-
+one dimensional Lieb-ladder network showing the central flat
+band and other two pairs of dispersive bands.
+reveals the entire band picture of the Lieb-ladder network
+as presented in Fig. 3. It clearly shows one momentum
+insensitive non-dispersive band at E = 0 with absolutely
+zero curvature and two pairs of Bloch bands carrying
+dispersive signature at E = ±
+�
+2(1 + cos ka) and E =
+±
+�
+2(2 + cos ka). The central flat band state confirms
+the existence of robust type of molecular state.
+Φ
+Φ
+Φ
+Φ
+Φ
+Φ
+Φ
+FIG. 4: (Color online) A quasi-one dimensional array of Lieb-
+ladder geometry with next nearest neighbor (NNN) hopping
+term incorporated by a diamond loop threaded by uniform
+magnetic flux Φ.
+III.
+DIAMOND-LIEB NETWORK
+In the previous description presented so far, the off-
+diagonal element, i.e., the hopping parameter is taken to
+be restricted within the nearest neighboring atomic sites
+only within the tight-binding formulation. We now con-
+sider the same quasi-one dimensional Lieb-ladder geom-
+etry with next nearest neighbor (NNN) hopping integral
+taken into consideration between the A types of sites as
+cited in the Fig. 4. With the inclusion of longer range
+connectivity the entire periodic geometry turns out to
+be quasi-one dimensional Lieb ladder with a rhombic ge-
+ometry embedded inside the skeleton.
+This additional
+overlap parameter introduces another closed loop within
+each unit cell where the impact of application of magnetic
+perturbation may be examined in details.
+(a)
+�2
+�1
+0
+1
+2
+�4
+�2
+0
+2
+4
+���0
+E
+(b)
+0
+0
+0
+0
+0
+0
+−1
+−1
++1
++1
+0
+0
+0
+0
++1
++1
+−1
+−1
+Φ
+Φ
+Φ
+FIG. 5: (Color online) (a) Presentation of allowed eigenspec-
+trum as a function of magnetic flux for diamond-Lieb net-
+work and (b) amplitude profile corresponding to the energy
+E = ǫ − 2t cos Θ.
+Before presenting the numerical results and discussion
+it is necessary to mention that uniform magnetic pertur-
+bation may also be applied within each rhombic plaque-
+tte. This can be feasible by an appropriate choice of the
+gauge. This can introduce additional externally tunable
+parameter which may lead to interesting band engineer-
+ing. This flux tunable localization of excitation will be
+discussed in the subsequent subsection.
+
+5
+A.
+Allowed eigenspectrum as a function of flux
+Now we analyze the impact of uniform magnetic per-
+turbation on the sustainability of the self-localized states.
+The magnetic flux is applied inside each embedded rhom-
+bic plaquette. As a result of this application of magnetic
+flux, the time reversal symmetry is broken (at least lo-
+cally) along the arm of the rhombic plaquette. This is
+considered by introducing a Peierls’ phase factor associ-
+ated with the hopping integral, viz., t → teiΘ, where,
+Θ = 2πΦ/4Φ0 and Φ0 = hc/e is termed as funda-
+mental flux quantum.
+The resultant nature of quan-
+tum interference happened due to multiple quantum dots
+is the ultimate determining factor for the sustainability
+of the self-localized modes after applying the perturba-
+tion. Here we have evaluated the allowed eigenspectrum
+(Fig. 5(a)) with respect to the applied flux for this flux
+included quasi-one dimensional diamond-Lieb geometry.
+The spectrum is inevitably flux periodic. Multiple band
+crossings, formation of several minibands and thus merg-
+ing of each other are seen in this quasi-continuous pat-
+tern.
+Here we should give emphasis on a pertinent issue.
+Fig. 5(b) shows a consistent demonstration of ampli-
+tude profile (satisfying the difference equation) for en-
+ergy E = ǫ − 2t cos Θ, ǫ being the uniform potential
+energy everywhere. One non-vanishing cluster is again
+isolated from the other by a physical barrier formed by
+the sites with zero amplitude as a direct consequence of
+phase cancellation at those nodes. This immediately tells
+us that the incoming electron coming with this particu-
+lar value of energy will be localized inside the trapping
+island. But now the energy eigenvalue is sensible to the
+applied flux which is an external agency. The central mo-
+tivation behind the application of this external parameter
+is that if possible, we may invite a comprehensive tun-
+ability of such bound states solely by manipulating the
+applied flux. We do not need to disturb any internal pa-
+rameter of the system, instead one can, in principle, con-
+trol the band engineering externally by a suitable choice
+of flux. The external perturbation can be tuned contin-
+uously satisfying the eigenvalue equation to control the
+position of the caged state.
+B.
+Density of states profile
+For the completeness of the analysis, we have com-
+puted the variation of density of states profile as a func-
+tion of energy of the incoming projectile for this quasi-one
+dimensional lattice with longer wavelength fluctuation
+using the standard green’s function technique both in
+the absence and presence of external perturbation. The
+variation with respect to the energy of the incoming pro-
+jectile for different values of magnetic flux is shown in the
+Fig. 6. The applied flux values are respectively Φ = 0,
+Φ = Φ0/4 and Φ = Φ0/2. All the variations are plot-
+ted for system size N = 753. As it is evident from the
+plots that there are different absolutely continuous sub-
+bands populated by extended kind of eigenfunctions. The
+existence of such dispersive modes is expected because
+of the inherent translational periodicity of the geometry.
+We have examined that for any mode belonging to the
+continuum zones the hopping integral shows oscillatory
+behavior which confirms the signature of the resonant
+modes. It is needless to say that the intricate nature of
+the DOS is highly sensitive on the external perturbation.
+Also the density of states plots as well as the allowed
+eigenspectrum support the existence of flux dependent
+caged state as discussed in the previous section.
+C.
+Band engineering
+In presence of uniform magnetic flux one can easily ex-
+press the Hamiltonian in the k-space language. The di-
+agonalization of this matrix will give the band dispersion
+as a function of flux. In this quasi-one dimensional dia-
+mond Lieb geometry we have got that, there are two flux
+independent dispersive bands E = ±
+�
+2(1 + cos ka) and
+three other flux sensible resonant bands. Therefore we
+should highlight a very pertinent issue here. For the last
+three flux dependent bands, one can easily control the
+group velocity of the wave train as well as the effective
+mass (equivalently the mobility) of the particle by tuning
+the external source of perturbation. This non-trivial ma-
+nipulation of the internal parameters of the system with
+the aid of flux makes this aspect of band engineering more
+challenging as well as interesting indeed.
+Before going to detailed discussion, it is important to
+be noted that, when an electron moves around a closed
+loop that traps a magnetic flux, the wave function picks
+up a phase related to the magnetic vector potential, viz.,
+ψ = ψ0ei
+�
+A.dr. The magnetic flux here plays an equiva-
+lent role as the wave vector [55]. One can thus think of a
+k−Φ/Φ0 diagram which is equivalent to a typical kx−ky
+diagram for electrons traveling in a two-dimensional pe-
+riodic lattice. The “Brillouin zone” equivalents are ex-
+pected to show up, across which variations of the group
+velocity will take place. This is precisely shown in the
+Fig. 7. In this plot, every contour presented corresponds
+to a definite value (positive or negative) of the group
+velocity of the wave packet. The red lines are the con-
+tours with zero mobility. Hence they are the equivalents
+of the boundaries of the Brillouin zone across which the
+group velocity reverts its sign if one moves parallel to
+the Φ-axis at any fixed value of the wave vector k, or vice
+versa. This essentially signifies that, we can, in principle,
+make an electron accelerate (or retard) without manipu-
+lating its energy by changing the applied magnetic flux
+only. The vanishing group velocity contours (marked by
+red) indicate that the associated wavefunctions are self-
+localized around finite size islands of atomic sites, making
+the eigenmode a non-dispersive one. As the curvature of
+the band is related to the mobility of the wave packet one
+can conclude from the Fig. 7 that tuning of the curva-
+
+6
+(a)
+�4
+�2
+0
+2
+4
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+E
+Ρ
+(b)
+�4
+�2
+0
+2
+4
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+E
+Ρ
+(c)
+�4
+�2
+0
+2
+4
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+E
+Ρ
+FIG. 6: (Color online) Variation of density of states ρ(E) as a function of energy E of the excitation. The external magnetic
+flux values are respectively (a) Φ = 0, (b) Φ = Φ0/4 and (c) Φ = Φ0/2.
+�2
+�1
+0
+1
+2
+�Π
+� Π
+2
+0
+Π
+2
+Π
+���0
+�
+FIG. 7: (Color online) k − Φ diagram showing different group
+velocity contours for electron moving in diamond embedded
+Lieb geometry. The red lines mark the zero group velocity
+of the wave packet. These red contours act as border lines
+showing a continuous change of vg with respect to flux.
+ture of the dispersive band is also possible with the help
+of external perturbation.
+IV.
+LIEB LADDER WITH QUASIPERIODIC
+NEXT NEAREST NEIGHBOR INTERACTION
+In the previous case the amplitude for E = 0 will be
+pinned at the top and down vertices of the diamond em-
+bedded. From this standpoint we now decorate each arm
+of the rhombic plaquette by a finite generation quasiperi-
+oidic fibonacci kind of geometry with two different hop-
+pings tx and ty respectively. The generation sequence for
+this quasiperiodic structure follows the standard inflation
+rule X → XY and Y → X. Based on this prescription
+regarding the anisotropy in off-diagonal term, there ex-
+ists three different types of atomic sites α (flanked by
+two X-bonds), β (in between X − Y pair) and γ (in be-
+tween Y − X pair).
+Here we should mention that we
+consider the generations with X type of bond at their
+extremities, i.e., G2n+1, (n being integer). This is only
+for convenience and does not alter the result Physics-wise
+as we go for thermodynamic limit.
+FIG. 8: (Color online) Distribution of self-localized modes
+showing a typical three-subband pattern for large enough gen-
+eration.
+Hence if we start with a odd generation Fibonacci seg-
+ment that decorates each arm of the diamond, then one
+can decimate the chain n-times by employing the RSRG
+method to get back the original diamond structure with
+renormalized parameters. The recursive flows of the pa-
+rameters are governed by the following equations, viz.,
+ǫα(n + 1) = ǫα(n) + t2
+x(n)
+∆(n)[2E − (ǫβ(n) + ǫγ(n))]
+ǫβ(n + 1) = ǫα(n) + (E − ǫβ(n))t2
+x(n)
+∆(n)
++
+t2
+x(n)
+(E − ǫβ(n))
+ǫγ(n + 1) = ǫγ(n) + (E − ǫγ(n))t2
+x(n)
+∆(n)
++
+t2
+y(n)
+(E − ǫβ(n))
+ǫC(n + 1) = ǫα(n) + 2t2
+x(n)
+∆(n) [2E − (ǫβ(n) + ǫγ(n))]
+tx(n + 1) = t2
+x(n)ty(n)
+∆(n)
+ty(n + 1) =
+tx(n)ty(n)
+(E − ǫβ(n))
+(8)
+where ∆(n) = [(E − ǫβ(n))(E − ǫγ(n))] − t2
+y(n)
+Obviously after decimation if we want to explore the
+same compact localized state (at E = ǫ) in this renor-
+malized lattice, then due to the iterative procedure, on-
+site potential is now a complicated function of energy.
+
+4
+***
+***
+3
++
++
+n
+2
+1
+0
+1
+1
+1
+1
+1
+1
+-3
+-2
+-1
+0
+2
+3
+1
+E7
+And if we now extract roots from the eigenvalue equa-
+tion (E−ǫα) = 0, all the roots will produce a multifractal
+distribution of the set of compact localized states. Obvi-
+ously as we increase the generation of the fibonacci struc-
+ture, in the thermodynamic limit, all the self-localized
+modes exhibit a global three subband structure. The pat-
+tern is already prominent in Fig. 8. Each subband can be
+fine scanned in the energy scale to bring out the inherent
+self-similarity and multifractality, the hallmark of the Fi-
+bonacci quasicrystals [56]. The self-similarity of the spec-
+trum have been checked by going over to higher enough
+generations, though we refrain from showing these data
+to save space here.
+V.
+LIEB LADDER WITH FRACTAL TYPE OF
+LONG RANGE CONNECTION
+FIG. 9: (Color online) An infinite array of Lieb strip with
+long range connectivity decorated by fractal object.
+We start this demonstration from the Fig. 9 where a fi-
+nite generation of self-similar Vicsek geometry [57, 58] is
+grafted inside the basic Lieb motif. The longer range con-
+nection is here established through the aperiodic object.
+Also a uniform magnetic flux Φ may be applied in each
+small plaquette of the fractal structure. It should be ap-
+preciated that while a Lieb geometry in its basic skeleton
+is known to support a robust type of central self-localized
+state, the inclusion of fractal structure of a finite genera-
+tion in each unit cell disturbs the translational ordering
+locally (though it is maintained on a global scale in the
+horizontal direction) in the transverse direction.
+This
+non-trivial competitive scenario makes the conventional
+methods of obtaining the self-localized states impossi-
+ble to be implemented, especially in the thermodynamic
+limit. We take the help of RSRG technique to bypass this
+issue and present an analytical formalism from which one
+can exactly determine the localized modes as a function
+of external flux. Starting from a finite generation of scale
+invariant fractal network, after suitable steps of decima-
+tion [57, 58] one can produce a Lieb ladder geometry
+with a diamond plaquette embedded into it (as discussed
+in the previous discussion). The renormalized potential
+of the top vertex of the diamond is now a complicated
+function of energy and flux. Therefore straightforward
+solving of the equation [E − ǫA(E, Φ)] = 0 gives us a in-
+teresting distribution of compact localized states in the
+E − Φ space.
+This non-trivial distribution of eigenvalues as a func-
+tion of flux may be considered an equivalent dispersion
+relation since for an electron moving round a closed path,
+FIG. 10: (Color online) Distribution of self-localized states
+with applied flux.
+the magnetic flux behaves the similar physical role as
+that of the wave vector [55]. The distribution of eigen-
+modes compose an interesting miniband-like structure as
+a function of external perturbation. The competition be-
+tween the global periodicity and the local fractal entity
+has a crucial impact on this spectrum. We can continu-
+ously engineer the magnetic flux to engineer the impris-
+onment of wave train with high selectivity.
+Moreover,
+there are a number of inter-twined band overlap, and a
+quite densely packed distribution of allowed modes, form-
+ing quasi-continuous E − Φ band structure. Close obser-
+vation of this eigenspectrum reveals the formation of in-
+teresting variants of the Hofstadter butterflies [59]. The
+spectral landscape is a quantum fractal, and encoding
+the gaps with appropriate topological quantum numbers
+remains an open problem for such deterministic fractals.
+Before ending this section we should put emphasis on
+a very pertinent point.
+An aperiodic fractal object is
+inserted in the unit cell of the periodic geometry. The
+self-similar pattern of the fractal entity will have the in-
+fluence on the spectrum. All such self-localized modes
+are the consequences of destructive quantum interfer-
+ence and the geometrical configuration of the underly-
+ing system. For this class of energy eigenvalue, the spa-
+tial span of the cluster of atomic sites containing non-
+vanishing amplitudes increases with the generation of the
+fractal geometry incorporated. Hence with an appropri-
+ate choice of the RSRG index n, the onset of localization
+and hence the spread of trapping island can be staggered,
+in space. This tunable delay of the extent of localization
+has already been studied for a wide varieties of fractal
+geometries [57, 58, 60, 61]. This comprehensive discus-
+sion regarding the manipulation of the geometry-induced
+localization makes the phenomenon of Aharonov-Bohm
+caging more interesting as well as challenging from the
+experimental point of view.
+
+-1.00
+0.75
+-0.50
+-0.25
+0.25
+0.50
+0.75
+0.00
+1.00
+Φ/Φ
+08
+(a)
+(b)
+1
+2
+N
+1
+2
+N
+Φ
+FIG. 11: (Color online) (a) Schematic diagram of elementary
+diamond-Lieb interferometer and (b) demonstrates the deco-
+ration of basic unit.
+VI.
+DIAMOND-LIEB INTERFEROMETER
+In this section we investigate the spectral character-
+istics of a quantum network in which each arm of the
+Lieb-ladder geometry is ‘decorated’ by diamond-shaped
+Aharonov-Bohm (AB) interferometer [37]. Each elemen-
+tary interferometer is pierced by a invariable magnetic
+perturbation applied perpendicular to the plane of the in-
+terferometer, and traps a flux Φ (in unit of Φ0 = hc/e).
+This type of diamond based interferometers have been
+formerly studied as the minimal prototypes of bipartite
+structures having nodes with different coordination num-
+bers, and representing a family of itinerant geometri-
+cally frustrated electronic systems [52]-[54]. We refer to
+Fig. 11(a). A standard diamond-Lieb AB interferome-
+ter is shown pictorially there whereas Fig. 11(b) demon-
+strates that each diamond loop can take a shape of a
+quantum ring consisting of multiple lattice points. Each
+arm of the diamond may be decorated by N number of
+atomic scatterers between the vertices, such that the to-
+tal number of single level quantum dots contained in a
+single interferometer is 4(N + 1). An uniform magnetic
+flux Φ may be allocated within each loop, and the elec-
+tron hopping is restricted to take the non-vanishing value
+for the nearest neighboring nodes only.
+To study the systematic spectral analysis we take the
+help of RSRG approach. Each elementary loop of the
+interferometer is properly renormalized to transform it
+into a simple diamond having just four sites.
+Due to
+this decimation process we will get three types of sites
+A, B and C (respectively marked by black, red and blue
+colored atomic sites in the Fig. 11(a)) with corresponding
+parameters given by
+˜ǫA = ǫ + 6tUN−1(x)
+UN(x)
+˜ǫB = ǫ + 4tUN−1(x)
+UN(x)
+˜ǫC = ǫ + 2tUN−1(x)
+UN(x)
+tF (B) = te±i(N+1)θ/UN(x)
+(9)
+Here, UN(x) is the N-th order Chebyshev polynomial of
+second kind, and x = (E −ǫ)/2t. The ‘effective’ diamond
+loops are then renormalized in a proper way (C types of
+sites are being decimated out) such that we will get back
+the Lieb ladder with renormalized on-site potential and
+overlap integral respectively given by
+˜ǫ4 =
+˜ǫB +
+4tFtB
+(E − ǫC)
+˜ǫ6 =
+˜ǫA +
+6tFtB
+(E − ǫC)
+˜t =
+2tF tB
+(E − ǫC)
+(10)
+We will now exploit all the above equations to extract the
+detailed information about the electronic spectrum and
+the nature of the eigenstates provided by such a model
+interferometer.
+A.
+Spectral landscape and inverse participation
+ratio
+To analyze we first put N = 0 here so that the quan-
+tum ring of elementary interferometer takes the form of a
+diamond (Fig. 11(a)). The density of states with energy
+for different values of magnetic flux enclosed within each
+elementary interferometer is shown in the upper panel of
+the Fig. 12. From the plot, we see that in absence of
+magnetic flux the density of states reflects the periodic
+nature of the geometry. It consists of absolutely contin-
+uous zones populated by resonant eigenstates with sharp
+spikes at E = 0 and ±2.
+But here it is to be noted
+that the localized character of those modes cannot be
+distinctly revealed because of its position within the con-
+tinuum of extended modes. But when we apply quarter
+flux quantum the central localized mode becomes iso-
+lated and prominent. It is also seen from the plots that
+with the gradual increment of flux value the window of
+resonant modes in the DOS profile shrinks along the en-
+ergy scale and ultimately leads to extreme localization of
+eigenstates for half flux quantum.
+Actually, the effec-
+tive overlap parameter between the two axial extremities
+of the interferometer vanishes for this special flux value
+and this makes the complete absence of resonant modes
+to be possible. This is the basic physical background of
+extreme localization of excitation. We should appreciate
+that this typical flux induced localization of wave train
+inside a charateristic trapping island is a subset of the
+usual Aharonov-Bohm caging [27]
+For the sake of completeness of the discussion related
+to the spectral property of such quantum interferometer
+model, we have also calculated the inverse participation
+ratio (IPR) to certify the above density of states plots.
+To formulate the localization of a normalized eigenstate
+the inverse participation ratio is defined as
+I =
+L
+�
+i=1
+|ψi|4
+(11)
+
+9
+(a)
+�4
+�2
+0
+2
+4
+0
+0.2
+0.4
+0.6
+0.8
+1
+E
+Ρ
+(b)
+�4
+�2
+0
+2
+4
+0
+0.2
+0.4
+0.6
+0.8
+1
+E
+Ρ
+(c)
+�4
+�2
+0
+2
+4
+0
+0.2
+0.4
+0.6
+0.8
+1
+E
+Ρ
+(d)
+�4
+�2
+0
+2
+4
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+0.035
+0.040
+E
+IPR
+(e)
+�4
+�2
+0
+2
+4
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+E
+IPR
+(f)
+�4
+�2
+0
+2
+4
+0.1
+0.2
+0.3
+0.4
+0.5
+E
+IPR
+FIG. 12: (Color online) (Upper panel) Variation of density of states ρ(E) as a function of energy E of the excitation and
+(lower panel) indicates the variation of inversion participation ratio (IPR) wth energy. The external magnetic flux values are
+respectively (a) Φ = 0, (b) Φ = Φ0/4 and (c) Φ = Φ0/2.
+It is known that for an extended mode IPR goes as 1/L,
+but it approaches to unity for a localized state. The lower
+panel of Fig. 12 describes the variation of IPR with the
+energy of the injected projectile for different flux values.
+It is evident from the plots that the IPR supports the
+spectral profile cited in the upper panel of Fig. 12. As
+we see that with nominal strength of perturbation the
+central gap opens up around E = 0, clearly indicating
+the central localized mode.
+The shrinking of resonant
+window with the gradual increment of flux is also ap-
+parent from the IPR plots. It is also interesting to ap-
+preciate that for half flux quantum IPR plot (Fig. 12f)
+also demonstrates the AB-caging leading to the extreme
+localization of eigenstates.
+B.
+Flux dependent eigenspectrum
+�2
+�1
+0
+1
+2
+�3
+�2
+�1
+0
+1
+2
+3
+���0
+E
+FIG. 13: (Color online) Flux dependent allowed eigenspec-
+trum for the diamond-Lieb AB-interferometer model.
+The
+pattern is flux periodic.
+Fig. 13 represents the essential graphical variation of
+allowed eigenspectrum for a diamond-Lieb AB interfer-
+ometer with N = 0 with respect to the external magnetic
+flux. With the increment of N, the number of scatterers
+in each elementary interferometer, the spectrum will be
+densely packed with several band crossings. The present
+variation is seen to be flux periodic of periodicity equal
+to one flux quantum. It is needless to say that the eigen-
+spectrum is inevitably sensitive to the numerical values
+of the parameters of the Hamiltonian. However, the pe-
+riodicity retains for such spectrum after every single flux
+quantum change of the external perturbation.
+It is observed that there is a tendency of clustering of
+the allowed eigenvalues towards the edges of the eigen-
+spectrum as is clear from the above-mentioned diagram.
+A number of band crossings are noticed and the spec-
+trum cites kind of a zero band gap semiconductor like
+behavior, mimicking Dirac point as observed in case of
+graphene, at Φ/Φ0 = ±i, i being an integer including
+zero. As we increase the complexity in each interferom-
+eter by increasing N, the central gap gets consequently
+filled up by more eigenstates, and the E −Φ contours get
+more flattened up forming a quasi-continuous spectrum,
+exotic in nature. The central eigenstate corresponding to
+eigenvalue E = 0 is a robust kind of mode irrespective of
+the application of perturbation, i.e., the existence of that
+state is insensitive to the value of the external flux. More-
+over, when the magnetic flux is set as Φ = (i+1/2)Φ0, we
+observe a spectral collapse. In that case one can easily
+identify the localization character of the central state.
+Most importantly, it is evident from the spectral land-
+scape that the it consists of a set of discrete points (eigen-
+values) for half flux quantum. This is the canonical case
+of extreme localization. For such special flux value the
+vanishing overlap parameter makes the geometry equiv-
+alent to discrete set of lattice points with zero connectiv-
+ity between them. This makes the excitation to be caged
+
+10
+within the trapping island. Further it is to be noted that
+this AB-caging [27] may happen for any value of N, the
+number of eigenvalues in the discrete set depends on the
+choice of N.
+VII.
+CLOSING REMARKS
+A methodical analysis of the flux induced tunable
+caging of excitation in a quasi-one dimensional Lieb net-
+work with long range connectivity is reported in this
+manuscript within the tight-binding framework.
+With
+the inclusion of second neighbor overlap integral in a dec-
+orated way, external source of perturbation can act as an
+important role for the selective caging of wave packet.
+Flux dependent band engineering and hence the com-
+prehensive control over the group velocity of the wave
+train as well as the band curvature are studied in de-
+tails. Decoration of the next nearest neighbor hopping
+in certain quasiperiodic fashion or by some determinis-
+tic fractal object is also demonstrated analytically. Real
+space renormalization group approach provides us a suit-
+able platform to obtain an exact prescription for the de-
+termination of self-localized modes induced by destruc-
+tive quantum interference effect. As we have seen that
+in the quasiperiodic Fibonacci variation the distribution
+of eigenstates shows a standard three-subband pattern
+while in case of fractal entity countably infinite number
+of localized modes cite an interesting quasi-continuous
+distribution against flux. We have also critically studied
+the spectral properties of a diamond Lieb interferome-
+ter. The energy spectrum shows an exotic feature com-
+prising extended, staggered and edge-localized eigenfunc-
+tions. The number of such states depend on the number
+of quantum dots present in each arm of the elementary
+diamond interferometer, and can populate the spectral
+landscape as densely as desired by the experimentalists.
+A constant magnetic perturbation can be utilized to con-
+trol the positions of all such states. Moreover at special
+flux value the spectrum describes the Aharonov-Bohm
+caging of eigenstates leading to an interesting spectral
+collapse.
+Acknowledgments
+The author is thankful for the stimulating discussions
+regarding the results with Dr. Amrita Mukherjee. The
+author also gratefully acknowledges the fruitful discus-
+sion made with Prof. A. Chakrabarti.
+[1] S. Mukherjee, A. Spracklen, D. Choudhury, N. Goldman,
+P. ¨Ohberg, E. Andersson, and R. R. Thomson, Phys. Rev.
+Lett. 114, 245504 (2015).
+[2] S. Mukherjee and R. R. Thomson, Opt. Lett. 40, 5443
+(2015).
+[3] R. A. Vicencio, C. Cantillano, L. Morales-Inostroza, B.
+Real, C. Mej´ıa-Cort`es, S. Weimann, A. Szameit, and M.
+I. Molina Phys. Rev. Lett. textbf114 245503 (2015).
+[4] P. W. Anderson, Phys. Rev. 109, 1492 (1958).
+[5] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys.
+80, 885 (2008).
+[6] D. N. Christodoulides, F. Lederer, and Y. Silberberg,
+Nature, 424, 817 (2003).
+[7] N. Masumoto, N. Y. Kim, T. Byrnes, K. Kusudo, A.
+L¨offler, S. H¨ofling, A. Forchel, and Y. Yamamoto, New.
+J. Phys. 14, 065002 (2012).
+[8] R.G. Dias, J.D. Gouveia, Sci. Rep. 5 16852 (2015).
+[9] M. Hyrk¨as, V. Apaja, M. Manninen, Phys. Rev. A 87,
+023614 (2013).
+[10] L. Morales-Inostroza, R.A. Vicencio, Phys. Rev. A 94,
+043831 (2016).
+[11] A. Ramachandran, A. Andreanov, S. Flach, Phys. Rev.
+B 96, 161104(R) (2017).
+[12] S. Flach, D. Leykam, J. D. Bodyfelt, P. Matthies, A. S.
+Desyatnikov, Europhys. Lett. 105, 30001 (2014).
+[13] J. D. Bodyfelt, D. Leykam, C. Danieli, X. Yu, S. Flach,
+Phys. Rev. Lett. 113, 236403 (2014).
+[14] W. Maimaiti, A. Andreanov, H. C. Park, O. Gendelman,
+S. Flach, Phys. Rev. B 95, 115135 (2017).
+[15] D. Leykam, A. Andreanov, S. Flach, Adv. Phys. X 3,
+1473052 (2018).
+[16] B. Sutherland, Phys. Rev. B 34, 5208 (1986).
+[17] M. Goda, S. Nishino, and H. Matsuda, Phys. Rev. Lett.
+96, 126401 (2006).
+[18] J. T. Chalker, T. S. Pickles, and P. Shukla, Phys. Rev.
+B 82, 104209 (2010).
+[19] H. Tasaki, Phys. Rev. Lett. 69, 1608 (1992).
+[20] M. Maksymenko, A. Honecker, R. Moessner, J. Richter,
+and O. Derzhko, Phys. Rev. Lett. 109, 096404 (2012).
+[21] V. J. Kauppila, F. Aikebaier, and T. T. Heikkil¨a, Phys.
+Rev. B 93, 214505 (2016).
+[22] S. Peotta and P. T¨orm¨a, Nat. Comm. 6, 8944 (2015).
+[23] Y.-F. Wang, Z.-C. Gu, C.-D. Gong, and D. N. Sheng,
+Phys. Rev. Lett. 107, 146803 (2011).
+[24] K. Kusakabe and H. Aoki, Phys. Rev. Lett. 72, 144
+(1994).
+[25] A. Mielke J. Phys. A-Math. Gen. 25, 4335 (1992).
+[26] E. H. Lieb, Phys. Rev. Lett. 62, 1201 (1989).
+[27] J. Vidal, R. Mosseri, and B. Doucot, Phys. Rev. Lett.
+81, 5888 (1998).
+[28] Y. Aharonov and D. Bohm, Phys. Rev. B 115, 485
+(1959).
+[29] S. Washburn and R. A. Webb, Adv. Phys. 35, 375 (1986).
+[30] M. B¨uttiker, Y. Imry, and M. Ya. Azbel, Phys. Rev. A
+30, 1982 (1984).
+[31] R. Landauer and M. B¨uttiker, Phys. Rev. Lett. 54, 2049
+(1985).
+[32] A. Levy Yeyati and M. B¨uttiker, Phys. Rev. B 52,
+R14360 (1995).
+[33] A. Yacoby, M. Heiblum, V. Umansky, H. Shtrikman, and
+D. Mahalu, Phys. Rev. Lett. 73, 3149 (1994).
+[34] A. Aharony, O. Entin-Wohlman, and Y. Imry, Phys. Rev.
+Lett. 90, 156802 (2003).
+[35] T. Kubo, Y. Tokusain, and S. Tarucha, J. Phys. A: Math.
+
+11
+Theo. 43, 354020 (2010).
+[36] M. Yamamoto, S. Takada, C. B¨auerle, K. Watanabe,
+A. D. Wieck, and S. Tarucha, Nature Nanotech. 7, 247
+(2012).
+[37] A. Aharony, S. Takada, O. Entin-Wohlman, M. Ya-
+mamoto, and S. Tarucha, New J. Phys. 16, 083015
+(2014).
+[38] J.S. Andrade Jr., H.J. Herrmann, R.F.S. Andrade, L.R.
+da Silva, Phys. Rev. Lett. 94, 018702 (2005).
+[39] A.L. Cardoso, R.F.S. Andrade, A.M.C. Souza, Phys.
+Rev. B 78, 214202 (2008).
+[40] A.A. Lopes, R.G. Dias, Phys. Rev. B 84, 085124 (2011).
+[41] D. Bercioux, M. Governale, V. Cataudella, V.M. Ra-
+maglia, Phys. Rev. Lett. 93, 056802 (2004).
+[42] S. Sil, S.K. Maiti, A. Chakrabarti, Phys. Rev. B 79,
+193309 (2009).
+[43] J.L. Movilla, J. Planelles, Phys. Rev. B 84 195110 (2011).
+[44] A. Aharony, Y. Tokura, G.Z. Cohen, O. Entin-Wohlman,
+S. Katsumoto, Phys. Rev. B 84 035323 (2011).
+[45] P.O. Lowdin, J. Mol. Spectrosc. 10, 12 (1963).
+[46] P.O. Lowdin, J. Math. Phys. 3, 969 (1962).
+[47] S. Datta, Electronic Transport in Mesoscopic Systems,
+Cambridge University Press, Cambridge, 1997.
+[48] S. Datta, Quantum Transport:
+Atom to Transistor,
+Cambridge University Press, Cambridge, 2005.
+[49] P. Dutta, S.K. Maiti, S.N. Karmakar, AIP Adv. 4,
+097126 (2014).
+[50] P. Dutta, S.K. Maiti, S.N. Karmakar, Org. Electron. 11
+1120 (2010).
+[51] P. Dutta, S.K. Maiti, S.N. Karmakar, J. Appl. Phys. 114
+034306 (2013).
+[52] Z. Gul´acsi, A. Kampf, and D. Vollhardt, Phys. Rev. Lett.
+99, 026404 (2007).
+[53] A. A. Lopes and R. G. Dias, Phys. Rev. B 84, 085124
+(2011).
+[54] A. A. Lopes, B. A. Z. Ant´onio, and R. G. Dias, Phys.
+Rev. B 89, 235418 (2014).
+[55] H.-F. Cheung, Y. Gefen, E. K. Riedel, W.-H. Shih, Phys.
+Rev. B 37, 6050 (1988).
+[56] M. Kohmoto, B. Sutherland, C. Tang, Phys. Rev. B 35,
+1020 (1987).
+[57] B. Pal, P. Patra, J. P. Saha, and A. Chakrabarti, Phys.
+Rev. A 87, 023814 (2013).
+[58] B. Pal and A. Chakrabarti, Phys. Rev. B 85, 214203
+(2012).
+[59] D. R. Hofstadter, Phys. Rev. B 14, 2239 (1976).
+[60] A. Nandy, B. Pal, and A. Chakrabarti, J. Phys.: Con-
+dens. Matt. 27, 125501 (2015).
+[61] A. Nandy, Phys. Scr. 96, 045802 (2021).
+

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+Analytical marginalisation over photometric redshift uncertainties in
+cosmic shear analyses
+Jaime Ruiz-Zapatero1 ★, Boryana Hadzhiyska2,3, David Alonso1, Pedro G. Ferreira1, Carlos García-García1
+and Arrykrishna Mootoovaloo1
+1Department of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK
+2Miller Institute for Basic Research in Science, University of California, Berkeley, CA, 94720, USA.
+3Physics Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720.
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+As the statistical power of imaging surveys grows, it is crucial to account for all systematic uncertainties. This is normally
+done by constructing a model of these uncertainties and then marginalizing over the additional model parameters. The resulting
+high dimensionality of the total parameter spaces makes inferring the cosmological parameters significantly more costly using
+traditional Monte-Carlo sampling methods. A particularly relevant example is the redshift distribution, 𝑝(𝑧), of the source
+samples, which may require tens of parameters to describe fully. However, relatively tight priors can be usually placed on these
+parameters through calibration of the associated systematics. In this paper we show, quantitatively, that a linearisation of the
+theoretical prediction with respect to these calibratable systematic parameters allows us to analytically marginalise over these
+extra parameters, leading to a factor ∼ 30 reduction in the time needed for parameter inference, while accurately recovering the
+same posterior distributions for the cosmological parameters that would be obtained through a full numerical marginalisation
+over 160 𝑝(𝑧) parameters. We demonstrate that this is feasible not only with current data and current achievable calibration
+priors but also for future Stage-IV datasets.
+Key words: cosmology: large-scale structure of Universe – gravitational lensing: weak – methods: data analysis
+1 INTRODUCTION
+In recent years unprecedentedly precise observations in cosmology
+have uncovered a number of tensions between datasets that may
+constitute both tantalising hints of new physics or a manifestation of
+a lack of control over theoretical systematics (Heymans et al. 2021;
+Riess et al. 2022).
+At its simplest, the current cosmological paradigm, the Λ (denoting
+the cosmological constant) cold dark matter model (ΛCDM), can be
+described by only five parameters: Ω𝑚, Ω𝑏, 𝐴𝑠, 𝑛𝑠 and ℎ (see e.g.
+Scott (2018) for a detailed review). However, in order to relate the
+theoretical predictions of this model to actual physical observables,
+it is necessary to extend it. Phenomenological models that describe
+the astrophysical systems that form the basis of our observations,
+as well as observational sources of systematic uncertainty, are then
+appended to the core ΛCDM model. In the presence of large statistical
+uncertainties, these models may consist of simple relationships in
+terms of a handful of parameters. However, more precise data requires
+an equally precise characterisation of these relationships, which leads
+to an increase in the complexity of the model. Thus, the number
+of parameters associated with these bridging models, colloquially
+referred to as “nuisance” parameters, has steadily grown over the
+years.
+The term “nuisance” is accurate when describing these parameters.
+★ E-mail: [email protected]
+Not only are they generally uninteresting by comparison with the
+fundamental cosmological parameters we aim to constraint, but the
+increase in parameter dimensionality of the model makes exploring
+their posterior distribution significantly more computationally costly.
+Standard Markov Chain Monte-Carlo (MCMC), and other rejection-
+based sampling methods (Metropolis et al. 1953; Foreman-Mackey
+et al. 2013; Alsing & Handley 2021, among others) suffer from the
+so-called “curse of dimensionality”, whereby the acceptance rate
+of new samples decreases sharply with the number of parameters
+(exponentially in the worst cases).
+Nuisance parameters can be divided into two groups based on their
+prior distributions: calibratable and non-calibratable parameters. The
+non-calibratable parameters can only be constrained by the data and,
+as such, typically have largely non-constraining priors. On the other
+hand, we can place tighter priors on the calibratable parameters,
+either by accurately characterising the instrument measurements or
+by using independent external observations. In the case of cosmic
+shear analyses, the impact of galaxy intrinsic alignments (Hirata &
+Seljak 2004) is a standard example of a non-calibratable systematic.
+On the calibratable side, the two best examples are multiplicative
+shape measurement systematics, and the uncertainties in the redshift
+distribution of the target source galaxies (Hoyle et al. 2018; Sánchez
+& Bernstein 2019; Hildebrandt et al. 2020a; Stölzner et al. 2021;
+Zhang et al. 2023).
+Of these calibratable systematics the dominant source of uncer-
+tainty in photometric surveys is the accuracy of redshift distributions,
+© 2022 The Authors
+arXiv:2301.11978v1  [astro-ph.CO]  27 Jan 2023
+
+2
+Ruiz-Zapatero et al.
+which are known to strongly affect the accuracy of cosmological con-
+straints. The vital quantity to determine is the redshift distribution
+of each tomographic sample of galaxies, 𝑝(𝑧). The fact that the un-
+certainties in 𝑝(𝑧) can be calibrated with external spectroscopic data
+(e.g. via direct calibration, (Lima et al. 2008; Wright et al. 2020),
+clustering redshifts (Schneider et al. 2006; Newman 2008; Matthews
+& Newman 2010; Schmidt et al. 2013), and shear ratios (Prat et al.
+2018; Sánchez et al. 2022)), enables us to place relatively strong
+priors on the redshift distribution, which in turn makes it possible to
+use approximate methods to efficiently marginalise over these uncer-
+tainties.
+Analytical marginalisation schemes for photometric redshift un-
+certainties have already been proposed in the literature. In Stölzner
+et al. (2021) an analytic marginalisation scheme for photometric red-
+shift uncertainties was proposed based on Gaussian mixture mod-
+els and applied to the analysis of KV450 data (Hildebrandt et al.
+2020b). Alternatively, in Zhang et al. (2023) a resampling approach
+to marginalize over these uncertainties was proposed and applied to
+the analysis HSC data. Here, we will explore the method initially pro-
+posed in Hadzhiyska et al. (2020), further exploited in García-García
+et al. (2023), and recently characterised in the context of the Laplace
+approximation in Hadzhiyska et al. (2023). The method is based on
+linearising the dependence of the theoretical prediction with respect
+to the parameters defining the redshift distribution around their cali-
+bration priors. This then allows one to analytically marginalise over
+these parameters by modifying the covariance matrix of the data,
+effectively assigning higher variance (as allowed by the calibration
+prior) to the data modes most sensitive to variations in the 𝑝(𝑧).
+The goal of this paper is to exhaustively validate this approximate
+marginalisation scheme in the context of cosmic shear analyses. We
+will do so by proving that we are able to obtain the same constraints
+on cosmological parameters using this scheme, as well as employing
+brute-force methods that sample the full parameter space exactly. We
+will show this for both simple parametrisations of the 𝑝(𝑧) uncertain-
+ties, in terms of shifts to the mean of the distribution, as well as using
+completely general “non-parametric” models that treat the amplitude
+of the 𝑝(𝑧) in narrowly-spaced intervals of 𝑧 as calibratable variables,
+leading to a model with more than ∼ 100 nuisance parameters. In or-
+der to numerically marginalize over such large parameter spaces we
+develop an auto-differentiable code to obtain theoretical predictions
+for the cosmic shear observables. This allows us to employ gradi-
+ent based sampling algorithms, such as Hamiltonian Monte Carlo,
+to beat the aforementioned curse of dimensionality. Finally, we will
+show that the method is valid not only for current data, but also for
+futuristic Stage-IV surveys, where photometric redshift uncertainties
+will likely make up a large fraction of the total error budget. Inter-
+estingly, our analysis will show that, in the context of cosmic shear
+data, relatively inexpensive parametrisations of photometric redshift
+uncertainties based on one free parameter per redshift bin (e.g. mean
+shifts, or ranked discrete realisations (Cordero et al. 2022)), return ef-
+fectively the same posterior distribution on cosmological parameters
+as the most general non-parametric models.
+This paper is structured as follows. In Section 2 we describe the
+methods used in this work including the theory behind weak lens-
+ing observables, the calibration of redshift distributions, and the
+mathematics of analytical marginalisation via first-order expansion.
+Section 3 presents the Dark Energy Survey data used to produce real-
+istic source redshift distributions and their associated uncertainties,
+as well as the models used to simulate future datasets. In Section
+4 we describe the likelihood used to analyse these data, as well as
+the different parametrisations used to describe 𝑝(𝑧) uncertainties.
+Section 5 presents our results, quantifying the performance of ana-
+lytical marginalisation methods. Finally, we present our conclusions
+in Section 6.
+2 METHODS
+2.1 Cosmic shear power spectra
+It is now commonplace to carry out the analysis of galaxy weak
+lensing data tomographically. The full sample is split into redshift
+bins and the two-point correlation functions of all pairs of bins are
+measured and compared with their theoretical prediction. Let 𝛾𝛼( ˆn)
+be a map of the spin-2 lensing shear field inferred from the sources in
+the ���-th redshift bin. Its relation with the three-dimensional matter
+overdensity 𝛿𝑚(x) is (Bartelmann & Schneider 2001; Krause et al.
+2017)
+𝛾𝛼( ˆn) =
+∫ 𝜒𝐻
+0
+𝑑𝜒 𝑞𝛼(𝜒)
+�
+−𝜒−2ðð∇−2𝛿𝑚(𝜒ˆn, 𝑧)
+�
+,
+(1)
+where ˆn is the sky direction, 𝜒 is the comoving radial distance at
+redshift 𝑧, 𝜒𝐻 is the distance to the horizon, 𝑞𝛼(𝜒) is the weak
+lensing radial kernel, and ð is the spin-raising differential operator,
+acting on a spin-𝑠 quantity as (Newman & Penrose 1966):
+ð 𝑠 𝑓 (𝜃, 𝜑) = −(sin 𝜃)𝑠
+� 𝜕
+𝜕𝜃 +
+𝑖
+sin 𝜃
+𝜕
+𝜕𝜑
+�
+(sin 𝜃)−𝑠 𝑠 𝑓
+(2)
+and turning it into a spin-(𝑠 + 1) quantity. The weak lensing kernel
+is1
+𝑞𝛼(𝜒) ≡ 3
+2 𝐻2
+0Ω𝑚
+𝜒
+𝑎(𝜒)
+∫ ∞
+𝑧(𝜒)
+𝑑𝑧′𝑝𝛼(𝑧′) 𝜒(𝑧′) − 𝜒
+𝜒(𝑧′)
+,
+(3)
+where 𝐻0 ≡ 𝐻(𝑧 = 0) is the Hubble expansion rate today, Ω𝑚 is
+the current matter density parameter and, most importantly for our
+discussion, 𝑝𝛼(𝑧) is the redshift distribution in bin 𝛼,
+The angular power spectrum of the 𝐸-mode components of two
+maps 𝛼 and 𝛽, 𝐶 𝛼𝛽
+ℓ
+can be related to the three-dimensional matter
+power spectrum 𝑃(𝑘, 𝑧) via:
+𝐶 𝛼𝛽
+ℓ
+= 𝐺2
+ℓ
+∫
+𝑑𝜒
+𝜒2 𝑞𝛼(𝜒) 𝑞𝛽(𝜒) 𝑃
+�
+𝑘 = ℓ + 1/2
+𝜒
+, 𝑧(𝜒)
+�
+,
+(4)
+where we have assumed the Limber approximation (Limber 1953;
+Afshordi et al. 2004), which is valid for the broad weak lensing ker-
+nels considered in this work. The scale-dependent lensing prefactor,
+𝐺ℓ ≡
+√︄
+(ℓ + 2)!
+(ℓ − 2)!
+1
+(ℓ + 1/2)2 ,
+(5)
+accounts for the difference between angular and three-dimensional
+derivatives in Eq. 1 (i.e. 𝜒2ð2∇−2 � 1). This prefactor leads to
+sub-percent differences for ℓ > 11 and can therefore be neglected
+on small scales (Kilbinger et al. 2017). In this work we will use
+the Halofit fitting function of Smith et al. (2003); Takahashi et al.
+(2012) to describe the matter power spectrum.
+The intrinsic alignment (IA) of galaxies due to local interactions
+(gravitational or otherwise), is an important contaminant for cosmic
+shear data that must be taken into account (Brown et al. 2002). For
+simplicity, however, and since the focus of this work is the impact of
+the marginalisation over redshift distribution uncertainties, we will
+ignore the contribution from intrinsic alignments in this analysis.
+1 Note that this is only strictly valid in ΛCDM (Ferreira 2019).
+MNRAS 000, 1–11 (2022)
+
+Analytical marginalisation over photo-𝑧 uncertainties
+3
+2.2 Redshift distribution uncertainties
+The sub-samples that make up the redshift bins used in the tomo-
+graphic cosmic shear analysis of an imaging survey are selected
+based on the source photometry, either by simple cuts in the in-
+ferred photometric redshifts (photo-𝑧), or by selecting directly in the
+magnitude-color space of the sample, bypassing photo-𝑧 estimation
+altogether. Regardless of the method used to select the sub-samples,
+their true redshift distributions are inevitably subject to some level
+of uncertainty, due to the lack of precise redshift measurements.
+The 𝑝(𝑧) can however be calibrated through various methods, e.g.:
+weighted direct calibration with a sufficiently complete spectroscopic
+sample (Lima et al. 2008; Wright et al. 2020), clustering redshifts
+(Schneider et al. 2006; Newman 2008; Matthews & Newman 2010;
+Schmidt et al. 2013), and shear ratios (Prat et al. 2018; Sánchez et al.
+2022). This typically leads to relatively tight priors on the 𝑝(𝑧), but
+the residual uncertainties in this prior must be propagated into the
+final parameter constraints.
+To characterise these uncertainties, we will make use of two dif-
+ferent methods, which encompass the range of model complexity we
+may reasonably expect from current and future data.
+• Method 1: 𝑧 shifts. Most cosmic shear analyses to date
+(Miyazaki et al. 2012; Hildebrandt et al. 2020b; Heymans et al.
+2021; Abbott et al. 2018a, 2022, among others) have summarised
+the uncertainty in the calibrated 𝑝𝛼(𝑧) into a single parameter Δ𝑧𝛼
+that shifts the mean of the redshift distribution. I.e. let ˆ𝑝𝛼(𝑧) be the
+best-guess redshift distribution. The true redshift distribution is then
+𝑝𝛼(𝑧) = ˆ𝑝𝛼(𝑧 + Δ𝑧𝛼).
+(6)
+A prior on Δ𝑧𝛼 can be derived using the calibration methods listed
+above. We will refer to this method as parametric.
+This simple model turns out to be relatively well suited to describe
+the impact of 𝑝(𝑧) uncertainties in the case of cosmic shear data.
+Since weak lensing is a radially cumulative effect, the amplitude
+of the weak lensing kernel (Eq. 3) is mostly sensitive to the mean
+redshift of the sample, and thus much of the effect on cosmic shear
+observables is well described by this parameter (Bonnett et al. 2016).
+Other modes of 𝑝(𝑧) uncertainty, such as the distribution width,
+may be more relevant for galaxy clustering observables, or for the
+intrinsic alignment contribution to cosmic shear. Near-future cosmic
+shear samples may indeed require a more sophisticated description
+of the 𝑝(𝑧) uncertainty, and thus we turn to a more general method.
+• Method 2: 𝑝(𝑧) bin heights. Most 𝑝(𝑧) calibration methods
+(e.g. direct calibration or clustering redshifts) will produce a binned
+measurement of the 𝑝(𝑧) with deterministic redshift bin ranges, and
+uncertain bin heights. The most general method to propagate these
+uncertainties is therefore to treat each bin height 𝑝𝑖 ≡ 𝑝(𝑧𝑖) as
+a free parameter in the model, with a prior given by the calibration
+uncertainties. The latter may be in the form of individual 1𝜎 errors for
+each bin height, if the uncertainties are approximately uncorrelated,
+or a full covariance matrix covering all bin heights.
+The resulting parametrisation thus sidesteps any attempt at sum-
+marising the uncertainty into effective parameters, and thus we will
+refer to this method as non-parametric. The method therefore fully
+propagates all calibration uncertainties into the final constraints with
+minimal approximations.
+The key practical difference between both methods, in the context
+of error propagation, is the additional complexity they incur. The
+parametric approach (Method 1) introduces one free parameter per
+redshift bin. For 𝑂(5) bins, this is already enough to significantly
+impact the performance of standard MCMC algorithms. In turn, the
+non-parametric approach (Method 2) introduces tens or hundreds of
+parameters per redshift bin, and one must resort to advanced sam-
+pling methods in order to fully explore the resulting model without
+assumptions.
+2.3 Linearisation and analytical marginalisation
+Let 𝛀 be the set of non-calibratable parameters of a model (in our
+case this is the set of cosmological and non-calibratable nuisance
+parameters) and 𝝂 the set of calibratable parameter such that the total
+set of parameters is given by 𝜽 = 𝛀 ∪ 𝝂. Now consider the general
+case of a Gaussian posterior distribution of the form
+−2 log 𝑃(𝛀, 𝝂|d) =(d − t)𝑇 C−1(d − t) + (𝝂 − ¯𝝂)𝑇 P−1(𝝂 − ¯𝝂)
+− 2 log 𝑃(𝛀) + const.,
+(7)
+where d is the data. We assume a Gaussian calibration prior with
+mean ¯𝝂 and covariance P, while 𝑃(𝛀) is the prior on 𝛀 (which is, as
+per our assumption, broad). t(𝛀, 𝝂) is the theoretical prediction for
+the data d which implicitly depends on both calibratable and non-
+calibratable parameters. C is the covariance matrix of d, which is
+parameter-independent.
+Assuming a tight prior on 𝝂, we start by expanding the theory
+prediction around ¯𝝂
+t ≃ ¯t + T(𝝂 − ¯𝝂),
+where ¯t ≡ t(𝛀, ¯𝝂),
+T ≡ 𝑑t
+𝑑𝝂
+����𝝂=¯𝝂
+.
+(8)
+Substituting this approximation in Eq. 7, the posterior becomes Gaus-
+sian in 𝝂, and thus the calibratable parameters can be marginalised
+analytically. As shown in Hadzhiyska et al. (2020), the resulting
+marginalised posterior is
+−2 log 𝑃(𝛀|d) ≃(d − ¯t)𝑇 ˜C−1(d − ¯t) − 2 log 𝑃(𝛀)
++ log
+�
+det
+�
+T𝑇 C−1T + P−1��
++ const.,
+(9)
+where the modified covariance is
+˜C ≡ C + TPT𝑇 .
+(10)
+Note that, strictly speaking, both the modified covariance and the
+term in the second line of Eq. 9 depend on𝛀, which would in principle
+complicate the evaluation of the likelihood. In practice, thisparameter
+dependence can be neglected such that the value of 𝛀 at which these
+terms are evaluated can be fixed during exploration of the posterior.
+However, fixing 𝛀 at values with a bad fit to the data will result
+in a mischaracterisation of the response of the theory vector to the
+nuisance parameters leading to inaccurate marginalised posteriors.
+Ideally, 𝛀 is fixed to its maximum a posteriori (MAP) value. However,
+as shown in Hadzhiyska et al. (2020) and in preliminary results, no
+appreciable differences are found in the marginalised posteriors for
+𝛀 within 2𝜎 of the MAP. Note that the size of the 2𝜎 region will
+depend on how constraining the data is.
+This result is intuitively simple to understand if we think of T as
+the response of the data to variations in the nuisance parameters.
+After marginalising over the calibratable parameters, the resulting
+distribution is a multi-variate Gaussian where the data covariance
+has been updated in Eq. 10 by increasing the uncertainty in the data
+modes that most prominently respond to variations in the nuisance
+parameters.
+In this work, 𝝂 corresponds to the parameters describing the red-
+shift distribution uncertainties, i.e. one shift parameter per redshift
+bin when using the parametric approach, or a set of 𝑝(𝑧) bin heights in
+the non-parametric scheme. The method described above, however,
+MNRAS 000, 1–11 (2022)
+
+4
+Ruiz-Zapatero et al.
+0.00
+0.05
+0.10
+0.15
+0.20
+p(z)0
+p(z)1
+p(z)2
+p(z)3
+0.5
+1.0
+1.5
+z
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+z
+0.5
+1.0
+1.5
+z
+0.5
+1.0
+1.5
+z
+0.5
+1.0
+1.5
+z
+10
+3
+10
+2
+10
+1
+100
+Figure 1. Top row: normalized galaxies’s redshift distributions for each of the 4 redshift bins. Bottom row: correlation matrix obtained using the DIR algorithm
+for each of the 4 galaxies’ redshift distributions. Note that for visualization purposes we display the absolute values of the each correlation matrix in logarithmic
+scale. In this plot we can see that the covariance matrices obtained through the DIR algorithm are mostly diagonal.
+is fully general and has in the past been applied to marginalise over
+other types of nuisance parameters, including multiplicative shape
+measurement biases (Hildebrandt et al. 2020b), as well as truly linear
+parameters such as shot-noise (García-García et al. 2021) or system-
+atic template amplitudes (Koukoufilippas et al. 2020). The aim of
+this paper is thus to determine the applicability of this method to the
+case of redshift distribution uncertainties.
+3 DATA
+In order to evaluate the performance of the analytical marginalisation
+approach described in the previous section in the context of current
+and future surveys, we make use of data from the first-year cosmic
+shear analysis of the Dark Energy Survey (DES-Y1, Abbott et al.
+(2018b)). The aim of this is twofold: first, to demonstrate that the
+method can be successfully implemented in real data, with real-life
+complications (e.g. noisy 𝑝(𝑧)s, numerical covariances, astrophysi-
+cal and observational systematics) and, second, to demonstrate this
+validity for future Stage-IV datasets in the presence of 𝑝(𝑧) cali-
+bration uncertainties already achieved on current data. This section
+describes the DES-Y1 data used, and the models used to generate
+simulated future Stage-IV data.
+3.1 DES-Y1 data and redshift distributions
+The Dark Energy Survey is a photometric, 5-year survey, that has
+observed 5000 deg2 of the sky using five different filter bands (grizY).
+The observations were made with the 4m Blanco Telescope, provided
+with the 570-Mpix Dark Energy Camera (DECam), from the Cerro
+Tololo Inter-American Observatory (CTIO), in Chile. In this paper
+we use cosmic shear data from the first data release (Abbott et al.
+2018b), which covers 1786 deg2 before masking. In particular, we
+use the public Metacalibration source catalog2, which is divided
+in four redshift bins covering the range 𝑧 ≲ 1.6 (Hoyle et al. 2018).
+We use the calibrated redshift distributions of the Metacalibra-
+tion sample provided by García-García et al. (2023). The 𝑝(𝑧)s were
+estimated via direct calibration (DIR Lima et al. (2008)), using the
+COSMOS 30-band catalog (Laigle et al. 2016) as a calibrating sam-
+ple. The uncertainties of the measured redshift distributions were
+estimated analytically, as described in García-García et al. (2023),
+accounting for both shot noise and sample variance, and represent
+a realistic level of 𝑝(𝑧) uncertainty achieved by current existing
+datasets. The redshift distributions were sampled on 40 bins of width
+𝛿𝑧 = 0.04 covering the range 0 ≤ 𝑧 ≤ 1.6. Fig. 1 shows, in the first
+row, the redshift distributions of the four Metacalibration samples
+and their statistical uncertainties. Note that we estimated the full co-
+variance matrix of the 𝑝(𝑧) bin heights. The covariance is dominated
+by the diagonal, as can be seen in the bottom panels of Fig. 1.
+We will also use the cosmic shear angular power spectra provided
+by Nicola et al. (2021). A full description of the methods used to
+estimate these power spectra, and their associated covariance matrix,
+from the DES-Y1 data is provided by the authors.
+3.2 Future Stage-IV data
+We generate a simulated data vector corresponding to a Stage-IV
+cosmic shear survey, such as the Legacy Survey of Space Time, at
+the Rubin Observatory (LSST Dark Energy Science Collaboration
+2012), or the Euclid survey (Spergel et al. 2015). Our aim is to ef-
+fectively test the analytical marginalisation method in the low-noise
+regime, where the inferred posterior is likely more sensitive to resid-
+ual 𝑝(𝑧) uncertainties, and the error budget may become dominated
+by these, rather than the statistical errors in the data themselves.
+2 https://desdr-server.ncsa.illinois.edu/despublic/y1a1_
+files/
+MNRAS 000, 1–11 (2022)
+
+Analytical marginalisation over photo-𝑧 uncertainties
+5
+For simplicity, we simulate the Stage-IV survey as having the
+same redshift distributions as the DES-Y1 sample. This includes
+both the 𝑝(𝑧)s themselves, and their calibration uncertainties. While
+it is possible that techniques for inferring redshifts from photometry,
+or the size and quality of calibrating spectroscopic samples, will
+improve substantially by the time Stage-IV data are available, we
+prefer to err on the side of caution and assume the same performance
+as currently achieved. For instance it is possible that redshift estimates
+will suffer commensurately with the increase in survey depth. The
+results presented here are therefore conservative, and their validity
+will only be reinforced if better 𝑝(𝑧) calibration samples are used in
+the future.
+We generate cosmic shear power spectra using CCL (Chisari et al.
+2019) for the best-fit Planck 2018 cosmological parameters (Planck
+Collaboration et al. 2020): Ω𝑏ℎ2 = 0.02237, Ω𝑐ℎ2 = 0.12, ℎ =
+0.6736, 109𝐴𝑠 = 2.0830, 𝑛𝑠 = 0.9649, 𝑤0 = −1, 𝑤𝑎 = 0. We use
+the same sampling in ℓ used for the DES-Y1 power spectra, and use
+only scales in the range ℓ ∈ [30, 2000].
+We compute the covariance matrix of these power spectra an-
+alytically, including a disconnected “Gaussian” component, and a
+connected super-sample covariance contribution (SSC).
+Cov
+�
+𝐶 𝛼
+ℓ , 𝐶𝜌𝜎
+ℓ′
+�
+= Cov𝐺
+�
+𝐶 𝛼𝛽
+ℓ
+, 𝐶𝜌𝜎
+ℓ′
+�
++ CovSSC
+�
+𝐶 𝛼𝛽
+ℓ
+, 𝐶 𝜎𝜌
+ℓ′
+�
+.
+(11)
+We estimate the Gaussian covariance using a simple mode-counting
+approximation (Efstathiou 2004) as
+Cov𝐺
+�
+𝐶 𝛼𝛽
+ℓ
+, 𝐶𝜌𝜎
+ℓ′
+�
+= 𝛿ℓℓ′
+𝐶 𝛼𝜌
+ℓ
+𝐶𝛽𝜎
+ℓ
++ 𝐶 𝛼𝜎
+ℓ
+𝐶𝛽𝜌
+ℓ
+(2ℓ + 1) Δℓ 𝑓sky
+,
+(12)
+where 𝑓sky is the fraction of the sky covered by the experiment. We
+assume 𝑓sky = 0.4, as in the case of LSST (LSST Dark Energy Sci-
+ence Collaboration 2012). The angular power spectra above contain
+the contribution from shape noise in the auto-correlation, of the form
+𝑁 𝛼𝛼
+ℓ
+=
+𝜎2𝛾
+¯𝑛𝛼
+.
+(13)
+Here 𝜎𝛾 = 0.28 is the per-component ellipticity dispersion in each
+source, and ¯𝑛𝛼 is the angular number density of sources in the 𝛼-th
+redshift bin. We assume 𝑛𝛼 = 4 arcmin−2 in each redshift bin.
+We compute the super-sample covariance contribution following:
+CovSSC(𝐶 𝛼𝛽
+ℓ
+, 𝐶𝜌𝜎
+ℓ′ ) =
+∫
+d𝜒 𝑞𝛼(𝜒)𝑞𝛽(𝜒)𝑞𝜌(𝜒)𝑞𝜎(𝜒)
+𝜒4
+×
+(14)
+𝜕𝑃(ℓ/𝜒, 𝑧)
+𝜕𝛿LS
+𝜕𝑃(ℓ′/𝜒, 𝑧)
+𝜕𝛿LS
+𝜎2
+LS(𝑧),
+(15)
+as in Nicola et al. (2021). 𝜕𝑃(𝑘, 𝑧)/𝜕𝛿LS is the response of the matter
+power spectrum to a large-scale density fluctuation 𝛿LS, and the
+quantity 𝜎2
+𝑏(𝑧) is the variance of the long wavelength mode over the
+survey footprint. We estimate the latter as in Krause & Eifler (2017),
+modelling the footprint simply as a circular cap of area 4𝜋 𝑓sky. We
+estimate the response function using perturbation theory and the halo
+model, as described in Krause & Eifler (2017), and as implemented
+in CCL.
+4 LIKELIHOOD
+We extract cosmological parameter constraints using a Gaussian like-
+lihood as described in Section 2.3. In order to validate the analytical
+Parameter priors
+Parameter
+Prior
+Parameter
+Prior
+Cosmology
+Redshift calibration
+Ωm
+𝑈 (0.1, 0.9)
+Δ𝑧1
+N0.0, 0.016)
+Ωb
+𝑈 (0.03, 0.07)
+Δ𝑧2
+N(0.0, 0.017)
+ℎ
+𝑈 (0.55, 0.91)
+Δ𝑧3
+N(0.0, 0.013)
+𝑛s
+𝑈 (0.87, 1.07)
+Δ𝑧4
+N(0.0, 0.015)
+𝜎8
+𝑈 (0.6, 0.9)
+𝑝i
+N( ¯𝑝𝑖, C)
+Shear multiplicative bias
+𝑚𝑖
+0.012
+Table 1. Prior distributions for the parameters considered in this work. Note
+that the redshift calibration section contains the priors for both the Δ𝑧 and
+𝑝𝛼 (𝑧) models which are not sampled simultaneously.
+marginalisation approach, we will either use the full posterior dis-
+tribution in Eq. 7, or the analytically marginalised version in Eq.
+93. In the first case, 𝝂 includes all nuisance parameters describing
+the redshift distribution uncertainties, and in both cases 𝛀 includes
+all other model parameters. Specifically, 𝛀 contains the five ΛCDM
+cosmological parameters (Ωm, Ωb, 𝜎8, 𝑛𝑠, ℎ).
+When marginalising over redshift distribution uncertainties, 𝝂 will
+contain either one redshift shift parameter Δ𝑧𝛼 for each redshift bin,
+when employing the parametric description of 𝑝(𝑧) uncertainties
+(Method 1), or a set of bin heights for each redshift bin determining
+𝑝𝛼(𝑧), when using the non-parametric approach (Method 2). The
+first case will introduce 4 new parameters to the model, while the
+latter will introduce 4 × 40 = 160 new amplitude parameters, as
+described in Section 3.1.
+Table 1 shows the parameter priors used in this work. All cosmo-
+logical parameters take uniform, largely uninformative priors. For
+simplicity, the multiplicative bias parameters were fixed at the center
+of the Gaussian priors from the official analysis of DES-Y1 (Ab-
+bott et al. 2018a). When using Method 1 to numerically marginalise
+over the 𝑝(𝑧) uncertainties, we used Gaussian priors on each of the
+shift parameters Δ𝑧𝛼, following those used by DES-Y1 (Abbott et al.
+2018a). When using Method 2 (marginalisation over 𝑝(𝑧) bin am-
+plitudes), we assume a multi-variate Gaussian prior, with the 𝑝(𝑧)
+covariance described in Sect. 3.1 and shown in Fig. 1.
+For both 𝑝(𝑧) uncertainty models, when using analytical marginal-
+isation, we use Eq. 9 and modify the covariance as in Eq. 10, with P
+given by the priors described above. When using numerical marginal-
+isation, we simply explore the posterior distribution of the full model,
+including all the 𝑝(𝑧), 𝑝𝑖, parameters. In the case of Method 2, this
+involves sampling a distribution with 165 parameters, of which the
+bulk (160 parameters) describe the 𝑝(𝑧) uncertainty. This is not fea-
+sible for standard Metropolis-Hastings MCMC methods Metropolis
+et al. (1953); Hastings (1970) due to the curse of dimensionality, and
+therefore we resort to a Hamiltonian Monte Carlo (HMC) approach.
+HMC (MacKay 2002; Betancourt 2017) uses notions of Hamil-
+tonian dynamics to draw trajectories on the parameter space along
+which the sampler moves. This results in a much greater accep-
+tance rate, and allows HMC to beat the dimensionality curse. HMC
+can thus efficiently explore parameter spaces with large numbers of
+dimensions in far less time than Metropolis-Hastings or nested sam-
+pling techniques (Alsing & Handley 2021). The main difficulty of
+3 Recall that we treat the term in the second line of Eq. 9 as a constant.
+MNRAS 000, 1–11 (2022)
+
+6
+Ruiz-Zapatero et al.
+0.2
+0.3
+0.4
+m
+0.70
+0.75
+0.80
+S8
+0.6
+0.7
+0.8
+0.9
+8
+0.6 0.7 0.8 0.9
+8
+0.71 0.75 0.79
+S8
+z - Analytical marg. - DESY1
+z - Numerical marg. - DESY1
+z - No marg. - DESY1
+Figure 2. Marginalised posterior distributions for the combination of parame-
+ters Ωm, 𝜎8 and 𝑆8 obtained when considering the Δ𝑧 model for photometric
+uncertainties for DES-Y1 data. The blue contours correspond to the case
+where the Δ𝑧 parameter are fixed. The magenta contours are obtained when
+numerically marginalizing over the Δ𝑧 parameters. Finally, the black dashed
+contours are obtained when analytically marginalizing over the Δ𝑧 parame-
+ters. We can observe that the analytical and numerical marginalisation return
+nearly identical posteriors.
+using HMC is the need to calculate gradients of the log-posterior
+to calculate the Hamiltonian equations of motion. The additional
+computational cost of obtaining these derivatives numerically (e.g.
+via adaptive finite differences) may outweigh the gains caused by the
+higher acceptance rates of HMC. To overcome this problem we make
+use of automatic differentiation (AD). To take advantage of AD, we
+have developed a cosmological theoretical prediction code natively
+written in the Julia programming language (Ruiz-Zapatero et al.
+2023). Julia is a just-in-time (JIT) compiled language with C-like
+performance and seamless AD integration, which can thus be used
+to efficiently sample complex cosmological posteriors using HMC.
+To sample the posterior distribution we use the No-U-Turns Sampler
+(NUTS Hoffman & Gelman (2011)) implementation of HMC within
+the Turing.jl package (Ge et al. 2018).
+5 RESULTS
+5.1 Linearising Δ𝑧
+Let us begin the discussion of our results by considering the simplest
+of the two models of the photometric uncertainties studied in this
+work, the Δ𝑧 model (called Method 1 above). As discussed in Section
+4, this model introduces 4 new shift parameters Δ𝑧 (one per redshift
+bin) in addition to the 5 ΛCDM parameters. All other nuisance
+parameters are kept fixed. For the DES-Y1 and LSST-like datasets,
+we will compare the result of analytically marginalizing over the Δ𝑧
+parameters against performing the full numerical marginalisation on
+the corresponding cosmological constraints. In order to quantify the
+contribution of redshift uncertainties to the total error budget, we will
+also present results for the case when the Δ𝑧 parameters are fixed (i.e.
+assuming perfect knowledge of the redshift distributions).
+0.30 0.35
+m
+0.82
+0.83
+0.84
+0.85
+S8
+0.75
+0.80
+0.85
+8
+0.75
+0.85
+8
+0.83
+0.85
+S8
+z - Analytical marg. - LSST
+z - Numerical marg. - LSST
+z - No marg. - LSST
+Figure 3. Marginalised posterior distributions for the combination of parame-
+ters Ωm, 𝜎8 and 𝑆8 obtained when considering the Δ𝑧 model for photometric
+uncertainties for futuristic LSST-like data. The green contours correspond to
+the case where the Δ𝑧 parameter are fixed. The orange contours are obtained
+when numerically marginalizing over the Δ𝑧 parameters. Finally, the black
+dashed contours are obtained when analytically marginalizing over the Δ𝑧 pa-
+rameters. We can observe that the analytical and numerical marginalisations
+return nearly identical posteriors.
+Our results for DES-Y1 data are shown in Fig. 2, with the er-
+rors on all parameters listed in Table 2. On the one hand, we find
+that marginalizing analytically or numerically over the Δ𝑧 parame-
+ters leads to the same marginalised posterior for the cosmological
+parameters. On the other hand, fixing the Δ𝑧 parameters returns a
+posterior distribution that is only mildly narrower than the marginal
+distribution. For the DES-Y1 data, the impact of redshift uncertain-
+ties in the final cosmological errors is relatively small (although not
+negligible). Thus, if we truly wish to study the effect of marginal-
+izing analytically as opposed to numerically over the Δ𝑧 parameters
+we will have to consider futuristic LSST-like data, where the impact
+of these uncertainties will likely be higher.
+We show results for futuristic LSST-like data on Fig. 3, with the
+parameter constraints listed in Table 2. First of all, in the case LSST-
+like data we observe that not marginalising over the Δ𝑧 parameters
+in the model results in significantly narrower posteriors, with the
+final uncertainties shrinking by a factor ∼ 2. The impact of redshift
+distribution uncertainties in this case is thus much more relevant,
+and the accuracy of the analytical marginalisation scheme becomes
+paramount. However, comparing the contours obtained by numerical
+and analytical marginalisation, we observe that both methods return
+largely equivalent posterior distributions, with the final uncertainties
+changing by much less than 10%. This holds even in the case the
+Δ𝑧 prior worsen by a factor 4 as seen in Figure A1, in Appendix A.
+Therefore, linearizing the likelihood around the Δ𝑧 parameters will be
+a good enough approximation for LSST-data, at least for relatively
+simple parametrisations of the 𝑝(𝑧) uncertainty, which will allow
+us to reduce the dimensionality of the model and make parameter
+inference more efficient.
+It is worth emphasizing that the results in this section are not
+meant to be interpreted as forecasts on the constraining power of
+MNRAS 000, 1–11 (2022)
+
+Analytical marginalisation over photo-𝑧 uncertainties
+7
+Δ𝑧 model
+Fixed
+Numerical
+Analytical
+Ωm
+DES-Y1
+0.333 ± 0.055
+0.3 ± 0.056
+0.306 ± 0.055
+LSST
+0.311 ± 0.011
+0.317 ± 0.02
+0.317 ± 0.02
+𝜎8
+DES-Y1
+0.724 ± 0.072
+0.765 ± 0.077
+0.758 ± 0.076
+LSST
+0.82 ± 0.015
+0.821 ± 0.027
+0.823 ± 0.027
+𝑆8
+DES-Y1
+0.753 ± 0.015
+0.756 ± 0.015
+0.756 ± 0.015
+LSST
+0.833 ± 0.002
+0.833 ± 0.005
+0.833 ± 0.006
+Table 2. Numerical values for the mean and 1𝜎 confidence intervals for
+the 1D marginalised posterior distributions of the cosmological parameters
+Ωm, 𝜎8 and 𝑆8 obtained when considering the first method (𝑧 shifts) to
+characterise the photometric redshift uncertainties. The first column shows
+the values obtained when the Δ𝑧 parameters were kept fixed, the second
+column when they were marginalised numerically and the third column when
+they were marginalised analytically. In each row we display the constraints
+obtained when using DES-Y1 or LSST-like data to constrain the models.
+LSST on cosmological parameters, but only on our ability to ana-
+lytically marginalize over photometric uncertainties in inferring the
+underlying cosmology. The recovered constraints depend strongly on
+assumptions such as the redshift calibration that LSST will be able to
+achieve for the different samples involved. As such, the results pre-
+sented here are only a conservative estimate of the effect of analytic
+marginalisation on cosmological constraints.
+5.2 Linearising 𝑝𝛼(𝑧)
+In the previous section we have shown that, even for futuristic LSST-
+like data, it is possible to marginalize over redshift uncertainties
+analytically, assuming a relatively simple parametrisation of these
+uncertainties. We now turn to more complex models to characterise
+these uncertainties.
+In order to do so we consider the previously discussed 𝑝𝛼(𝑧)
+model (called Method 2 above), which turns the height of each bin
+in the redshift distribution histograms into a free parameter. This
+results in 40 new free parameters per redshift bin with a total of 160
+parameters for the data considered in this work.
+We start by revisiting the DES-Y1 data analysis, presenting our
+results in Fig. 4. As we observed in the previous section, we find that
+even for the far more general 𝑝𝛼(𝑧) model there is no significant
+difference between numerically marginalizing over the 𝑝𝛼(𝑧), or
+doing so through our approximate analytical approach. Furthermore,
+as before, fixing the shape of the redshift distribution leads to only
+mildly tighter constraints. On the one hand, this means that the result
+found for the Δ𝑧 model is not reliant on the simplicity of the model,
+but instead inherent to the sensitivity of DES-Y1 data. On the other
+hand, this also means that we must turn once again to futuristic LSST-
+like data to study the impact of a more general parametrisation of
+photometric uncertainties.
+The results for futuristic LSST-like data are shown in Fig. 5. As in
+the case of the Δ𝑧 parametrisation, we find that, in the case LSST-
+like data, not including the 𝑝𝛼(𝑧) parameters in the model results
+in significantly narrower posteriors. By looking at the corresponding
+numerical values in Tab. 3, we see that the 𝑆8 constraints become
+twice as tight when the 𝑝𝛼(𝑧) parameters are fixed. Most importantly,
+we find that marginalizing over the 𝑝���(𝑧) parameters analytically
+or numerically yields almost indistinguishable posteriors. Thus, the
+results found in Sect. 5.1 for the simple Δ𝑧 parametrisation, in fact
+hold for significantly more general models of the uncertainty in the
+galaxy redshift distributions.
+Finally, in Fig. 6 we present the constraints obtained for the 160
+0.2
+0.3
+0.4
+m
+0.70
+0.74
+0.78
+S8
+0.6
+0.7
+0.8
+0.9
+8
+0.6 0.7 0.8 0.9
+8
+0.71 0.75 0.79
+S8
+p (z) - Analytical marg. - DESY1
+p (z) - Numerical marg. - DESY1
+p (z) - No marg. - DESY1
+Figure 4. Marginalised posterior distributions for the combination of pa-
+rameters Ωm, 𝜎8 and 𝑆8 obtained when considering the 𝑝𝛼 (𝑧) model for
+photometric uncertainties for DES-Y1 data. The blue contours correspond to
+the case where the 𝑝𝛼 (𝑧) parameter are fixed. The magenta contours are ob-
+tained when numerically marginalising over the 𝑝𝛼 (𝑧) parameters. Finally,
+the black dashed contours are obtained when analytically marginalizing over
+the 𝑝𝛼 (𝑧) parameters. We can observe that the analytical and numerical
+marginalisation return nearly identical posteriors.
+0.30 0.35
+m
+0.82
+0.83
+0.84
+0.85
+S8
+0.75
+0.80
+0.85
+0.90
+8
+0.75
+0.85
+8
+0.83
+0.85
+S8
+p (z) - Analytical marg. - LSST
+p (z) - Numerical marg. - LSST
+p (z) - No marg. - LSST
+Figure 5. Marginalised posterior distributions for the combination of pa-
+rameters Ωm, 𝜎8 and 𝑆8 obtained when considering the 𝑝𝛼 (𝑧) model for
+photometric uncertainties for LSST-like futuristic data. The green contours
+correspond to the case where the 𝑝𝛼 (𝑧) parameter are fixed. The orange
+contours were obtained when numerically marginalizing over the 𝑝𝛼 (𝑧) pa-
+rameters. Finally, the black dashed contours were obtained when analytically
+marginalizing over the 𝑝𝛼 (𝑧) parameters. We can observe that the analytical
+and numerical marginalization return nearly identical posteriors.
+MNRAS 000, 1–11 (2022)
+
+8
+Ruiz-Zapatero et al.
+𝑝𝛼 (𝑧) model
+Fixed
+Numerical
+Analytical
+Ωm
+DES-Y1
+0.333 ± 0.056
+0.308 ± 0.055
+0.312 ± 0.057
+LSST
+0.311 ± 0.011
+0.317 ± 0.02
+0.317 ± 0.021
+𝜎8
+DES-Y1
+0.723 ± 0.073
+0.755 ± 0.075
+0.75 ± 0.077
+LSST
+0.824 ± 0.015
+0.816 ± 0.026
+0.815 ± 0.027
+𝑆8
+DES-Y1
+0.753 ± 0.015
+0.755 ± 0.015
+0.755 ± 0.015
+LSST
+0.838 ± 0.002
+0.837 ± 0.006
+0.837 ± 0.006
+Table 3. Numerical values for the mean and 1𝜎 confidence intervals for the
+1D marginalised posterior distributions of the cosmological parameters Ωm,
+𝜎8 and 𝑆8 obtained when considering the second method (𝑝(𝑧) bin heights)
+to characterise the photometric redshift uncertainties. The first column shows
+the values obtained when the 𝑝𝛼 (𝑧) parameters are kept fixed, the second
+column when they are marginalised numerically, and the third column when
+they are marginalised analytically. In each row we display the constraints
+obtained when using DES-Y1 or LSST-like data to constrain the models.
+𝑝𝛼(𝑧) parameters for both the DES-Y1 (top panel) and LSST-like
+data (bottom panel) in color bands. We observe that the posterior
+distributions are largely dominated by the prior (shown in dashed
+black line with error bars) and, thus, the redshift distribution is not
+significantly self-calibrated by the data in either case.
+Before moving to the next Section, it is worth stressing that con-
+straining such a large parameter space has only been possible thanks
+to the auto-differentiable nature of the code used to obtain theoret-
+ical predictions, allowing us to use gradient-based samplers, much
+more efficient that standard samplers. The development of such auto-
+differentiable codes will therefore become imperative in the near fu-
+ture given the increasing complexity of models used in cosmological
+analyses.
+5.3 Δ𝑧 vs 𝑝𝛼(𝑧)
+In the previous sections we have focused in the impact of how we
+marginalize over the different parametrisations of photometric red-
+shift uncertainties. In this section we will focus instead on what we
+marginalize over, i.e. the impact of the choice of parametrisation.
+The question is then: Can a one-parameter-per-bin model (Δ𝑧 model)
+capture all the meaningful modifications to photometric redshift dis-
+tributions?
+In order to answer this question, we constrain the cosmological
+parameters for the Δ𝑧 and 𝑝𝛼(𝑧) models in the case with futuristic
+LSST-like data. In both cases, we marginalize numerically over their
+respective nuisance parameters. As shown in Fig. 7 and Tables 2
+and 3, both methods recover the same posterior distributions with
+small differences. Thus, it is in principle possible that even Stage-IV
+surveys will be able to use relatively simple models to describe the
+redshift distribution of cosmic shear samples4.
+6 CONCLUSIONS
+One of the most significant obstacles to overcome in photometric
+weak lensing surveys is the accurate modeling of redshift distribu-
+tions, 𝑝(𝑧). Not only are our measurements prone to error, which can
+bias the inferred cosmological parameters, but accounting for these
+4 Note, however, this is likely not the case for photometric galaxy clustering
+studies where other properties of the redshift distribution (e.g. its width) have
+a stronger impact on the theoretical prediction (Nicola et al. 2020).
+0.00
+0.05
+0.10
+0.15
+0.20
+p (z)
+DESY1
+0.0
+0.5
+1.0
+1.5
+z
+0.00
+0.05
+0.10
+0.15
+0.20
+p (z)
+LSST
+Figure 6. Posterior distributions for the 𝑝𝛼 (𝑧) parameters when considering
+DES-Y1 data (top row) and futuristic LSST-like data (bottom row). The black
+dashed line shows the mean of the Gaussian prior of the 𝑝𝛼 (𝑧) parameters.
+The error bars show their corresponding error.
+uncertainties is also a major inhibitor of efficient parameter inference.
+In this paper, we investigate the impact of analytically marginalizing
+over the uncertainties in the redshift distribution of galaxies in weak
+lensing surveys, as initially proposed in Hadzhiyska et al. (2020).
+In particular, we thoroughly quantify the validity of this approach
+for a current weak lensing survey, DES, as well as for a futuristic
+LSST-like survey, testing whether a fast analytic method proposed
+in this work is capable of reproducing the posterior distributions and
+constraints one arrives at when adopting the traditional method of
+diligently varying tens or hundreds of nuisance parameters.
+Our results show that, for present surveys, marginalizing over the
+uncertainty in the redshift distribution of galaxies has only a mild
+impact on the constraints on cosmological parameters, although one
+that our analytical approximation is able to reproduce accurately. This
+is true for the two parametrisations of the uncertainties considered
+in this work, in terms of mean redshift shifts or redshift distribution
+histogram heights. However, the impact of redshift distribution un-
+certainties changes dramatically for future LSST-like surveys. In this
+case, redshift uncertainties commensurate with current calibration
+samples lead to an degradation in the final constraints on cosmo-
+logical parameters of up to a factor ∼ 2. Capturing this effect for
+an arbitrarily complex parametrisation of the redshift distribution
+uncertainties is an a priori difficult task without resorting to a full
+exploration of the parameter space. Nevertheless, we find that the
+analytical approximate scheme explored here is still able to recover
+the marginalised constraints on cosmological parameters to high
+fidelity, even after marginalising over more than 100 nuisance pa-
+MNRAS 000, 1–11 (2022)
+
+Analytical marginalisation over photo-𝑧 uncertainties
+9
+0.30 0.35
+m
+0.82
+0.83
+0.84
+0.85
+S8
+0.75
+0.80
+0.85
+0.90
+8
+0.6
+0.7
+0.8
+0.9
+h
+0.95
+1.00
+ns
+0.025
+0.050
+0.075
+b
+0.04 0.06
+b
+0.95
+1.00
+ns
+0.6 0.7 0.8 0.9
+h
+0.74 0.80 0.86
+8
+0.82
+0.84
+S8
+z - Numerical marg. - LSST
+p (z) - Numerical marg. - LSST
+Figure 7. Comparison between the obtained marginalised posterior distributions of the cosmological parameters when numerically marginalizing over the Δ𝑧
+(black dash-dotted) and 𝑝𝛼 (𝑧) (orange) photometric uncertainties models when applied to LSST-like futuristic data. We can observe that both prametrizations
+of the photometric redshift uncertainties return identical posteriors for the cosmological parameters.
+rameters. This means that, while future surveys will certainly have
+to account for these uncertainties, they will be able to do so using
+fast marginalisation methods without increasing the dimensionality
+of their astrophysical and cosmological models.
+Our results have also shown that simple parametrisations of the
+redshift distribution for cosmic shear samples, in terms of shifts in
+the mean redshift, are, surprisingly, able to reproduce the impact of
+the full uncertainty on 𝑝(𝑧) on the final constraints to high precision.
+Although this result will likely not hold for other probes (e.g. tomo-
+graphic galaxy clustering), it should certainly simplify the analysis
+of future cosmic shear data.
+It is worth emphasizing that our work has focused exclusively on
+the case of cosmic shear data, and that our conclusions only apply in
+this context. The validity of the analytical approximation employed
+here for general tomographic tracers of structure with uncertain ra-
+dial kernels is not guaranteed, and future work should quantify its
+performance on photometric clustering data – the other key probe
+of the flagship “3×2pt” analysis of imaging surveys – and its cross
+correlation with cosmic shear and CMB lensing data (Heymans et al.
+2021; Abbott et al. 2022; García-García et al. 2021; White et al.
+2022).
+MNRAS 000, 1–11 (2022)
+
+10
+Ruiz-Zapatero et al.
+ACKNOWLEDGEMENTS
+We would like to thank Aně Slosar and Marius Millea for useful dis-
+cussions. DA is supported by the Science and Technology Facilities
+Council through an Ernest Rutherford Fellowship, grant reference
+ST/P004474. PGF, CGG and AM are supported by European Re-
+search Council Grant No: 693024 and the Beecroft Trust. JRZ is
+supported by an STFC doctoral studentship. We made extensive use
+of computational resources at the University of Oxford Department
+of Physics, funded by the John Fell Oxford University Press Research
+Fund.
+We made extensive use of the numpy (Oliphant 2006; Van Der Walt
+et al. 2011), scipy (Virtanen et al. 2020), astropy (Astropy Col-
+laboration et al. 2013, 2018), healpy (Zonca et al. 2019), GetDist
+Lewis (2019), and matplotlib (Hunter 2007) python packages. We
+also make use of the Julia packages ForwardDiff.jl (Revels et al.
+2016) and Turing.jl (Ge et al. 2018).
+DATA AVAILABILITY
+The code developed for this work as well as the derived datasets
+produced (power spectra and covariances) are available upon request.
+The catalogues and maps used were made publicly available by the
+authors of the relevant papers, as described in the text.
+REFERENCES
+Abbott T. M. C., et al., 2018a, Phys. Rev. D, 98, 043526
+Abbott T. M. C., et al., 2018b, ApJS, 239, 18
+Abbott T. M. C., et al., 2022, Phys. Rev. D, 105, 023520
+Afshordi N., Loh Y.-S., Strauss M. A., 2004, Phys. Rev. D, 69, 083524
+Alsing J., Handley W., 2021, MNRAS, 505, L95
+Astropy Collaboration et al., 2013, A&A, 558, A33
+Astropy Collaboration et al., 2018, AJ, 156, 123
+Bartelmann M., Schneider P., 2001, Phys. Rep., 340, 291
+Betancourt M., 2017, arXiv e-prints, p. arXiv:1701.02434
+Bonnett C., et al., 2016, Phys. Rev. D, 94, 042005
+Brown M. L., Taylor A. N., Hambly N. C., Dye S., 2002, MNRAS, 333, 501
+Chisari N. E., et al., 2019, ApJS, 242, 2
+Cordero J. P., et al., 2022, MNRAS, 511, 2170
+Efstathiou G., 2004, MNRAS, 349, 603
+Ferreira P. G., 2019, ARA&A, 57, 335
+Foreman-Mackey D., Hogg D. W., Lang D., Goodman J., 2013, PASP, 125,
+306
+García-García C., Ruiz-Zapatero J., Alonso D., Bellini E., Ferreira P. G.,
+Mueller E.-M., Nicola A., Ruiz-Lapuente P., 2021, J. Cosmology As-
+tropart. Phys., 2021, 030
+García-García C., Alonso D., Ferreira P. G., Hadzhiyska B., Nicola A.,
+Sánchez C., Slosar A., 2023, J. Cosmology Astropart. Phys., 2023, 025
+Ge H., Xu K., Ghahramani Z., 2018, in International Conference on Ar-
+tificial Intelligence and Statistics, AISTATS 2018, 9-11 April 2018,
+Playa Blanca, Lanzarote, Canary Islands, Spain. pp 1682–1690, http:
+//proceedings.mlr.press/v84/ge18b.html
+Hadzhiyska B., Alonso D., Nicola A., Slosar A., 2020, J. Cosmology As-
+tropart. Phys., 2020, 056
+Hadzhiyska B., et al., 2023, In prep,
+Hastings W. K., 1970, Biometrika, 57, 97
+Heymans C., et al., 2021, A&A, 646, A140
+Hildebrandt H., et al., 2020a, A&A, 633, A69
+Hildebrandt H., et al., 2020b, A&A, 633, A69
+Hirata C. M., Seljak U., 2004, Phys. Rev. D, 70, 063526
+Hoffman M. D., Gelman A., 2011, arXiv e-prints, p. arXiv:1111.4246
+Hoyle B., et al., 2018, MNRAS, 478, 592
+Hunter J. D., 2007, Computing in Science & Engineering, 9, 90
+Kilbinger M., et al., 2017, MNRAS, 472, 2126
+Koukoufilippas N., Alonso D., Bilicki M., Peacock J. A., 2020, MNRAS,
+491, 5464
+Krause E., Eifler T., 2017, MNRAS, 470, 2100
+Krause E., et al., 2017, arXiv e-prints, p. arXiv:1706.09359
+LSST Dark Energy Science Collaboration 2012, arXiv e-prints, p.
+arXiv:1211.0310
+Laigle C., et al., 2016, ApJS, 224, 24
+Lewis A., 2019, arXiv e-prints, p. arXiv:1910.13970
+Lima M., Cunha C. E., Oyaizu H., Frieman J., Lin H., Sheldon E. S., 2008,
+MNRAS, 390, 118
+Limber D. N., 1953, ApJ, 117, 134
+MacKay D. J. C., 2002, Information Theory, Inference & Learning Al-
+gorithms. Cambridge University Press, Cambridge University Press,
+Shaftesbury Road Cambridge, CB2 8BS, United Kingdom
+Matthews D. J., Newman J. A., 2010, ApJ, 721, 456
+Metropolis N., Rosenbluth A. W., Rosenbluth M. N., Teller A. H., Teller E.,
+1953, J. Chem. Phys., 21, 1087
+Miyazaki S., et al., 2012, in McLean I. S., Ramsay S. K., Takami H., eds,
+Vol. 8446, Ground-based and Airborne Instrumentation for Astronomy
+IV. SPIE, p. 84460Z, doi:10.1117/12.926844, https://doi.org/10.
+1117/12.926844
+Newman J. A., 2008, ApJ, 684, 88
+Newman E. T., Penrose R., 1966, Journal of Mathematical Physics, 7, 863
+Nicola A., et al., 2020, J. Cosmology Astropart. Phys., 2020, 044
+Nicola A., García-García C., Alonso D., Dunkley J., Ferreira P. G., Slosar A.,
+Spergel D. N., 2021, J. Cosmology Astropart. Phys., 2021, 067
+Oliphant T. E., 2006, A guide to NumPy. Vol. 1, Trelgol Publishing USA
+Planck Collaboration et al., 2020, A&A, 641, A6
+Prat J., et al., 2018, Phys. Rev. D, 98, 042005
+Revels J., Lubin M., Papamarkou T., 2016, arXiv:1607.07892 [cs.MS]
+Riess A. G., et al., 2022, ApJ, 934, L7
+Ruiz-Zapatero J., et al., 2023, In prep,
+Sánchez C., Bernstein G. M., 2019, MNRAS, 483, 2801
+Sánchez C., et al., 2022, Phys. Rev. D, 105, 083529
+Schmidt S. J., Ménard B., Scranton R., Morrison C., McBride C. K., 2013,
+MNRAS, 431, 3307
+Schneider M., Knox L., Zhan H., Connolly A., 2006, ApJ, 651, 14
+Scott D., 2018, arXiv e-prints, p. arXiv:1804.01318
+Smith R. E., et al., 2003, MNRAS, 341, 1311
+Spergel D., et al., 2015, arXiv e-prints, p. arXiv:1503.03757
+Stölzner B., Joachimi B., Korn A., Hildebrandt H., Wright A. H., 2021, A&A,
+650, A148
+Takahashi R., Sato M., Nishimichi T., Taruya A., Oguri M., 2012, ApJ, 761,
+152
+Van Der Walt S., Colbert S. C., Varoquaux G., 2011, Computing in Science
+& Engineering, 13, 22
+Virtanen P., et al., 2020, Nature Methods, 17, 261
+White M., et al., 2022, J. Cosmology Astropart. Phys., 2022, 007
+Wright A. H., Hildebrandt H., van den Busch J. L., Heymans C., 2020, A&A,
+637, A100
+Zhang T., Rau M. M., Mandelbaum R., Li X., Moews B., 2023, MNRAS,
+518, 709
+Zonca A., Singer L., Lenz D., Reinecke M., Rosset C., Hivon E., Gorski K.,
+2019, Journal of Open Source Software, 4, 1298
+APPENDIX A: STRESS-TESTING THE APPROXIMATION
+As described in Sect. 2, the approximation used here to analytically
+marginalise over the redshift calibration parameters assumes a suffi-
+ciently tight prior on these parameters, such that the dependence of
+the theory prediction on them can be linearised. Testing whether this
+assumption might break in a realistic scenario, is therefore essential.
+This is important in the context of Stage-IV since, even though it is
+expected that spectroscopic samples and the associated calibration
+MNRAS 000, 1–11 (2022)
+
+Analytical marginalisation over photo-𝑧 uncertainties
+11
+0.3
+0.4
+m
+0.80
+0.85
+S8
+0.8
+0.9
+8
+0.6
+0.7
+0.8
+0.9
+h
+0.95
+1.00
+ns
+0.04
+0.06
+b
+0.04
+0.06
+b
+0.95 1.00
+ns
+0.6 0.7 0.8 0.9
+h
+0.8
+0.9
+8
+0.80
+0.85
+S8
+z - Analytical marg. - LSST - 4
+z - Numerical marg. - LSST- 4
+Figure A1. Shows a comparison between the obtained marginalised posterior distributions of the cosmological parameters when analytically marginalizing over
+the Δ𝑧 (black dashed) and when performing the full numerical marginalisation (orange) when analyzing LSST-like data. In both cases the Δ𝑧 prior distributions
+where made 4 times wider. We can observe that despite significantly broadening the prior distributions the analytical marginalisation returns virtually identical
+posteriors for the cosmological parameters.
+techniques will improve over time, the increase in depth that LSST-
+like surveys will represent may make the calibration of the faintest
+samples in the survey particularly challenging.
+To further stress-test our approximate method, we repeat our anal-
+ysis of the LSST-like futuristic data using the Δ𝑧 model for redshift
+uncertainties with priors 4 times larger than used in our fiducial
+analysis (which themselves were based on existing calibration sam-
+ples). The result of this test is shown in Fig. A1. Reassuringly, the
+results show that, despite quadrupling the uncertainty in the redshift
+nuisance parameters, the analytic marginalisation method yields vir-
+tually the same constraints on the cosmological parameters as the
+brute-force marginalisation, in spite of the significantly broader pos-
+terior contours. This implicit validates the approximation that a first-
+order expansion of the theory data vector with respect to a change
+in redshift distribution is sufficient over a conservative range of cali-
+bration priors.
+This paper has been typeset from a TEX/LATEX file prepared by the author.
+MNRAS 000, 1–11 (2022)
+

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+Synthesis and processing of lithium-loaded plastic 
+scintillators on the kilogram scale 
+Michael J. Forda*, Elisabeth Aigeldingera, Felicia Sutantoa, Natalia P. Zaitsevaa, 
+Viacheslav A. Lia, M. Leslie Carmana, Andrew Glenna, Cristian R. Catalaa, Steven A. 
+Dazeleya, Nathaniel Bowdena*1 
+ 
+aLawrence Livermore National Laboratory 
+7000 East Avenue, Livermore, CA 94550 
+ 
+Abstract 
+Plastic scintillators that can discriminate between gamma rays, fast neutrons, and 
+thermal neutrons were synthesized and characterized while considering the balance 
+between processing and performance at the kilogram scale. These trade-offs were 
+necessitated by the inclusion of 0.1 wt. % lithium-6 to enable detection of thermal 
+neutrons. The synthesis and processing of these plastic scintillators on the kilogram 
+scale required consideration of many factors. First, a comonomer (methacrylic acid) 
+was used to solubilize salts of lithium-6, which allow for a thermal-neutron capture 
+reaction that produces scintillation light following energy transfer. Second, scintillation 
+performance and processability were considered because the increasing content of the 
+comonomer resulted in a sharp decrease in the light output. The use of small amounts 
+of comonomer (≤3 wt. %) resulted in better performance but required high processing 
+temperatures. At large scales, these high temperatures could initiate an exothermic 
+polymerization that results in premature curing and/or defects. The deleterious effects 
+of the comonomer may be mitigated by using m-terphenyl as a primary dye rather than 
+2,5-diphenyloxazole (PPO), which has been traditionally used in organic scintillators. 
+Finally, the curing environment was controlled to avoid defects like cracking and 
+discoloration while maintaining solubility of dopants during curing. For scintillators 
+that were produced from kilogram-scale batches of precursors, the effective attenuation 
+of scintillation light was characterized.  
+ 
+* This is to indicate the corresponding author. 
+ Email address: [email protected]; [email protected] 
+ 
+
+Keywords: pulse-shape discrimination, plastic scintillators, inverse beta decay, neutron 
+detection, large-scale detectors 
+ 
+1. Introduction 
+The rising cost of fossil fuels and concerns about greenhouse gas emissions 
+motivate a revival of nuclear power generation.[1] However, the expansion of nuclear 
+power throughout the world and the construction of novel reactor types may challenge 
+the resources available for implementing conventional safeguards. Direct and 
+nonintrusive measurements of reactor operation using offer one approach to address 
+this concern. 
+These measurements can be facilitated by the development of novel detectors 
+to enable near-field (ca. 10-100 m) monitoring of antineutrinos produced by a 
+reactor.[2–10] Previous reports highlight the ability of organic scintillators to monitor 
+this antineutrino flux. For example, the Precision Reactor Oscillation and Spectrum 
+Experiment (PROSPECT)[11] recently demonstrated the measurement of the 
+antineutrino spectrum from 235U at the High Flux Isotope Reactor at Oak Ridge 
+National Laboratory. Like in many experiments that use organic scintillators to monitor 
+antineutrino flux, PROSPECT utilized about 4 tons of a liquid scintillator loaded with 
+a neutron capture agent (6Li in this case), which can detect signatures of inverse beta 
+decay. In this experiment, the scintillation light is associated with signatures of inverse 
+beta decay using a measurement scheme called pulse-shape discrimination (PSD).[12] 
+After monitoring a scintillation pulse over time, a prompt signal of scintillation light 
+can be associated with the antineutrino energy, and the delayed signal of scintillation 
+light can be associated with neutron capture by 6Li. Detectors like PROSPECT contain 
+scintillator with this capability of PSD, making them useful for uniquely identifying 
+capture reactions and rejecting fast-neutron background events. 
+For continued development of novel detectors, the phase of the detector 
+material as it relates to the mobility of the detector may be considered. Liquid 
+scintillators are relatively inexpensive, are easy to manufacture, and have good 
+
+performance. However, liquid scintillators may require consideration of potential 
+hazards (e.g., flammability), handling, and storage when used in mobile detectors. 
+Conversely, plastic scintillators are less hazardous than liquid scintillators since they 
+are in the solid state, and plastic scintillators are self-supporting, which is useful for 
+mobility. However, plastic scintillators have not been widely available as materials 
+capable of PSD until recently[13,14]. Additionally, the PSD performance of plastic 
+scintillators worsens as the length of the scintillator increases due to light attenuation; 
+this attenuation is detrimental to the performance of large-volume (ton-scale) 
+detectors.[15] Thus, a mobile detector may require further consideration of these trade-
+offs and further development of detector materials like plastic scintillators.  
+One development that is important for plastic scintillators is the capability of 
+these materials to discriminate between gamma rays, fast neutrons, and thermal 
+neutrons (Figure 1a). This capability necessitates the inclusion of a neutron capture 
+agent like 155Gd, 157Gd, 10B, or 6Li, and various reports describe attempts to incorporate 
+neutron capture agents into plastic scintillators. In particular, the doping of scintillators 
+with 6Li may be preferable since the 6Li(n,t) capture reaction produces a localized, 
+mono-energetic energy deposition that can be efficiently identified via PSD and energy 
+selections.[16,17]  
+Carboxylate salts of 6Li have been previously used to incorporate 6Li into 
+plastic scintillators while obtaining materials that are transparent. In one example, a 6Li 
+salt of methacrylic acid (MAA) was copolymerized with styrene. The 6Li salt of MAA 
+was not soluble in the plastic scintillator precursors at appreciable amounts; additional 
+MAA was needed to dissolve the 6Li salt of MAA and increase the 6Li content. This 
+necessity of additional MAA highlights how solubility of the polar 6Li compounds in 
+the nonpolar matrix must be considered to produce plastic scintillators. Despite the need 
+for additional MAA, these scintillators were promising for thermal-neutron detection.  
+At a small scale (≈ 1 cm), the plastic scintillator was responsive to an incident beam of 
+thermal neutrons from a research reactor.[18] However, the optical attenuation 
+properties were not assessed for this material but become important for large-volume 
+detectors, where the longest side of a single scintillator may be on the order of 10-100 
+cm.  
+
+Other developments of these scintillators that contain 6Li focused on 
+exploration of additional carboxylates.[19–22] One report described an investigation of 
+16 different 6Li salts. These 6Li salts were dissolved in a comonomer mixture of styrene 
+and methacrylic acid (90:10 styrene:MAA) to determine the maximum solubility of 6Li 
+at 60 oC. Additional 6Li could be dissolved as the MAA content increased, but the 
+scintillation light output and figure of merit (FoM) for PSD decreased as MAA and 6Li 
+content increased.[22] Thus, improvements in scintillation performance should 
+consider the balance between processing and performance. This consideration is 
+especially important in large plastic scintillators where thermal runaway[23] becomes 
+a concern for processing, and attenuation becomes a concern for performance. 
+In this report, we describe the synthesis and processing of lithium-loaded 
+plastic scintillators on the kilogram scale. We considered aspects related to the 
+composition of the plastic scintillator like the primary dye, secondary dye, monomers, 
+and lithium salts (Figure 1b) as well as aspects related to processing and curing like 
+dissolution and temperature of cure (Figure 1c). We first synthesized various 6Li salts 
+and characterized their solubility at different temperatures and with various 
+concentrations of comonomers. Then, we considered trade-offs in processing and 
+performance at the 10 g scale by evaluating the scintillation performance upon addition 
+of comonomer and 6Li salt in plastic scintillators that contained 2,5-diphenyloxazole 
+(PPO) as the primary dye. The light output of scintillation was reduced as the content 
+of the comonomer MAA increased in these scintillators.  
+We then compared the performance of plastic scintillators that contain PPO 
+vs. m-terphenyl (mTP) as the primary dye. The performance of plastic scintillators that 
+contain mTP as the primary dye is less sensitive to the addition of MAA. When scaling 
+our synthesis to 1 kg (Figure 1d), we targeted compositions that were available at scale 
+and allowed for lower temperatures of processing to avoid thermally initiated 
+polymerization and thermal runaway. All of these measures considered the potential for 
+production of large-scale plastic scintillators, which will be useful for applications like 
+mobile antineutrino detectors.  
+
+ 
+2. Materials and methods 
+Styrene (99%, Sigma), vinyl toluene (99%, Sigma), divinyl benzene (Sigma, 
+technical grade), and methyl methacrylate (99%, VWR) were passed through an 
+alumina column to remove inhibitor. Methacrylic acid (99%, Sigma) was dried over 
+sodium chloride and distilled under vacuum to remove inhibitor. All monomers were 
+sparged with nitrogen for > 30 min before being stored in a nitrogen-filled glovebox. 
+Figure 1. a) Plastic scintillators that are capable of PSD and contain lithium-6 distinguish between gamma rays, fast 
+neutrons, and thermal neutrons, as shown in this PSD plot. The gradient scale bar represents a relative population of data 
+points. b) The synthesis of plastic scintillators requires consideration of many components, as shown in this schematic. 
+The chemical structures, starting from the bottom left and going clockwise, are those of m-terphenyl, 2,5-diphenyloxazole, 
+Exalite 404, styrene, methacrylic acid, divinylbenzene, and three 6Li salts of carboxylic acids. c) The processing of the 
+precursors requires control of dissolution and curing conditions. d) Control of synthesis and processing has enabled the 
+production of large plastic scintillators, scaling to rectangular prisms like the ones in this photograph. The large 
+scintillators are about 0.41 m in length (total mass of about 1.5 kg).  
+ 
+
+0.8
+.0
+0.0
+0.5
+0.8
+Thermal neutrons
+0.7
+Fas, neurone
+0.3
+0.4
+Gamma raye
+0.2
+0.2
+0.1
+100 200
+300
+4010
+Approt, enengy
+e!All monomers except methacrylic acid were stored in an inert atmosphere at −20 oC. 
+Methacrylic acid was stored at room temperature. m-terphenyl (mTP, Smolecule) was 
+purified by recrystallization from toluene. L-231 (Luperox, 1,1-di(t-butylperoxy)-3,3,5-
+trimethylcyclohexane) was used as a radical initiator after sparging for > 30 min with 
+dry nitrogen and kept at −20 °C until needed. 2,5-diphenyloxazole (PPO, scintillation 
+grade from Sigma), 1,4-bis(2-methylstyryl)benzene (bisMSB, Luxottica/Exciton), and 
+1,4-bis(9,9-diethyl-7-(tert-pentyl)-9H-fluoren-2-yl)benzene (E404, Luxottica/Exciton) 
+were used as received without further purification.  
+6Li carboxylate salts were synthesized by first suspending 6Li2CO3 (National 
+Isotope Development Center) in a 1:1 mixture of water (deionized) and methanol 
+(99.8%, Sigma). Excess carboxylic acid (1.02 equivalent excess) was mixed into a 1:1 
+mixture of water and methanol. The basic suspension was slowly added to the acid 
+solution, and this mixture was heated to reflux for > 4 hours. The solution was filtered, 
+and the 6Li salt was precipitated by adding excess volume of cold acetone. The 6Li salt 
+was collected by vacuum filtration and washed with acetone, followed by drying under 
+vacuum at 80 oC. This procedure was used for carboxylate salts of pentanoic acid, 
+hexanoic acid, octanoic acid, 2-methylpropanoic acid, 2-methylbutanoic acid, 3-
+methylbutanoic acid, and 2-ethylhexanoic acid. All acids were purchased from Sigma 
+or VWR and used as received. 
+Plastics were synthesized in a dry nitrogen environment. For initial evaluation, 
+plastics were synthesized using 10 g of precursor materials. For the production of large 
+plastic scintillators, the amount of precursors used was up to 2.7 kg. All materials that 
+were not stored in a dry nitrogen environment were dried under vacuum. 6Li salts were 
+easiest to process when they were first dissolved in a 1:1 mixture by weight of 
+styrene:MAA at elevated temperatures (about 60-80 oC) before adding the remainder 
+of the plastic composition. A typical synthesis would involve dissolving the primary 
+dye (e.g., PPO) and the secondary dye (e.g., bis-MSB or E404) in styrene or vinyl 
+toluene (VT). The monomer VT was used as the polymer matrix for plastics that used 
+mTP as the primary dye. VT was used for these plastics as we observed less consistency 
+in solubility for mTP-based plastics that used styrene. The performance of plastic 
+scintillators that used styrene and VT were compared, and we observed no meaningful 
+
+difference in performance for these plastics. This precursor solution was heated to about 
+60-80 oC and mixed with a solution of the 6Li salt in 1:1 mixture by weight of 
+styrene:MAA. DVB (typically 5 wt. %) and initiator (0.08 wt. % for 10 g plastics; 0.01 
+wt. % for plastics batches greater than 400 g) were added. This mixture that contains 
+the precursor solution mentioned earlier and the solution that contains 6Li was poured 
+into a mould and sealed. The mould was placed in a nitrogen-filled oven and cured at 
+elevated temperatures. In one experiment, the viscosity of the precursor solution was 
+monitored using a rotary viscometer (Brookfield DV2T). 
+A typical curing profile would consist of heating for 7 days at 60 oC, followed 
+by a temperature ramp to 75 oC over one day. The scintillators were cured in convection 
+ovens (Cascade TEK) that were fitted with gas lines. Dry nitrogen flowed into solvent-
+resistance plastic bags that contained the mould inside the oven. The bags maintained 
+a positive pressure of nitrogen. The plastic would stay at 75 oC for four days and then 
+cool to room temperature over the course of one day. Following curing, the scintillators 
+were removed from the moulds, then machined and polished. All photographs of 
+samples were taken using a Nikon D750 and were globally edited in Adobe Lightroom 
+for colour and exposure corrections.  
+For initial scintillator characterization, samples of mass equal to 10 g were 
+measured. The outer edge and one face of the scintillators were wrapped and covered 
+with Teflon tape. The exposed face was coupled with optical grease to a Hamamatsu 
+R6231-100-SEL photomultiplier tube (PMT). Signals from the PMT were recorded at 
+a sampling rate of 200 MS/s using a 14-bit CompuScope 14200 waveform digitizer. A 
+relative quantification of light output (LO) was measured using ionizing radiation from 
+137Cs incident upon the plastic scintillator. The values of LO that we report in this 
+manuscript are specific to our measurement system and thus should only be used for 
+relative comparison. We normalized the value of LO to measurements of the 
+commercial scintillator EJ-200. The location of 500 keVee was defined by the value of 
+the pulse integral at 50% of the height of the 137Cs Compton edge. For many 
+measurements, duplicate samples were synthesized, and averages are reported. For one 
+condition, 9 samples were replicated in multiple batches and were measured. The 
+standard deviation of these measurements was within 7% of the average value, which 
+
+could be representative of the standard deviation related to contributions from 
+measurement and synthesis. Where standard deviation is not reported, a conservative 
+value of 10% of the value given could be assumed.  
+The measurement of effective attenuation length was performed with a longer 
+scintillator bar (1″ x 1″ x 16″). This measurement employed a setup that was identical 
+to the setup used to characterize scintillator bars in an antineutrino detector called 
+SANDD (Segmented AntiNeutrino Directional Detector).[24] This bar was wrapped 
+with polytetrafluoroethylene tape (POLY-TEMP PN-16050), and a pair of 1" 
+Hamamatsu R1924A-100 PMTs were mounted at either end of the scintillator bar using 
+EJ-550 silicone optical grease. The PMT operating voltage was set at -1100 V, and 
+signals were digitized using a Struck SIS3316 digitizer module (250 MS/s, 14 bit, 5 V 
+dynamic range). The energy threshold was set at approximately 0.1 MeVee, and 1600 
+ns-long waveforms were sent to disk and stored in ROOT data format. The charge 
+response difference between the two PMTs due to gain and optical coupling variation 
+was corrected using a collimated 137Cs gamma-ray source directed at the center of the 
+plastic bar. Here, lead bricks were used for shaping the gamma-ray source into a fan 
+beam of about 0.5 cm width.   
+To obtain the effective attenuation length of the scintillator bar, we performed 
+a series of collimated 22Na measurements at regular intervals along the length of the 
+bar. In each measurement, we identified the location of the Compton continuum 
+maximum of the 1.275 MeV gamma-ray by fitting the energy response with a Gaussian 
+profile while varying the range of the fit to find the minimum χ2 (best fit). The Compton 
+continuum maximum position was identified as the mean of the Gaussian profile that 
+yielded the minimum χ2. The associated uncertainty was estimated by varying the range 
+of the fit until the χ2 value exceeded the 68% confidence level (CL) of the minimum χ2; 
+the uncertainty was the corresponding range of the mean of the Gaussian profile. 
+To measure PSD in smaller plastics, plastic scintillators were exposed to a 
+252Cf source. The source was shielded behind 5.1 cm of lead to reduce the gamma-ray 
+flux. To obtain a flux of thermal neutrons, high density polyethylene was also used as 
+a moderator for 252Cf. The measurements of scintillation from plastic scintillators 
+
+exposed to 252Cf were integrated over time to determine the total charge (Qtotal). The 
+charge of the delayed component of the signal (Qtail) was determined from a delayed 
+fraction of the scintillation pulse. Scintillation pulses due to interactions of the 
+scintillator with neutrons have a larger fraction of Qtail relative to Qtotal; therefore, a 
+comparison of Qtail relative to Qtotal can be used to distinguish between scintillation due 
+to neutrons vs. gamma rays. The PSD was quantified using a figure of merit (FoM) that 
+is determined from histograms of the ratio of the charge of the delayed component 
+relative to the total charge, as described in previous reports[15,25]. Briefly, the FoM is: 
+𝐹𝑜𝑀 = 
+〈𝑛,𝑡〉−〈𝛾〉
+𝐹𝑊𝐻𝑀𝑛,𝑡+𝐹𝑊𝐻𝑀𝛾  
+ 
+ 
+(2) 
+In this equation, 〈n,t〉-〈γ〉 represents the difference between the average value of the 
+neutron and gamma-ray signals, and FWHMn,t +FWHMγ represents the sum of the full-
+width at half of the maximum value of the distributions of the thermal-neutron and 
+gamma-ray signals at the electron-equivalent energy of the thermal-neutron spot. For 
+plastics that don’t contain 6Li, the same equation was used for FoM, but the position 
+and FWHM of the neutron peak is used at an electron-equivalent energy near the 137Cs 
+Compton edge.  
+For PSD of the larger scintillator bar (1″ × 1″ × 16″), an identical setup as the 
+measurement of effective attenuation length was used. The bar was irradiated with an 
+uncollimated 252Cf source, and lead bricks with a total thickness of 6″ were placed 
+between the detector and the 252Cf source to reduce the gamma-ray flux. The charge 
+integration limits were optimized, and the best parameters were found to be [tL-20 ns ≤ 
+Qtotal ≤ tL+1300 ns] and [tL+24 ns ≤ Qtail ≤ tL+1300 ns], where tL is the leading edge of 
+the waveform. Assuming light transport behaves exponentially along the length of the 
+scintillator bar, we can eliminate the dependence of energy on event position by 
+reconstructing the energy as 𝐸 = √𝐸𝐴𝐸𝐵 , where EA and EB are the charges collected 
+by the two PMTs.  
+
+3. Results and discussion 
+3a. Selection of lithium-6 salt based on solubility at moderate 
+temperatures 
+ 
+To make large-scale production of plastic scintillators easier, our selection of 
+materials focused on those materials that had simple processing requirements (e.g., 
+temperatures below 80 oC). This requirement is necessary since high processing 
+temperatures for large plastic scintillators could thermally initiate the polymerization. 
+After dissolution of all dopants, we monitored the viscosity over time of liquid 
+precursors with and without the crosslinker divinylbenzene (DVB) at a temperature of 
+50 oC (Figure 2). The liquid precursors did not contain a radical initiator that initiates 
+polymerization; thus, any increase in viscosity is due to thermally initiated 
+polymerization. For precursor liquids that contained DVB, the viscosity began to 
+increase to measurable values above 0.1 Pa s after about 15600 s (4.3 hours) at 50 oC. 
+The viscosity further increased, reaching values greater than 40 Pa s after about 37400 
+s (10.4 hours) at 50 oC. While these timescales may be appropriate to process plastic 
+Figure 2. The viscosity of liquid precursors containing 30 wt. % PPO and 0.2 wt. % 
+bis-MSB can increase over time as polymerization occurs, and this increase in viscosity 
+prevents processing of the liquid into a mould. In this case, precursors with 8 wt. % 
+DVB and without DVB are compared while held at a temperature of 50 oC.  
+
+scintillators with industrial equipment, we observed that complete dissolution of all 
+components often required > 12 hours of stirring at elevated temperatures.  
+The premature onset of polymerization caused this increase in viscosity, which 
+could prevent trapped air bubbles from escaping or even prohibit transfer of the liquid 
+precursor to a mould. For precursor liquids that did not contain DVB, the viscosity 
+began to increase to measurable values above 0.1 Pa s after about 175000 s (48.6 hours) 
+at 50 oC. As before, the viscosity further increased and reached values greater than 40 
+Pa s after 275000 s (76.4 hours) 50 oC. Without DVB, the working time of the liquid 
+precursors can be increased. Still, the limited working time of these materials highlights 
+the need for simple processing requirements like low temperatures.  
+Table 1. Summary of solubility tests. Note that some salts that formed a gel phase 
+were initially soluble. For solubility tests at 23 oC, the solubility was observed after 20 
+hours of mixing. For solubility tests at 65 oC, the solubility was observed after 30 
+minutes at 65 oC following 2 hours at 50 oC. 
+Acid used for 6Li salt 
+Solubility, 23 oC, 85:15 styrene:MAA 
+Solubility, 65 oC, 85:15 styrene:MAA 
+Pentanoic acid 
+Insoluble 
+Soluble 
+Hexanoic acid 
+Insoluble 
+Soluble 
+Octanoic acid 
+Insoluble 
+Soluble 
+2-methylpropanoic acid 
+Formed gel 
+Soluble 
+2-methylbutanoic acid 
+Formed gel 
+Soluble 
+3-methylbutanoic acid 
+Soluble 
+Soluble 
+2-ethylhexanoic acid 
+Formed gel 
+Soluble 
+ 
+ Another requirement for simple processing relates to the relative solubility of 
+the 6Li salts. Thus, we evaluated the solubility of various carboxylate salts of 6Li while 
+
+considering the need for simple processing requirements. We synthesized 6Li salts of 
+pentanoic acid, hexanoic acid, octanoic acid, 2-methylpropanoic acid, 2-
+methylbutanoic acid, 3-methylbutanoic acid, and 2-ethylhexanoic acid (Figure 3a). 
+The synthesis and solubility of these salts have been described previously[22], but our 
+focus is on solubility for synthesis of large plastics, which requires further 
+considerations related to processing that have not been reported. We added the 6Li salts 
+to liquid precursors that contained all monomers and dopants. The composition of the 
+liquid precursor was as follows: 30 wt. % 2,5-diphenyloxazole (PPO); 0.2 wt. % 1,4-
+bis(2-methylstyryl)benzene (bisMSB); 5 wt. % DVB; an equivalent amount of 6Li salt 
+to obtain 0.1 wt. % 6Li; and the remainder was a mixture of 85 wt. % styrene and 15 
+wt. % methacrylic acid (MAA). The MAA is necessary to dissolve the 6Li salt; other 
+monomers like methyl methacrylate and methyl acrylate do not dissolve the 6Li salt.  
+All precursors were mixed into a single vial and allowed to equilibrate for 20 
+hours at room temperature (23 oC). Lithium-6 2-methylbutanoate was readily soluble 
+in the mixture initially. However, an opaque gel formed within 2 hours and persisted 
+after 20 hours (Figure 3b). Lithium-6 2-ethylhexanoate also formed a gel after initial 
+dissolution. For lithium-6 2-methylpropanoate, an opaque gel was observed after 20 
+hours, but this mixture never fully dissolved, suggesting low solubility of this 6Li salt. 
+Similarly, 6Li salts of the linear alkyl carboxylic acids (pentanoic acid, hexanoic acid, 
+and octanoic acid) never fully dissolved at 23 oC. For lithium-6 3-methylbutanoate, the 
+liquid precursor remained clear after 20 hours at 23 oC.  
+The opaque gels that we observed could be destabilized after heating to 
+elevated temperatures. All vials were heated to 50 oC for 2 hours, which improved 
+dissolution of all components that were insoluble at room temperature. For example, 
+the liquids that contained 6Li salts of 2-methylbutanoic acid and 2-ethylhexanoic acid 
+became transparent. Further heating to 65 oC for 30 minutes improved dissolution; all 
+
+liquid precursors with different 6Li salts were transparent after this heating step except 
+for the precursor that contained lithium-6 2-methylpropanoate (Figure 3c).  
+Based on this analysis, we selected the 6Li salt of 3-methylbutanoic acid for 
+the synthesis of large plastics; however, other 6Li salts like lithium-6 pentanoate and 
+lithium-6 2-methylbutanoate may also be suitable given that they form transparent 
+precursors after dissolution at 65 oC. Furthermore, we sometimes observed the 
+formation of a gel phase for precursors that contained lithium-6 3-methylbutanoate 
+when using less MAA, which highlights how plastic scintillators that contain this 6Li 
+salt are not immune to this processing challenge.  
+To avoid the formation of this gel phase, the 6Li salt could be dissolved 
+separately from the rest of the dopants. A 1:1 mixture of styrene and MAA was 
+sufficient to avoid thermally initiated homopolymerization of MAA during dissolution; 
+homopolymerization of MAA results in an opaque material. The remainder of the 
+styrene needed to form the final plastic was used to dissolve the primary and secondary 
+Figure 3. a) Chemical structures of 6Li salts that were studied; from left to right the structures correspond to 
+6Li salts of pentanoic acid, hexanoic acid, octanoic acid, 2-methylpropanoic acid, 2-methylbutanoic acid, 3-
+methylbutanoic acid, and 2-ethylhexanoic acid. b) Photographs of liquid precursors in vials following 
+equilibration for 20 hours at 23 oC. c) Photographs of liquid precursors in vials following equilibration at an 
+additional 2 hours at 50 oC plus 30 minutes at 65 oC. All liquid precursors contain all components of a plastic 
+scintillator except the radical initiator. The precursors differ in the 6Li salt that was added; from left to right, the 
+vials contain 6Li salts of pentanoic acid, hexanoic acid, octanoic acid, 2-methylpropanoic acid, 3-methylbutanoic 
+acid, 2-methylbutanoic acid, and 2-ethylhexanoic acid. The vials are 28 mm in diameter.  
+
+b)23 °C; 85:15 styrene:MAA
+c)65C:85:15styrene:MAAdye in a separate container. Then, the two separate mixtures could be heated to 60-80 
+oC and mixed at elevated temperatures before adding DVB and the radical initiator and 
+casting in a mould.  
+3b. Effect of comonomer on scintillation performance 
+Importantly, the effect of the comonomer that solubilizes the 6Li salts on 
+scintillation performance should be evaluated. The addition of non-aromatic 
+comonomers like methyl methacrylate (MMA) or methacrylic acid (MAA) can reduce 
+scintillation performance.[22,26] When 26 wt. % of MMA was used in a plastic 
+scintillator, the light yield reduced by 5% when compared to a plastic scintillator that 
+contained only polystyrene as the matrix. When 58 wt. % of MMA was used in a plastic 
+scintillator, the light yield reduced by 18%.[26] The total amount of the comonomer 
+Figure 4. a) A detrimental effect of MAA and 6Li salts reduce light output (LO) of scintillators that contain PPO 
+as a primary dye. 6Li salt may also influence light output. b) The LO decreases as the content of MAA increases, 
+which can be observed in the histograms that show the 137Cs Compton edge. c) The photograph of these vials highlights 
+the solubility threshold of lithium-6 3-methylbutanoate at 65 oC. The vials are 28 mm in diameter. d) Less substantial 
+effect on LO by MAA and 6Li salt for scintillators that contain mTP instead of PPO. The 6Li salt used for all samples 
+referenced in this figure was lithium-6 3-methylbutanoate. For a) and b), the average (dashed line) and standard 
+deviation (grey shaded region) of scintillators that do not contain any comonomer are shown for reference. 
+
+65°C:6LitO
+MAA weight content:
+6.3%
+6.9%
+7.6%
+8.2%
+9.5%
+Precipitate still
+presentMAA added for dissolution of 6Li salts is typically less than 20 wt. % of the total 
+material, so we instead focused on comonomer addition at these lower concentrations.  
+Table 2. Summary of effect of composition on light output.  
+Primary dye 
+Co-monomer 
+Co-monomer content 
+Lithium salt content 
+Light output 
+PPO 
+N/A 
+0 
+0 
+1.05 
+mTP 
+N/A 
+0 
+0 
+1.12 
+ 
+ 
+ 
+ 
+ 
+PPO 
+MMA 
+0.6 
+0 
+1.03 
+PPO 
+MMA 
+3 
+0 
+1.07 
+PPO 
+MMA 
+6 
+0 
+1.05 
+PPO 
+MMA 
+13 
+0 
+1.09 
+ 
+ 
+ 
+ 
+ 
+PPO 
+MAA 
+0.6 
+0 
+0.96 
+PPO 
+MAA 
+2 
+0 
+0.81 
+PPO 
+MAA 
+3 
+0 
+0.78 
+PPO 
+MAA 
+5 
+0 
+0.77 
+PPO 
+MAA 
+6 
+0 
+0.73 
+PPO 
+MAA 
+13 
+0 
+0.64 
+ 
+ 
+ 
+ 
+ 
+PPO 
+MAA 
+3 
+1.7 
+0.61 
+PPO 
+MAA 
+5 
+1.7 
+0.57 
+PPO 
+MAA 
+6 
+1.7 
+0.56 
+PPO 
+MAA 
+13 
+1.7 
+0.51 
+ 
+ 
+ 
+ 
+ 
+mTP 
+MAA 
+3 
+1.7 
+0.93 
+mTP 
+MAA 
+5 
+1.7 
+0.95 
+mTP 
+MAA 
+6 
+1.7 
+0.93 
+mTP 
+MAA 
+13 
+1.7 
+0.92 
+ 
+Even though MMA does not solubilize 6Li salts that we studied, we used this 
+comonomer as a non-aromatic additive to compare its effect on performance to 
+scintillators that contain the solubilizing comonomer, MAA. We also compared the 
+performance of plastic scintillators that did not contain any comonomer. The average 
+light output (LO) of three separate samples that did not contain any comonomer was 
+1.05 with a standard deviation of 0.07 (Figure 4a, Table 2). With 0.6 wt. % MMA 
+added, the LO was 1.03. As the amount of MMA increased, there was no clear trend in 
+
+the LO. At 13 wt. % MMA, the LO was 1.09, and all measured values of LO were 
+within a standard deviation of the average value of plastic scintillators that did not 
+contain a comonomer. Thus, at these concentrations of MMA comonomer, the energy 
+transfer and light emission do not appear to be affected.  
+The same trend did not persist when using MAA. At 0.6 wt. % MAA, the LO 
+was 0.96, which corresponds to a 9% reduction in LO when compared to plastic 
+scintillators without this comonomer. At 1.6% MAA, the LO was further reduced to 
+0.81, which is a 23% reduction. The LO continues to decrease as MAA content 
+increases (Figure 4b, Table 2), but the magnitude of reduction appears to taper as the 
+MAA content exceeds 3 wt. %. At 13 wt. % MAA, the LO was reduced to 0.64, which 
+corresponds to a 40% reduction. The discrepancy between the effects of MMA and 
+MAA on LO indicates that the decrease in LO upon addition of MAA does not result 
+from simple dilution by a non-aromatic material. Rather, this decrease in LO suggests 
+that MAA may be detrimental to processes that affect scintillation like energy transfer 
+or emission. Although the exact mechanism is not fully elucidated, it is possible that 
+the heteroatoms on PPO (N, O) may interact with the polar acid functional group on 
+MAA.  
+Notably, the LO further decreases upon addition of 6Li salts to plastics that 
+contain PPO as the primary dye along with MAA as a comonomer when compared to 
+Figure 5. PSD distributions used to calculate FoM for comparison of plastics that contain PPO (a) and mTP (b). 
+Note that the FoMs for 6Li-plastics are given for discrimination between thermal neutrons and gamma rays whereas 
+FoMs for plastics without 6Li correspond to discrimination between fast neutrons and gamma rays. 
+
+the addition of MAA alone. At 13 wt. % MAA, the LO decreased from 0.64 to 0.52 
+when adding the 6Li salt. The reduction in LO with increasing MAA content and 
+addition of 6Li salt poses a challenge related to processing: higher content of MAA 
+allows for processing of the plastic scintillator at lower temperatures but reduces LO. 
+Plastic scintillators with lower content of MAA are more difficult to process due to poor 
+solubility of the 6Li salt. For example, lithium-6 3-methylbutanoate is not fully soluble 
+at 65 oC when the MAA content is equal to or less than 8 wt. % (Figure 4c). The cause 
+of the reduction in performance upon addition of 6Li salt may be similar to the reduction 
+in performance upon addition of MAA; PPO may have unfavourable interactions and/or 
+reactivity with these polar molecules.  
+To test this idea, we compared the scintillation performance when using 30 wt. 
+% m-terphenyl (mTP) as a primary dye instead of PPO. The chemical structure of mTP 
+(Figure 1b) does not contain any heteroatoms (i.e., only contains C, H). Plastics that 
+used mTP as the primary dye but did not contain MAA or a 6Li salt had an average LO 
+of 1.12 (Figure 4d, Table 2). Plastics that contained MAA and 6Li had a slight 
+reduction in LO with values of 0.93, 0.95, 0.93, and 0.92 at 3 wt. %, 5 wt. %, 6 wt. %, 
+and 13 wt. % MAA. At 13 wt. % MAA, the LO of a plastic that contains 6Li was reduced 
+by 18% when compared to a plastic that contained no MAA or 6Li. This value of LO is 
+nearly double the LO of an equivalent plastic that contained PPO instead of mTP as the 
+primary dye.  
+We also compared the FoM for PSD. For plastics that contain PPO but no 6Li, 
+the FoM is 3.05, which provides a baseline for comparison after addition of MAA and 
+6Li. Note that this FoM compares discrimination between gamma rays and fast neutrons 
+whereas the FoM for plastics that contain 6Li compares discrimination between gamma 
+rays and thermal neutrons. The energy range used to determine FoM was 450-550 
+keVee, but that range was adjusted to capture the thermal neutron peak. Upon addition 
+of 13 wt. % MAA and 6Li, the FoM decreases to 2.24 and decreases for all plastics that 
+were measured that contained MAA (Figure 5a).  
+These PSD results can also be compared to plastics that contain mTP instead 
+of PPO (Figure 5b). For plastics that contain mTP but no 6Li, the FoM is 2.73. Upon 
+
+addition of MAA and 6Li, the FoM is between 2.95 and 3.24. This increase is mostly 
+attributed to the high value of Qtail/Qtotal for the thermal-neutron capture spot along with 
+the observation that plastics that contain mTP are less sensitive to the addition of the 
+polar compounds that enable thermal-neutron capture. The FoM for plastics that 
+contained PPO decreased when MAA and 6Li were added whereas the FoM for plastics 
+that contained mTP increased when MAA and 6Li were added.  
+3c. Production of large plastic scintillators 
+As shown above, mTP may be promising for high performance scintillators 
+that contain 6Li; however, the inconsistencies with solubility and the current availability 
+at large scales in sufficient purity from commercial suppliers make mTP less suitable 
+Figure 6. a,b) Conditions that are not optimized for the production of large plastics produce defects like cracks and 
+bubbles (a) and discoloration (b). The left-most scintillator in (a) does not contain defects and serves as a reference. 
+These scintillators in (a) have a diameter of about 5.5 cm. The plastic scintillator in (b) is 40 cm in length. c) When the 
+curing conditions and environment are controlled, plastics scintillators can be produced in large scale, as shown in this 
+photograph of a scintillator that is 5 cm in width atop a sheet of paper. d) The effective attenuation length was measured 
+by placing a collimated gamma-ray source at set distances away from two PMTs and measuring the PMT response 
+(black). The data were fitted with an exponential profile to estimate the effective attenuation length (19-21cm). e) Large 
+plastic scintillators are capable of PSD, as shown in this distribution that demonstrates an ability to separate signals from 
+thermal neutrons and gamma rays.  
+
+b)for the production of large plastic scintillators. For these large plastic scintillators, we 
+selected PPO despite its lower LO and lower FoM at smaller scales.  
+Similarly, other secondary dyes like Exalite 404 (E404) may have the best 
+performance in the plastic scintillators that we evaluated[27], but the cost of E404 might 
+be prohibitive when compared to a secondary dye like 1,4-bis(2-methylstyryl)benzene 
+(bisMSB).  
+When processing the plastics, all precursors except DVB and the radical 
+initiator are slowly heated to temperatures between 60 and 80 oC until full dissolution. 
+The plastics are cured in glass or aluminium moulds. To control the rate of 
+polymerization, the radical initiator is added at concentrations of 0.01 wt. %, and 
+plastics are cured at an initial temperature of 60 oC. Various times of curing were used, 
+and a typical recipe would involve curing for 7 days at 60 oC followed by an additional 
+4 days of curing at 75 oC. Excessive radical concentration and/or heating during 
+processing and curing could lead to defects like cracks and bubbles (Figure 6a). 
+Precipitation of precursors that have lower solubility may occur if temperatures are too 
+low. When curing the plastics, oxygen is displaced by a steady flow of nitrogen; without 
+nitrogen flow, discoloration can occur (Figure 6b). These precautionary measures  
+allow us to produce plastic scintillators without defects and with minimal 
+discoloration (Figure 6c). The effective attenuation of a plastic scintillator that was 16” 
+long was measured by placing a collimated gamma-ray source near the scintillator and 
+measuring the total light detected by PMTs that are mounted on each end of the plastic 
+scintillator (Figure 6d). The effective attenuation was determined to be about 19-21 
+cm, which is comparable to the value obtained in previous experiments.[24] This plastic 
+also had PSD capability; a thermal-neutron spot is clearly separated from the gamma 
+signal (Figure 6e). After we optimized our process for synthesis of these large plastics, 
+we outsourced production to Eljen Technologies who is currently producing 6Li-loaded 
+prototypes with dimensions exceeding 0.5 m; full characterization of scintillator 
+performance at these large scales will be the subject of a future publication.   
+
+4. Conclusion 
+By careful control of composition and processing, plastic scintillators that can 
+discriminate between gamma rays, fast neutrons, and thermal neutrons can be produced 
+at a scale of 1 kg or greater. The solubility of dopants that enable scintillation 
+functionality and solubilizing additives like methacrylic acid (MAA) that may 
+negatively affect performance were considered. Synthesis and processing procedures 
+were developed for large plastic scintillators containing 0.1 wt. % 6Li and high 
+concentration (30 wt. %) of PPO used as a primary dye. These scintillators were capable 
+of PSD. In these studies, various 6Li salts of aliphatic carboxylic acids were evaluated, 
+and many were found to be suitable for the production of large plastic scintillators with 
+the addition of MAA. The amount of MAA that was added to solubilize 6Li salts 
+affected the scintillation performance but also determined the temperature that plastic 
+scintillators could be produced at. An alternative way to avoid the deleterious effects of 
+MAA was discovered; use of m-terphenyl instead of PPO improved plastic scintillators. 
+However, m-terphenyl may have limitations like availability at large volumes.  
+With these considerations in mind, methods for the preparation of plastic 
+scintillators loaded with 6Li were established and demonstrated. Large-volume pieces 
+that could be used for large detectors were produced.[24,28] Such detectors will be 
+important for future safeguards related to nuclear power production and for unravelling 
+unknown aspects of particle physics.   
+Acknowledgements 
+This work was performed under the auspices of the U.S. Department of Energy by 
+Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 and 
+was supported by the LLNL-LDRD Program under Project No. 20-SI-003, release 
+number LLNL-JRNL-839909. We would like to thank Jacob Kim for careful reading 
+and discussion of this manuscript.  
+ 
+The authors declare no competing interests.  
+
+References 
+[1] 
+W.J. Nuttall, Nuclear Renaissance: Technologies and Policies for the Future of 
+Nuclear Power, CRC Press, 2022. 
+[2] 
+J.F. Cavaignac, A. Hoummada, D.H. Koang, B. Vignon, Y. Declais, H. de Kerret, 
+H. Pessard, J.M. Thenard, Indication for neutrino oscillation from a high statistics 
+experiment at the bugey reactor, Phys. Lett. B. 148 (1984) 387–394. 
+https://doi.org/10.1016/0370-2693(84)90109-6. 
+[3] 
+B. Achkar, R. Aleksan, M. Avenier, G. Bagieu, J. Bouchez, R. Brissot, J.-F. 
+Cavaignac, J. Collot, M.-C. Cousinou, J.P. Cussonneau, Y. Declais, Y. Dufour, 
+J. Favier, F. Garciaz, E. Kajfasz, H. de Kerret, D.H. Koang, B. Lefièvre, E. 
+Lesquoy, J. Mallet, A. Metref, E. Nagy, H. Pessard, F. Pierre, M. Obolensky, A. 
+Stutz, J.P. Wuthrick, Search for neutrino oscillations at 15, 40 and 95 meters from 
+a nuclear power reactor at Bugey, Nucl. Phys. B. 434 (1995) 503–532. 
+https://doi.org/10.1016/0550-3213(94)00513-E. 
+[4] 
+M. Abbes, B. Achkar, S. Ait-Boubker, R. Aleksan, M. Avenier, G. Bagieu, J. 
+Ballansat, C. Barnoux, R. Bazzoli, J. Berger, M. Bermond, P. Besson, M. 
+Billault, J. Boucher, J. Bouchez, M. Bouriant, R. Brissot, B. Camberlin, J.F. 
+Cavaignac, P. Charvin, J. Collot, A. Commerçon, M.-C. Cousinou, J.P. 
+Cussonneau, G. Daguin-Moynot, Y. Declais, T. Desanlis, J.-M. Dubois, Y. 
+Dufour, G. Farrache, J. Favier, Y. Gally, F. Garciaz, L. Giacobone, B. Guerre-
+Chaley, J.-P. Jobez, D. Jourde, E. Kajfasz, H. de Kerret, D.H. Koang, B. Lefièvre, 
+F. Léon, E. Lesquoy, J. Mallet, A. Menthe, A. Metref, J. Mullié, E. Nagy, M. 
+Obolensky, P. Ollive, A. Oriboni, H. Pessard, F. Pierre, J. Poinsignon, R. 
+Potheau, R. Provasi, A. Stutz, J. Thion, J.-F. Thomas, J.P. Wuthrick, The Bugey 
+3 neutrino detector, Nucl. Instrum. Methods Phys. Res. Sect. Accel. 
+Spectrometers 
+Detect. 
+Assoc. 
+Equip. 
+374 
+(1996) 
+164–187. 
+https://doi.org/10.1016/0168-9002(96)00220-3. 
+[5] 
+Daya Bay Collaboration, A Precision Measurement of the Neutrino Mixing 
+Angle theta_13 using Reactor Antineutrinos at Daya Bay, arXiv, 2007. 
+https://doi.org/10.48550/arXiv.hep-ex/0701029. 
+[6] 
+A. Bernstein, N. Bowden, B.L. Goldblum, P. Huber, I. Jovanovic, J. Mattingly, 
+Colloquium: Neutrino detectors as tools for nuclear security, Rev. Mod. Phys. 92 
+(2020) 011003. https://doi.org/10.1103/RevModPhys.92.011003. 
+[7] 
+N.S. Bowden, A. Bernstein, M. Allen, J.S. Brennan, M. Cunningham, J.K. 
+Estrada, C.M.R. Greaves, C. Hagmann, J. Lund, W. Mengesha, T.D. Weinbeck, 
+C.D. Winant, Experimental results from an antineutrino detector for cooperative 
+monitoring of nuclear reactors, Nucl. Instrum. Methods Phys. Res. Sect. Accel. 
+Spectrometers 
+Detect. 
+Assoc. 
+Equip. 
+572 
+(2007) 
+985–998. 
+https://doi.org/10.1016/j.nima.2006.12.015. 
+[8] 
+Y. Kuroda, S. Oguri, Y. Kato, R. Nakata, Y. Inoue, C. Ito, M. Minowa, A mobile 
+antineutrino detector with plastic scintillators, Nucl. Instrum. Methods Phys. Res. 
+Sect. Accel. Spectrometers Detect. Assoc. Equip. 690 (2012) 41–47. 
+https://doi.org/10.1016/j.nima.2012.06.040. 
+[9] 
+Nucifer Collaboration, G. Boireau, L. Bouvet, A.P. Collin, G. Coulloux, M. 
+Cribier, H. Deschamp, V. Durand, M. Fechner, V. Fischer, J. Gaffiot, N. Gérard 
+Castaing, R. Granelli, Y. Kato, T. Lasserre, L. Latron, P. Legou, A. Letourneau, 
+
+D. Lhuillier, G. Mention, Th.A. Mueller, T.-A. Nghiem, N. Pedrol, J. Pelzer, M. 
+Pequignot, Y. Piret, G. Prono, L. Scola, P. Starzinski, M. Vivier, E. Dumonteil, 
+D. Mancusi, C. Varignon, C. Buck, M. Lindner, J. Bazoma, S. Bouvier, V.M. 
+Bui, V. Communeau, A. Cucoanes, M. Fallot, M. Gautier, L. Giot, G. Guilloux, 
+M. Lenoir, J. Martino, G. Mercier, T. Milleto, N. Peuvrel, A. Porta, N. Le Quéré, 
+C. Renard, L.M. Rigalleau, D. Roy, T. Vilajosana, F. Yermia, Online monitoring 
+of the Osiris reactor with the Nucifer neutrino detector, Phys. Rev. D. 93 (2016) 
+112006. https://doi.org/10.1103/PhysRevD.93.112006. 
+[10] A. Haghighat, P. Huber, S. Li, J.M. Link, C. Mariani, J. Park, T. Subedi, 
+Observation of Reactor Antineutrinos with a Rapidly Deployable Surface-Level 
+Detector, 
+Phys. 
+Rev. 
+Appl. 
+13 
+(2020) 
+034028. 
+https://doi.org/10.1103/PhysRevApplied.13.034028. 
+[11] J. Ashenfelter, B. Balantekin, H.R. Band, G. Barclay, C.D. Bass, D. Berish, N.S. 
+Bowden, A. Bowes, C.D. Bryan, J.P. Brodsky, J.J. Cherwinka, R. Chu, T. 
+Classen, K. Commeford, D. Davee, D. Dean, G. Deichert, M.V. Diwan, M.J. 
+Dolinski, J. Dolph, J.K. Gaison, A. Galindo-Uribarri, K. Gilje, A. Glenn, B.W. 
+Goddard, M. Green, K. Han, S. Hans, K.M. Heeger, B. Heffron, D.E. Jaffe, D. 
+Jones, T.J. Langford, B.R. Littlejohn, D.A.M. Caicedo, R.D. McKeown, M.P. 
+Mendenhall, P. Mueller, H.P. Mumm, J. Napolitano, R. Neilson, D. Norcini, D. 
+Pushin, X. Qian, E. Romero, R. Rosero, B.S. Seilhan, R. Sharma, S. Sheets, P.T. 
+Surukuchi, R.L. Varner, B. Viren, W. Wang, B. White, C. White, J. Wilhelmi, C. 
+Williams, T. Wise, H. Yao, M. Yeh, Y.-R. Yen, G. Zangakis, C. Zhang, X. 
+Zhang, The PROSPECT Physics Program, J. Phys. G Nucl. Part. Phys. 43 (2016) 
+113001. https://doi.org/10.1088/0954-3899/43/11/113001. 
+[12] PROSPECT Collaboration, J. Ashenfelter, A.B. Balantekin, H.R. Band, C.D. 
+Bass, D.E. Bergeron, D. Berish, N.S. Bowden, J.P. Brodsky, C.D. Bryan, J.J. 
+Cherwinka, T. Classen, A.J. Conant, A.A. Cox, D. Davee, D. Dean, G. Deichert, 
+M.V. Diwan, M.J. Dolinski, A. Erickson, M. Febbraro, B.T. Foust, J.K. Gaison, 
+A. Galindo-Uribarri, C.E. Gilbert, K.E. Gilje, B.T. Hackett, S. Hans, A.B. 
+Hansell, K.M. Heeger, J. Insler, D.E. Jaffe, X. Ji, D.C. Jones, O. Kyzylova, C.E. 
+Lane, T.J. Langford, J. LaRosa, B.R. Littlejohn, X. Lu, D.A.M. Caicedo, J.T. 
+Matta, R.D. McKeown, M.P. Mendenhall, J.M. Minock, P.E. Mueller, H.P. 
+Mumm, J. Napolitano, R. Neilson, J.A. Nikkel, D. Norcini, S. Nour, D.A. Pushin, 
+X. Qian, E. Romero-Romero, R. Rosero, D. Sarenac, P.T. Surukuchi, A.B. 
+Telles, M.A. Tyra, R.L. Varner, B. Viren, C. White, J. Wilhelmi, T. Wise, M. 
+Yeh, Y.-R. Yen, A. Zhang, C. Zhang, X. Zhang, Measurement of the 
+Antineutrino Spectrum from $^{235}$U Fission at HFIR with PROSPECT, 
+Phys. 
+Rev. 
+Lett. 
+122 
+(2019) 
+251801. 
+https://doi.org/10.1103/PhysRevLett.122.251801. 
+[13] F.D. Brooks, R.W. Pringle, B.L. Funt, Pulse Shape Discrimination in a Plastic 
+Scintillator, 
+IRE 
+Trans. 
+Nucl. 
+Sci. 
+7 
+(1960) 
+35–38. 
+https://doi.org/10.1109/TNS2.1960.4315733. 
+[14] N. Zaitseva, B.L. Rupert, I. PaweŁczak, A. Glenn, H.P. Martinez, L. Carman, M. 
+Faust, N. Cherepy, S. Payne, Plastic scintillators with efficient neutron/gamma 
+pulse shape discrimination, Nucl. Instrum. Methods Phys. Res. Sect. Accel. 
+Spectrometers 
+Detect. 
+Assoc. 
+Equip. 
+668 
+(2012) 
+88–93. 
+https://doi.org/10.1016/j.nima.2011.11.071. 
+
+[15] N.P. Zaitseva, A.M. Glenn, A.N. Mabe, M.L. Carman, C.R. Hurlbut, J.W. Inman, 
+S.A. Payne, Recent developments in plastic scintillators with pulse shape 
+discrimination, Nucl. Instrum. Methods Phys. Res. Sect. Accel. Spectrometers 
+Detect. 
+Assoc. 
+Equip. 
+889 
+(2018) 
+97–104. 
+https://doi.org/10.1016/j.nima.2018.01.093. 
+[16] G.H.V. Bertrand, M. Hamel, S. Normand, F. Sguerra, Pulse shape discrimination 
+between (fast or thermal) neutrons and gamma rays with plastic scintillators: 
+State of the art, Nucl. Instrum. Methods Phys. Res. Sect. Accel. Spectrometers 
+Detect. 
+Assoc. 
+Equip. 
+776 
+(2015) 
+114–128. 
+https://doi.org/10.1016/j.nima.2014.12.024. 
+[17] I.A. Pawełczak, A.M. Glenn, H.P. Martinez, M.L. Carman, N.P. Zaitseva, S.A. 
+Payne, Boron-loaded plastic scintillator with neutron-γ pulse shape 
+discrimination capability, Nucl. Instrum. Methods Phys. Res. Sect. Accel. 
+Spectrometers 
+Detect. 
+Assoc. 
+Equip. 
+751 
+(2014) 
+62–69. 
+https://doi.org/10.1016/j.nima.2014.03.027. 
+[18] R.D. Breukers, C.M. Bartle, A. Edgar, Transparent lithium loaded plastic 
+scintillators for thermal neutron detection, Nucl. Instrum. Methods Phys. Res. 
+Sect. Accel. Spectrometers Detect. Assoc. Equip. 701 (2013) 58–61. 
+https://doi.org/10.1016/j.nima.2012.10.080. 
+[19] N. Zaitseva, A. Glenn, H. Paul Martinez, L. Carman, I. Pawełczak, M. Faust, S. 
+Payne, Pulse shape discrimination with lithium-containing organic scintillators, 
+Nucl. Instrum. Methods Phys. Res. Sect. Accel. Spectrometers Detect. Assoc. 
+Equip. 729 (2013) 747–754. https://doi.org/10.1016/j.nima.2013.08.048. 
+[20] N.J. Cherepy, R.D. Sanner, P.R. Beck, E.L. Swanberg, T.M. Tillotson, S.A. 
+Payne, C.R. Hurlbut, Bismuth- and lithium-loaded plastic scintillators for gamma 
+and neutron detection, Nucl. Instrum. Methods Phys. Res. Sect. Accel. 
+Spectrometers 
+Detect. 
+Assoc. 
+Equip. 
+778 
+(2015) 
+126–132. 
+https://doi.org/10.1016/j.nima.2015.01.008. 
+[21] A.N. Mabe, A.M. Glenn, M.L. Carman, N.P. Zaitseva, S.A. Payne, Transparent 
+plastic scintillators for neutron detection based on lithium salicylate, Nucl. 
+Instrum. Methods Phys. Res. Sect. Accel. Spectrometers Detect. Assoc. Equip. 
+806 (2016) 80–86. https://doi.org/10.1016/j.nima.2015.09.111. 
+[22] C. Frangville, M. Hamel, G.H. V. Bertrand, E. Montbarbon, A. Grabowski, C. 
+Lynde, Large solubility of lithium carboxylates reaching high rates of 6 Li 
+incorporation in polystyrene-based plastic scintillators for fast/thermal neutron 
+and gamma ray detection, Mater. Chem. Front. 3 (2019) 1626–1631. 
+https://doi.org/10.1039/C9QM00153K. 
+[23] A. Husain, A.E. Hamielec, Thermal polymerization of styrene, J. Appl. Polym. 
+Sci. 22 (1978) 1207–1223. https://doi.org/10.1002/app.1978.070220505. 
+[24] F. Sutanto, T.M. Classen, S.A. Dazeley, M.J. Duvall, I. Jovanovic, V.A. Li, A.N. 
+Mabe, E.T.E. Reedy, T. Wu, SANDD: A directional antineutrino detector with 
+segmented 6Li-doped pulse-shape-sensitive plastic scintillator, Nucl. Instrum. 
+Methods Phys. Res. Sect. Accel. Spectrometers Detect. Assoc. Equip. 1006 
+(2021) 165409. https://doi.org/10.1016/j.nima.2021.165409. 
+[25] A.N. Mabe, A.M. Glenn, M.L. Carman, N.P. Zaitseva, S.A. Payne, Transparent 
+plastic scintillators for neutron detection based on lithium salicylate, Nucl. 
+
+Instrum. Methods Phys. Res. Sect. Accel. Spectrometers Detect. Assoc. Equip. 
+806 (2016) 80–86. https://doi.org/10.1016/j.nima.2015.09.111. 
+[26] H. Paul Martinez, I. Pawelczak, A.M. Glenn, M. Leslie Carman, N. Zaitseva, S. 
+Payne, Pulse shape discrimination in non-aromatic plastics, Nucl. Instrum. 
+Methods Phys. Res. Sect. Accel. Spectrometers Detect. Assoc. Equip. 771 (2015) 
+28–31. https://doi.org/10.1016/j.nima.2014.10.023. 
+[27] N.P. Zaitseva, A.M. Glenn, M.L. Carman, A.N. Mabe, S.A. Payne, N. Marom, 
+X. Wang, Multiple dye interactions in plastic scintillators: Effects on pulse shape 
+discrimination, Nucl. Instrum. Methods Phys. Res. Sect. Accel. Spectrometers 
+Detect. 
+Assoc. 
+Equip. 
+978 
+(2020) 
+164455. 
+https://doi.org/10.1016/j.nima.2020.164455. 
+[28] V.A. Li, T.M. Classen, S.A. Dazeley, M.J. Duvall, I. Jovanovic, A.N. Mabe, 
+E.T.E. Reedy, F. Sutanto, A prototype for SANDD: A highly-segmented pulse-
+shape-sensitive plastic scintillator detector incorporating silicon photomultiplier 
+arrays, Nucl. Instrum. Methods Phys. Res. Sect. Accel. Spectrometers Detect. 
+Assoc. Equip. 942 (2019) 162334. https://doi.org/10.1016/j.nima.2019.162334. 
+ 
+ 
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+Contextual Autonomy Evaluation of Unmanned
+Aerial Vehicles in Subterranean Environments
+Ryan Donald
+Peter Gavriel
+Adam Norton
+S. Reza Ahmadzadeh
+PeARL lab and NERVE Center
+University of Massachusetts Lowell
+Lowell, USA
+Ryan [email protected]
+Peter [email protected]
+Adam [email protected]
+Reza [email protected]
+Abstract—In this paper we focus on the evaluation of con-
+textual autonomy for robots. More specifically, we propose a
+fuzzy framework for calculating the autonomy score for a
+small Unmanned Aerial Systems (sUAS) for performing a task
+while considering task complexity and environmental factors.
+Our framework is a cascaded Fuzzy Inference System (cFIS)
+composed of combination of three FIS which represent dif-
+ferent contextual autonomy capabilities. We performed several
+experiments to test our framework in various contexts, such as
+endurance time, navigation, take off/land, and room clearing,
+with seven different sUAS. We introduce a predictive measure
+which improves upon previous predictive measures, allowing for
+previous real-world task performance to be used in predicting
+future mission performance.
+Index Terms—Contextual Autonomy, Unmanned Aerial Vehi-
+cles, Fuzzy Systems
+I. INTRODUCTION
+In today’s world, robots are expected to become increasingly
+present by assisting humans in performing various tasks in
+different environments. While some robots have been designed
+for a single purpose, others can accomplish a variety of tasks
+with different levels of autonomy. Measuring robot autonomy
+is an important and ever evolving concept and existing meth-
+ods for evaluating robot autonomy can be categorized into
+two main families: contextual and non-contextual. While the
+former methods consider mission and task-specific measures
+(e.g., ALFUS [1], ACL [2]), the latter only rely on implicit
+system capabilities and do not consider the mission and
+environment features (e.g., NCAP [3], [4]).
+Our study in this paper focuses on evaluating the contextual
+autonomy for small Unmanned Aerial Systems (sUAS). Exist-
+ing methods such as ALFUS [1] and MPP [5] share a similar
+shortcoming in that neither provides a simple implementation
+for use with real-world systems. Another drawback of existing
+methods that our approach addresses is the lack of a consistent
+process for breaking down tasks into sub-tasks and combining
+scores calculated for sub-tasks into a unified score for the given
+task. In this paper we propose a method for evaluating the
+contextual autonomy of sUAS based on a fuzzy interface that
+allows the operator to design and modify the evaluation system
+using linguistic reasoning. We designed four indoor tasks
+(endurance time, navigation, takeoff/land, and room clearing)
+and tested our interface in various experiments with seven
+different sUAS. Our results show that the proposed approach
+calculates a contextual autonomy score that can be used to
+rank the systems for each context.
+II. RELATED WORK
+Some of the first and more simplistic methods of categoriz-
+ing autonomous systems are the Levels of Automation (LOA)
+proposed by Sheridan [6] and its later expansion [7]. LOA
+defines automation as “the full or partial replacement of a
+function previously carried out by the human operator” in a 1
+to 10 range; 1 being full control by the human and 10 being
+full control by the computer. LOA does not accurately describe
+how outside factors can affect the autonomous capability of a
+system. While it could theoretically be applied to a robot, it
+would not be accurate as it fails to accommodate for differing
+degrees of difficulty in tasks, and environmental factors.
+Another evaluation method is known as the Autonomy
+Control Levels (ACL) [2]. ACL is designed for Unmanned
+Aerial Vehicles (UAV), and operates on a similar basis of uti-
+lizing autonomy levels from 0-10, with 0 being fully remotely
+controlled by a pilot, and 10 being a human-like system.
+These levels closely resemble the 10 LOA, following the same
+concept. The ACL characterizes each system according to
+four metrics, which attempt to categorize different areas of
+autonomous behaviors for the system. In each of these, an
+autonomy level from 0-10 is given based upon these behaviors.
+This system has a similar drawback, in that it does not account
+for difficulties in the mission itself.
+Another method is the Autonomy and Technological Readi-
+ness Assessment (ATRA) [8]. ATRA attempts to combine both
+the basic theory behind the Autonomy Level, and the Tech-
+nology Readiness Level (TRL) metric into one framework [8].
+TRL utilizes these two metrics in an attempt to evaluate the
+autonomy level provided by different technologies onboard the
+UAS. This is emphasized as a solution for the gap between
+existing theoretical work and technological advances in the
+UAS autonomy space.
+Autonomy Levels for Unmanned Systems (ALFUS) is a
+method for defining the autonomy of a system in terms of
+three different axes [9]. ALFUS has a strong theoretical basis,
+but somewhat impractical in the real-world implementation
+due to the lack of maturity in some of these systems, as well
+arXiv:2301.02603v1  [cs.RO]  6 Jan 2023
+
+as the inability of most, if not all, available systems to reach
+the upper levels of the three axes.
+The three axes mentioned are known as the Mission Com-
+plexity (MC), Environmental Complexity (EC), and Human
+Independence (HI) axes. Each one of these axes pertains itself
+to a different aspect of the contextual autonomy of a system.
+The MC axis pertains mostly towards the difficulty of the tasks
+and movements required of the system to complete the task
+(e.g. maneuvers, speed, searching). Alternatively, the EC axis
+concerns itself with the difficulty in the performance of the
+task caused by environmental factors (e.g. Lighting, Obstacles,
+Enclosed Spaces). Lastly, the HI axis is representative of the
+level of independence between the user and the system (e.g.
+task planning, task execution).
+Due to the ability to split the representation of a system’s
+autonomy into these three axes, it allows for the character-
+ization and evaluation of system’s autonomy in real world
+tests, including the impact that both the environment and the
+mission profile can have on the system’s autonomy. Our work
+in this paper is based off many of the ideas put forward
+through ALFUS, and we utilize it as a foundational part of
+our contextual autonomy evaluation.
+The Mission Performance Potential is proposed as a method
+for the evaluation of a unmanned system’s autonomous per-
+formance, as well as a predictor for future missions [5].
+This method provides a metric which represents the max-
+imum performance of a system in a given mission at a
+given autonomy level. Uniquely, this method includes both
+non-contextual autonomy metrics, and contextual autonomy
+metrics, and provides a single output prediction based on both
+types of data.
+One of the drawbacks of MPP is that it only provides
+a prediction of the performance of a system at a specified
+autonomy level for a specified mission. In other words, this
+does not evaluate how a real system performs, but rather the
+maximum potential for a system to perform. Our approach
+instead calculated the actual autonomy of a system based on
+actual data from real-world experiments.
+Fig. 1: Our cascaded Fuzzy Inference System used for calcu-
+lating a contextual autonomy score for a performed task.
+III. FRAMEWORK
+ALFUS’ summary model works with a set of metrics for
+each of its three axes, as well as a system of levels from
+0 to 10. These levels are based upon possible answers from
+those metrics, to provide a level evaluation of a system. As
+a generic framework, ALFUS tends to have a very broad,
+and somewhat open to interpretation, definition of metrics.
+For instance, in the case of the EC axis, it ranges from a
+“simple environment,” to an “extreme environment.” However,
+the summary model describes the system in terms of an
+autonomy level for each axis, while the Contextual Autonomy
+Capability within ALFUS provides an actual score for each
+axis. Due to the autonomy level evaluation, there is some
+ambiguity when characterizing systems. This is one of the
+main concerns with ALFUS, in that while it does provide a
+strong theoretical background, the actual implementation of
+the ideas with real-world systems is not as clear.
+We utilize Takagi-Sugeno Fuzzy Inference Systems (FIS)
+as a means to combine different metrics in an evaluation of
+an sUAS which is both easy to use, and allows us to use
+some data which is either not easily defined numerically, or
+inherently qualitative about the environment, combined with
+standard quantitative metrics. Fuzzy inferences also allow for
+slight deviations in a metric to not cause a drastic change
+in the evaluation of that sUAS. We designed a set of tests
+with various mission and environment complexity levels (see
+Section V), and defined a fuzzy inference system for each test.
+Unlike MPP [5], our fuzzy inference systems are based on the
+three-axis model used in ALFUS, by creating an individual FIS
+for metrics associated with each axis (i.e., MC, EC, HI), and
+an additional FIS which combines these three outputs into a
+single score. This structure representing a cascaded FIS (cFIS)
+is illustrated in Fig. 1. For each test, the outcome of the FIS
+for all three axis is fed into a combining FIS that produces a
+final autonomy score. Each FIS in our cFIS is a Sugeno-type
+FIS with multiple inputs and one output. For each input of an
+FIS, we consider three membership functions (MFs) labeled
+as low, medium, and high. Without loss of generality, we used
+triangular MFs, however, other types of MF can be used. The
+input variables used in different tests and their corresponding
+MF parameters have been reported in Table II. The output of
+each Sugeno-type FIS has five singleton MFs (i.e., constant):
+very bad, bad, medium, good, very good. Our FIS’ use a
+triangular fuzzifier and a Sugeno defuzzifier (i.e., weighted
+average output). For each FIS, we defined a rule base (i.e., a
+set of linguistic rules).
+In the cFIS structure in Fig. 1, the defuzzified output of
+each FIS is a value in the range of [0, 1]. For the initial three
+FIS, 0 and 1 represent the lowest and highest complexity,
+respectively. In the case of the final FIS, 0 and 1 represent the
+lowest and the highest autonomy, respectively. If we define the
+singleton value of each output function as zi, and the degree
+to which each output is weighted based upon the ruleset as
+wi, then the output final score can be calculated as follows:
+s =
+�N
+i=1 wizi
+�N
+i=1 wi
+(1)
+where N represents the number of rules in the rule base.
+Table I reports an example of the fuzzy ruleset we used. The
+advantage of this system is that we can utilize many different
+
+Human
+Independence
+FIS
+Environmental
+Output
+Input Data
+Complexity
+TestFIS
+Score
+FIS
+Mission
+Complexity
+FISFig. 2: From left to right, top to bottom: Cleo Robotics Dronut,
+Flyability Elios 2, Lumenier Nighthawk 3, Parrot ANAFI USA GOV,
+Skydio X2D, Teal Drones Golden Eagle, Vantage Robotics Vesper
+types of data, and clearly define the ranges for each value,
+allowing the pilots performing the tests to provide feedback
+on the membership functions and rulesets.
+Mission Complexity Axis
+Low
+Medium
+High
+Environment
+Complexity
+Axis
+Low
+Very Bad
+Bad
+Medium
+Medium
+Bad
+Medium
+Good
+High
+Medium
+Good
+Very good
+TABLE I: Fuzzy Ruleset utilized in our final combinational FIS
+IV. UAS PLATFORMS
+Fig. 2 illustrates seven sUAS platforms evaluated in our ex-
+periments. The platforms include: the Cleo Robotics Dronut1,
+Flyability Elios 22, Lumenier Nighthawk 33, Parrot ANAFI
+USA GOV4, Skydio X2D5, Teal Drones Golden Eagle6, and
+Vantage Robotics Vesper7. These platforms provide a wide
+ranging set of capabilities and use cases. For instance, Parrot,
+Skydio X2D, Golden Eagle, and Vesper were developed for
+outdoor reconnaissance, whereas the Dronut and Elios 2 were
+developed for indoor reconnaissance and inspection, specif-
+ically in urban and industrial environments. Previously, we
+have used the same set of sUAS for a non-contextual bench-
+marking [4], [10]. In our evaluations, we have anonymized the
+data by assigning the platforms labels A through G without
+any specific ordering or correlation.
+V. TEST DESIGN
+To evaluate the contextual autonomy of our platforms, we
+have designed several tests across a spectrum of areas. The
+variables for which we collected data for each test is reported
+in Table II. In this section, we describe each test briefly.
+As shown in Fig. 3 all tests have been designed for indoor
+environments.
+1https://cleorobotics.com/
+2https://www.flyability.com/elios-2
+3https://www.lumenier.com/
+4https://www.parrot.com/us/drones/anafi
+5https://www.skydio.com/skydio-x2
+6https://tealdrones.com/suas-golden-eagle/
+7https://vantagerobotics.com/vesper/
+A. Runtime Endurance
+This family of tests focuses on the battery life of the
+system in various operational profiles. As shown in Fig. 3a,
+the specific test we use from this group focuses on the
+system flying continuously in a figure-8 pattern. The main
+performance metric for the test is the test duration.
+B. Navigation
+We have designed two main types of navigation tests, each
+with several profiles defined based on the type of movement
+(horizontal, vertical, or both) and the type of confinement
+(horizontal, vertical, or both). As shown in Fig. 3b, navigation
+through confined spaces involves traversal into and out of a
+continuously confined space, with tests for hallway, tunnel,
+stairwell, and shaft. Navigation through apertures involves
+transient traversal through an opening, with tests for doorway
+and window. Each navigation environment is characterized
+according to the dimensions of the confined space or aperture,
+lighting, surface textures, and the presence of obstructions on
+either side of the confined space or aperture. The main metrics
+of performance are efficacy and average navigation time.
+C. Room Clearing
+In this test method, the system performs a visual inspection
+of an example room whose walls, floor, and ceiling are out-
+fitted with visual acuity targets which contain nested Landolt
+C symbols of decreasing size. As shown in Fig. 3c, the test
+was performed under two conditions: with and without using
+camera zoom. The main performance metrics are duration,
+coverage, and average acuity.
+D. Takeoff and Land/Perch
+As illustrated in Fig. 3d, these tests evaluate the system’s
+ability to takeoff and land or perch in various environments
+that may be affected by stabilization issues or preventative
+safety checks from the system. The conditions tested vary the
+angle of the ground plane (flat, 5° and 10° pitch and roll)
+and the presence of obstructions (1.2-2.4m overhead, 0.6-1.2m
+lateral). The main performance metric is efficacy.
+VI. RESULTS
+Utilizing our framework outlined in Section III, we calculate
+a performance score for each sUAS based upon the conditions
+and performance metrics detailed below. As mentioned above,
+our system provides a single score for each of the three
+attributes, the EC, MC and HI axes, and utilizes those scores
+to provide a single score for the entire test. It should be noted
+that although we consider all three axes in our cFIS structure,
+due to lack of data, we consider the lowest level (i.e., full
+tele-operation) for the HI axis across all tests. The test-specific
+details of the structure is discussed in corresponding sections.
+Another factor to note is that for some of these experiments,
+some sUAS were not available at the time of testing, due to the
+sUAS being repaired, or other circumstances. In this case, we
+attempt to remedy this by calculating a partial point achieved
+by the sUAS. For situations where data for an entire test is
+
+(a) Runtime Endurance test showing a system performing a specified
+movement
+(b) Navigation tests (left to right, top to bottom): hallway, tunnel,
+stairwell, shaft, door, window.
+(c) Room clearing test showing a system inspecting surfaces for
+visual acuity targets.
+(d) Takeoff and land/perch tests showing variations in ground plane
+angle (top row) and nearby obstructions (bottom row).
+Fig. 3: Tests designed for the evaluation of sUAS contextual autonomy.
+missing, we cannot fully evaluate the sUAS. However, we have
+evaluated sUAS for individual tests for which the data was
+recorded. Despite these edge cases, most sUAS had available
+data, which was used in our evaluations.
+As mentioned before, the proposed structure in Fig. 1,
+was adapted to each specific test. The resulting cascaded FIS
+are depicted in Fig. 6 each of which represents a cFIS for
+a specific test. More specifically, each sub-figure shows the
+inputs and outputs of each cFIS, as well as how the FIS
+modules are connected. This is meant to provide a visual
+aid, which can be useful to keep track of each FIS, as we
+discuss the results. Associated membership functions for each
+input, and the outputs, can be found in Table II. Some of
+the FIS surfaces, which show the relationship between two
+input values in an FIS, and the corresponding output value, are
+shown in Fig. 4. Additionally, numerical results are reported
+in Fig.5 and Table III.
+A. Test Results
+Runtime Endurance: The runtime endurance test is likely
+the simplest of the tests performed, however, it is still a useful
+test in gauging how a sUAS might perform in a real mission.
+Fig. 3a illustrates the runtime endurance test design including
+the navigation path and two stands. The adapted cFIS for
+this test is shown in Fig. 6a with four inputs: number of
+obstructions, number of crashes, light level and speed.
+Despite the simplicity of this test, some sUAS did not
+performed well largely due to slow speed. Our results indicate
+that some sUAS may have trouble in portions of this real-
+world mission which requires both speed and maneuverability.
+(a) Final FIS Surface
+(b) Through Aperture test EC
+(c) Takeoff and Land/Perch EC
+(d) Through Corridor MC
+Fig. 4: FIS surfaces for different tests.
+In should be noted that during this test only four sUAS were
+available.
+Navigation: In the navigation tests, due to the differences
+between the through corridor and through apertures tests, two
+slightly different cascaded FIS were designed. These can be
+found in Fig. 6b and Fig. 6d. The input to the through corri-
+dors cFIS includes area (cross section), light level, verticality,
+coverage, number of crashes, and duration. The inputs to the
+through apertures cFIS include area, light level, number of
+
+ 10 degrees
+10 degrees
+5 degrees
+5degree
+1.2 mCutput_Score
+10.5
+0
+0.5
+0.5
+MC
+0
+ECOutput_Score
+0.6
+0.4
+0.2
+0:
+0
+2
+200
+400
+4
+600
+Area
+Light0.7
+10.5
+0.3
+10
+10
+5
+5
+Pitch
+0
+RollOutput_Score
+0.5
+00
+2
+0.5
+Crashes
+3
+CompletionXVariables
+Description
+MFs: Triangular{Low, Medium, High}
+Area
+Aperture/Hallway Cross-Section (m2)
+[0, 0, 2.7]
+[0.6, 3, 5.4]
+[3.3, 6, 6]
+Light
+Ambient Light Level (Lux)
+[0, 0, 337.5]
+[75, 375, 675]
+[412.5, 750, 750]
+Vert
+Verticality (°)
+[0, 0, 37.5]
+[7.5, 45, 82.5]
+[52.5, 90, 90]
+Crash
+Number of Crashes
+[0, 0, 1.25]
+[0.5, 1.5, 2.5]
+[1.75, 3, 3]
+Rollovers
+Number of Rollovers
+[0, 0, 1.25]
+[0.5, 1.5, 2.5]
+[1.75, 3, 3]
+Comp. %
+Completion Percentage
+[0, 0, 0.55]
+[0.15, 0.6, 0.92]
+[0.7, 1, 1]
+Yaw/Pitch
+Static Yaw/Pitch Angle (°)
+[0, 0, 4.17]
+[0.83, 5, 9.12]
+[5.83, 10, 10]
+VR
+Static Vertical Obstruction (m)
+[0.6, 0.6, 1.1]
+[0.7, 1.2, 1.7]
+[1.3, 1.8, 1.8]
+LR
+Static Lateral Obstruction (m)
+[1.2, 1.2, 2.2]
+[1.4, 2.4, 3.4]
+[2.6, 3.6, 3.6]
+Coverage
+Coverage Percentage
+[0, 0, 0.55]
+[0.15, 0.6, 0.92]
+[0.7, 1, 1]
+Cs Detected
+Landolt C Depth Detected
+[0, 0, 50]
+[10, 50, 90]
+[50, 100, 100]
+Duration
+Duration of Test (Minutes)
+[2.5, 2.5, 5.25]
+[3.05, 5.25, 7.45]
+[5.25, 8, 8]
+Obs.
+Number of Obstructions
+[0, 0, 2.5]
+[1, 3, 5]
+[3.5, 6, 6]
+Output Variable
+Description
+Sugeno MFs: Constant {Very Low to Very High}
+Score
+Combined Defuzzified Score
+[0, 0.25, 0.5, 0.75, 1]
+TABLE II: Membership Functions (MFs) for each input and output variable used in an FIS in the evaluation of these sUAS.
+Fig. 5: Scores for each sUAS as a percentage of the maximum score
+possible on the y-axis, with each test on the x-axis
+crashes, and completion percentage. As shown above in Fig. 5,
+each sUAS that performed the through apertures test, achieved
+a maximum score, besides sUAS G, which performed slightly
+worse, due to both issues in correctly traversing the aperture,
+as well as being the only sUAS to suffer a crash during the test.
+Next, for the through corridors test, As shown in Fig. 5, there
+is more variance in the performance between each sUAS, with
+UAS A performing the best, and UAS E performing the worst,
+even though sUAS E tied for best in through apertures. This
+is important, as there is more room for error while traversing
+corridors, than there is traversing an aperture.
+Takeoff and Land/Perch: For the takeoff and land/perch
+test, the cFIS diagram can be found in Fig. 6e. The inputs
+include pitch, yaw, number of crashes, completion percentage,
+vertical and lateral obstruction, and number of rollovers. As
+can be seen in Fig. 5 and Table III, sUAS A and sUAS G
+perform best across both sections of the test and thus provide
+the highest level of autonomy. Likewise, sUAS B performs the
+worst in both portions of the test, showcasing a lower level of
+autonomy, compared to the other sUAS. This evaluation allows
+for the characterization of how an sUAS may perform during
+portions of a mission which requires the system to takeoff or
+land in a specified spot, of varying difficulty.
+Room Clearing: Since the room clearing test is done in
+a static environment, we included a time constraint in our
+testing making the evaluation to focus on the performance
+of the sUAS in regards to the Mission Complexity axis. The
+designed cFIS can be seen in Fig. 6c. Inputs include light level,
+number of obstructions, number of crashes, duration, coverage,
+and landolt C depth detected. Results are found in Fig. 5, and
+Table III. In this test, the strongest performer was UAS E;
+however, most of the UAS performed closely to each other.
+Surprisingly, a strong performance in the runtime endurance
+test did not necessarily correlate to a strong performance in
+this test. Both of these tests require a system with good maneu-
+verability capabilities, but this test also requires a controller
+which allows the user to visually identify different landmarks.
+B. Final Results
+UAS
+T.C.
+T.A.
+Takeoff
+Land
+R.E.
+R.C.
+Predictive
+Score
+A
+1.0
+1.0
+1.0
+1.0
+0.5
+0.76
+0.85
+B
+0.90
+1.0
+0.71
+0.87
+0.76
+0.73
+0.82
+C
+0.84
+1.0
+1.0
+0.87
+-
+-
+0.92
+D
+-
+-
+0.75
+0.97
+0.65
+0.75
+0.77
+E
+0.80
+1.0
+0.82
+0.89
+-
+0.85
+0.87
+F
+-
+-
+0.99
+0.91
+-
+-
+0.95
+G
+0.83
+0.83
+1.0
+1.0
+0.5
+0.79
+0.80
+TABLE III: Scores of each sUAS for each test, as well as a weighted
+multiple which allows for an overall evaluation of each sUAS
+Contextual autonomy evaluations are concerned with the
+performance of a system within an environment while per-
+forming a specific task with a known level of complexity.
+However, calculating an overall score that represents an av-
+erage autonomy for a given system in a spectrum of tests
+and environment is desirable. To combine the test scores into
+a single score, we utilize a weighted product, with equal
+weightings for each test, as we did previously in our non-
+contextual evaluation [4]. The weighted product represented
+as
+P =
+M
+�
+i
+φwi
+i ,
+(2)
+where M is the number of individual tests, φi represents an
+individual test score, and wi the weight assigned to that test.
+
+1.0
+T
+T
+0.9
+UASA
+0.8 -
+UAS B
+UAS C
+0.7 -
+UAS D
+UAS E
+0.6
+UAS F
+UAS G
+0.5
+T.C.
+T.A.
+R.E.
+R.C.
+Takeoff
+Land(a) Runtime Endurance (R.E.) test FIS
+(b) Through Apertures (T.A.) test FIS
+(c) Room Clearing (R.C.) test FIS
+(d) Through Corridors (T.C.) test FIS
+(e) Takeoff and Land/Perch test FIS
+Fig. 6: Diagrams of each system of cascaded FIS utilized to
+calculate scores for each test
+has several benefits including that different test results can
+be combined without requiring normalization or scaling. The
+results for each sUAS are shown in Table III. It is important to
+note that many of the sUAS perform better than the others in
+some tests, but worse in others. One example of this is UAS
+A, which has the fourth highest weighted multiple (overall
+score), while performing the best in four out of six tests. This
+is due to the sUAS’s relatively poor performance in both the
+runtime endurance test, as well as in the room clearing test.
+The use case of a singular score like this presents itself when
+a user would like to know which sUAS is likely to provide
+the most overall autonomy, across multiple tests and different
+environment.
+VII. CONCLUSIONS AND DISCUSSIONS
+In this paper, we proposed a framework for evaluation of
+contextual autonomy for robotic systems. Our framework con-
+sists of a cascaded Fuzzy Inference System (cFIS) that com-
+bines test results over three axes of evaluation (mission com-
+plexity, environment complexity and human independence)
+introduced by the ALFUS framework. We have designed
+four tests with different mission complexity and environment
+complexity levels and performed several experiments with
+several sUAS, and we have shown that our modular framework
+is adaptable to different tests. For future work, we plan to
+extend our framework for performance evaluation.
+To achieve this, a desired mission can be decomposed into
+base tasks, such as takeoff/landing, traversing through environ-
+ments/apertures, clearing rooms, and general maneuverability.
+The user then can define a set of weights and calculate a
+potential performance score of a sUAS for the target mission.
+Unlike MPP [5], however, we suggest a method which is based
+upon performance in set tasks, rather than a combination of
+non-contextual attributes, and environmental factors.
+ACKNOWLEDGMENT
+This work is sponsored by the Department of the Army, U.S.
+Army Combat Capabilities Development Command Soldier
+Center, award number W911QY-18-2-0006. Approved for
+public release #PR2022 88282
+REFERENCES
+[1] P. J. Durst and W. Gray, “Levels of autonomy and autonomous system
+performance assessment for intelligent unmanned systems,” Engineer-
+ing research and development center Vicksburg Ms Geotechnical and
+Structures lab, Tech. Rep., 2014.
+[2] B. T. Clought, “Metrics, schmetrics! how the heck do you determine
+a uav’s autonomy anyway?” Air Force Research Laboratory, Wright-
+Pattterson Air Force Base, OH, Tech. Rep., August 2002.
+[3] P. J. Durst, W. Gray, and M. Trentini, “A non-contextual model for
+evaluating the autonomy level of intelligent unmanned ground vehicles,”
+in Proceedings of the 2011 Ground Vehicle Systems Engineering and
+Technology Symposium, 2011.
+[4] B. Hertel, R. Donald, C. Dumas, and S. R. Ahmadzadeh, “Methods
+for combining and representing non-contextual autonomy scores for
+unmanned aerial systems,” International Conference on Automation,
+Robotics, and Applications (ICARA), vol. 8th, pp. 145–149, 2022.
+[5] P. Durst, W. Gray, A. Nikitenko, J. Caetano, M. Trentini, and R. King,
+“A framework for predicting the mission-specific performance of au-
+tonomous unmanned systems,” International Conference on Intelligent
+Robots and Systems, pp. 1962–1969, 2014.
+[6] T. B. Sheridan, “Automation, authority, and angst – revisited,” vol. 35,
+September 1991, pp. 2–6.
+[7] R. Parasuraman, T. B. Sheridan, and C. D. Wickens, “A model for types
+and levels of human interaction with automation,” IEEE Transactions on
+Systems, Man, and Cybernetics–Part A: Systems and Humans, vol. 30,
+pp. 286–297, May 2000.
+[8] F. Kendoul, “Towards a unified framework for uas autonomy and
+technology readiness assessment (ATRA),” pp. 55–71, 2 2013.
+[9] H.-M. Huang, “Autonomy levels for unmanned systems framework;
+volume II: Framework models,” NIST Special Publication: Gaithersburg,
+MD, USA, p. 30, 2007.
+[10] A. Norton, R. Ahmadzadeh, K. Jerath, P. Robinette, J. Weitzen, T. Wick-
+ramarathne, H. Yanco, M. Choi, R. Donald, B. Donoghue et al.,
+“Decisive test methods handbook: Test methods for evaluating suas in
+subterranean and constrained indoor environments, version 1.1,” arXiv
+preprint arXiv:2211.01801, 2022.
+
+Obs.
+Crashes
+Environmental
+Mission
+Complexity
+Complexity
+Light
+FIS
+FIS
+Level
+Speed
+Output
+Test FIS
+ScoreArea
+Crashes
+Environmental
+Mission
+Complexity
+Complexity
+Light
+FIS
+FIS
+Level
+Comp. %
+Output
+Test FIS
+ScoreCoverage
+Cs
+Detected
+Crashes
+Light
+Environmental
+Mission
+Complexity
+Complexity
+Duration
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1dE0T4oBgHgl3EQfuQGc/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf,len=471
+page_content='Contextual Autonomy Evaluation of Unmanned  Aerial Vehicles in Subterranean Environments  Ryan Donald  Peter Gavriel  Adam Norton  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Reza Ahmadzadeh  PeARL lab and NERVE Center  University of Massachusetts Lowell  Lowell, USA  Ryan Donald@student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='uml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='edu  Peter Gavriel@uml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='edu  Adam Norton@uml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='edu  Reza Ahmadzadeh@uml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='edu  Abstract—In this paper we focus on the evaluation of con-  textual autonomy for robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' More specifically, we propose a  fuzzy framework for calculating the autonomy score for a  small Unmanned Aerial Systems (sUAS) for performing a task  while considering task complexity and environmental factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  Our framework is a cascaded Fuzzy Inference System (cFIS)  composed of combination of three FIS which represent dif-  ferent contextual autonomy capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' We performed several  experiments to test our framework in various contexts, such as  endurance time, navigation, take off/land, and room clearing,  with seven different sUAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' We introduce a predictive measure  which improves upon previous predictive measures, allowing for  previous real-world task performance to be used in predicting  future mission performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  Index Terms—Contextual Autonomy, Unmanned Aerial Vehi-  cles, Fuzzy Systems  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' INTRODUCTION  In today’s world, robots are expected to become increasingly  present by assisting humans in performing various tasks in  different environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' While some robots have been designed  for a single purpose, others can accomplish a variety of tasks  with different levels of autonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Measuring robot autonomy  is an important and ever evolving concept and existing meth-  ods for evaluating robot autonomy can be categorized into  two main families: contextual and non-contextual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' While the  former methods consider mission and task-specific measures  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=', ALFUS [1], ACL [2]), the latter only rely on implicit  system capabilities and do not consider the mission and  environment features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=', NCAP [3], [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  Our study in this paper focuses on evaluating the contextual  autonomy for small Unmanned Aerial Systems (sUAS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Exist-  ing methods such as ALFUS [1] and MPP [5] share a similar  shortcoming in that neither provides a simple implementation  for use with real-world systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Another drawback of existing  methods that our approach addresses is the lack of a consistent  process for breaking down tasks into sub-tasks and combining  scores calculated for sub-tasks into a unified score for the given  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' In this paper we propose a method for evaluating the  contextual autonomy of sUAS based on a fuzzy interface that  allows the operator to design and modify the evaluation system  using linguistic reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' We designed four indoor tasks  (endurance time, navigation, takeoff/land, and room clearing)  and tested our interface in various experiments with seven  different sUAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Our results show that the proposed approach  calculates a contextual autonomy score that can be used to  rank the systems for each context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' RELATED WORK  Some of the first and more simplistic methods of categoriz-  ing autonomous systems are the Levels of Automation (LOA)  proposed by Sheridan [6] and its later expansion [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' LOA  defines automation as “the full or partial replacement of a  function previously carried out by the human operator” in a 1  to 10 range;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 1 being full control by the human and 10 being  full control by the computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' LOA does not accurately describe  how outside factors can affect the autonomous capability of a  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' While it could theoretically be applied to a robot, it  would not be accurate as it fails to accommodate for differing  degrees of difficulty in tasks, and environmental factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  Another evaluation method is known as the Autonomy  Control Levels (ACL) [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' ACL is designed for Unmanned  Aerial Vehicles (UAV), and operates on a similar basis of uti-  lizing autonomy levels from 0-10, with 0 being fully remotely  controlled by a pilot, and 10 being a human-like system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  These levels closely resemble the 10 LOA, following the same  concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' The ACL characterizes each system according to  four metrics, which attempt to categorize different areas of  autonomous behaviors for the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' In each of these, an  autonomy level from 0-10 is given based upon these behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  This system has a similar drawback, in that it does not account  for difficulties in the mission itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  Another method is the Autonomy and Technological Readi-  ness Assessment (ATRA) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' ATRA attempts to combine both  the basic theory behind the Autonomy Level, and the Tech-  nology Readiness Level (TRL) metric into one framework [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  TRL utilizes these two metrics in an attempt to evaluate the  autonomy level provided by different technologies onboard the  UAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' This is emphasized as a solution for the gap between  existing theoretical work and technological advances in the  UAS autonomy space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  Autonomy Levels for Unmanned Systems (ALFUS) is a  method for defining the autonomy of a system in terms of  three different axes [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' ALFUS has a strong theoretical basis,  but somewhat impractical in the real-world implementation  due to the lack of maturity in some of these systems, as well  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='02603v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='RO]  6 Jan 2023  as the inability of most, if not all, available systems to reach  the upper levels of the three axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  The three axes mentioned are known as the Mission Com-  plexity (MC), Environmental Complexity (EC), and Human  Independence (HI) axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Each one of these axes pertains itself  to a different aspect of the contextual autonomy of a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  The MC axis pertains mostly towards the difficulty of the tasks  and movements required of the system to complete the task  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' maneuvers, speed, searching).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Alternatively, the EC axis  concerns itself with the difficulty in the performance of the  task caused by environmental factors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Lighting, Obstacles,  Enclosed Spaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Lastly, the HI axis is representative of the  level of independence between the user and the system (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  task planning, task execution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  Due to the ability to split the representation of a system’s  autonomy into these three axes, it allows for the character-  ization and evaluation of system’s autonomy in real world  tests, including the impact that both the environment and the  mission profile can have on the system’s autonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Our work  in this paper is based off many of the ideas put forward  through ALFUS, and we utilize it as a foundational part of  our contextual autonomy evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  The Mission Performance Potential is proposed as a method  for the evaluation of a unmanned system’s autonomous per-  formance, as well as a predictor for future missions [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  This method provides a metric which represents the max-  imum performance of a system in a given mission at a  given autonomy level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Uniquely, this method includes both  non-contextual autonomy metrics, and contextual autonomy  metrics, and provides a single output prediction based on both  types of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  One of the drawbacks of MPP is that it only provides  a prediction of the performance of a system at a specified  autonomy level for a specified mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' In other words, this  does not evaluate how a real system performs, but rather the  maximum potential for a system to perform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Our approach  instead calculated the actual autonomy of a system based on  actual data from real-world experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 1: Our cascaded Fuzzy Inference System used for calcu-  lating a contextual autonomy score for a performed task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' FRAMEWORK  ALFUS’ summary model works with a set of metrics for  each of its three axes, as well as a system of levels from  0 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' These levels are based upon possible answers from  those metrics, to provide a level evaluation of a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' As  a generic framework, ALFUS tends to have a very broad,  and somewhat open to interpretation, definition of metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  For instance, in the case of the EC axis, it ranges from a  “simple environment,” to an “extreme environment.” However,  the summary model describes the system in terms of an  autonomy level for each axis, while the Contextual Autonomy  Capability within ALFUS provides an actual score for each  axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Due to the autonomy level evaluation, there is some  ambiguity when characterizing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' This is one of the  main concerns with ALFUS, in that while it does provide a  strong theoretical background, the actual implementation of  the ideas with real-world systems is not as clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  We utilize Takagi-Sugeno Fuzzy Inference Systems (FIS)  as a means to combine different metrics in an evaluation of  an sUAS which is both easy to use, and allows us to use  some data which is either not easily defined numerically, or  inherently qualitative about the environment, combined with  standard quantitative metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Fuzzy inferences also allow for  slight deviations in a metric to not cause a drastic change  in the evaluation of that sUAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' We designed a set of tests  with various mission and environment complexity levels (see  Section V), and defined a fuzzy inference system for each test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  Unlike MPP [5], our fuzzy inference systems are based on the  three-axis model used in ALFUS, by creating an individual FIS  for metrics associated with each axis (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=', MC, EC, HI), and  an additional FIS which combines these three outputs into a  single score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' This structure representing a cascaded FIS (cFIS)  is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' For each test, the outcome of the FIS  for all three axis is fed into a combining FIS that produces a  final autonomy score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Each FIS in our cFIS is a Sugeno-type  FIS with multiple inputs and one output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' For each input of an  FIS, we consider three membership functions (MFs) labeled  as low, medium, and high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Without loss of generality, we used  triangular MFs, however, other types of MF can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' The  input variables used in different tests and their corresponding  MF parameters have been reported in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' The output of  each Sugeno-type FIS has five singleton MFs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=', constant):  very bad, bad, medium, good, very good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Our FIS’ use a  triangular fuzzifier and a Sugeno defuzzifier (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=', weighted  average output).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' For each FIS, we defined a rule base (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=', a  set of linguistic rules).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  In the cFIS structure in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 1, the defuzzified output of  each FIS is a value in the range of [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' For the initial three  FIS, 0 and 1 represent the lowest and highest complexity,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' In the case of the final FIS, 0 and 1 represent the  lowest and the highest autonomy, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' If we define the  singleton value of each output function as zi, and the degree  to which each output is weighted based upon the ruleset as  wi, then the output final score can be calculated as follows:  s =  �N  i=1 wizi  �N  i=1 wi  (1)  where N represents the number of rules in the rule base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  Table I reports an example of the fuzzy ruleset we used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' The  advantage of this system is that we can utilize many different  Human  Independence  FIS  Environmental  Output  Input Data  Complexity  TestFIS  Score  FIS  Mission  Complexity  FISFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 2: From left to right, top to bottom: Cleo Robotics Dronut,  Flyability Elios 2, Lumenier Nighthawk 3, Parrot ANAFI USA GOV,  Skydio X2D, Teal Drones Golden Eagle, Vantage Robotics Vesper  types of data, and clearly define the ranges for each value,  allowing the pilots performing the tests to provide feedback  on the membership functions and rulesets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  Mission Complexity Axis  Low  Medium  High  Environment  Complexity  Axis  Low  Very Bad  Bad  Medium  Medium  Bad  Medium  Good  High  Medium  Good  Very good  TABLE I: Fuzzy Ruleset utilized in our final combinational FIS  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' UAS PLATFORMS  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 2 illustrates seven sUAS platforms evaluated in our ex-  periments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' The platforms include: the Cleo Robotics Dronut1,  Flyability Elios 22, Lumenier Nighthawk 33, Parrot ANAFI  USA GOV4, Skydio X2D5, Teal Drones Golden Eagle6, and  Vantage Robotics Vesper7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' These platforms provide a wide  ranging set of capabilities and use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' For instance, Parrot,  Skydio X2D, Golden Eagle, and Vesper were developed for  outdoor reconnaissance, whereas the Dronut and Elios 2 were  developed for indoor reconnaissance and inspection, specif-  ically in urban and industrial environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Previously, we  have used the same set of sUAS for a non-contextual bench-  marking [4], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' In our evaluations, we have anonymized the  data by assigning the platforms labels A through G without  any specific ordering or correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' TEST DESIGN  To evaluate the contextual autonomy of our platforms, we  have designed several tests across a spectrum of areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' The  variables for which we collected data for each test is reported  in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' In this section, we describe each test briefly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 3 all tests have been designed for indoor  environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  1https://cleorobotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='com/  2https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='flyability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='com/elios-2  3https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='lumenier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='com/  4https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='parrot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='com/us/drones/anafi  5https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='skydio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='com/skydio-x2  6https://tealdrones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='com/suas-golden-eagle/  7https://vantagerobotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='com/vesper/  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Runtime Endurance  This family of tests focuses on the battery life of the  system in various operational profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 3a,  the specific test we use from this group focuses on the  system flying continuously in a figure-8 pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' The main  performance metric for the test is the test duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Navigation  We have designed two main types of navigation tests, each  with several profiles defined based on the type of movement  (horizontal, vertical, or both) and the type of confinement  (horizontal, vertical, or both).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 3b, navigation  through confined spaces involves traversal into and out of a  continuously confined space, with tests for hallway, tunnel,  stairwell, and shaft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Navigation through apertures involves  transient traversal through an opening, with tests for doorway  and window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Each navigation environment is characterized  according to the dimensions of the confined space or aperture,  lighting, surface textures, and the presence of obstructions on  either side of the confined space or aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' The main metrics  of performance are efficacy and average navigation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Room Clearing  In this test method, the system performs a visual inspection  of an example room whose walls, floor, and ceiling are out-  fitted with visual acuity targets which contain nested Landolt  C symbols of decreasing size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 3c, the test  was performed under two conditions: with and without using  camera zoom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' The main performance metrics are duration,  coverage, and average acuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Takeoff and Land/Perch  As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 3d, these tests evaluate the system’s  ability to takeoff and land or perch in various environments  that may be affected by stabilization issues or preventative  safety checks from the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' The conditions tested vary the  angle of the ground plane (flat, 5° and 10° pitch and roll)  and the presence of obstructions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='2-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='4m overhead, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='6-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='2m  lateral).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' The main performance metric is efficacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' RESULTS  Utilizing our framework outlined in Section III, we calculate  a performance score for each sUAS based upon the conditions  and performance metrics detailed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' As mentioned above,  our system provides a single score for each of the three  attributes, the EC, MC and HI axes, and utilizes those scores  to provide a single score for the entire test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' It should be noted  that although we consider all three axes in our cFIS structure,  due to lack of data, we consider the lowest level (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=', full  tele-operation) for the HI axis across all tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' The test-specific  details of the structure is discussed in corresponding sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  Another factor to note is that for some of these experiments,  some sUAS were not available at the time of testing, due to the  sUAS being repaired, or other circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' In this case, we  attempt to remedy this by calculating a partial point achieved  by the sUAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' For situations where data for an entire test is  (a) Runtime Endurance test showing a system performing a specified  movement  (b) Navigation tests (left to right, top to bottom): hallway, tunnel,  stairwell, shaft, door, window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  (c) Room clearing test showing a system inspecting surfaces for  visual acuity targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  (d) Takeoff and land/perch tests showing variations in ground plane  angle (top row) and nearby obstructions (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 3: Tests designed for the evaluation of sUAS contextual autonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  missing, we cannot fully evaluate the sUAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' However, we have  evaluated sUAS for individual tests for which the data was  recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Despite these edge cases, most sUAS had available  data, which was used in our evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  As mentioned before, the proposed structure in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 1,  was adapted to each specific test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' The resulting cascaded FIS  are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 6 each of which represents a cFIS for  a specific test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' More specifically, each sub-figure shows the  inputs and outputs of each cFIS, as well as how the FIS  modules are connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' This is meant to provide a visual  aid, which can be useful to keep track of each FIS, as we  discuss the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Associated membership functions for each  input, and the outputs, can be found in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Some of  the FIS surfaces, which show the relationship between two  input values in an FIS, and the corresponding output value, are  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Additionally, numerical results are reported  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='5 and Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Test Results  Runtime Endurance: The runtime endurance test is likely  the simplest of the tests performed, however, it is still a useful  test in gauging how a sUAS might perform in a real mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 3a illustrates the runtime endurance test design including  the navigation path and two stands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' The adapted cFIS for  this test is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 6a with four inputs: number of  obstructions, number of crashes, light level and speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  Despite the simplicity of this test, some sUAS did not  performed well largely due to slow speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Our results indicate  that some sUAS may have trouble in portions of this real-  world mission which requires both speed and maneuverability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  (a) Final FIS Surface  (b) Through Aperture test EC  (c) Takeoff and Land/Perch EC  (d) Through Corridor MC  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 4: FIS surfaces for different tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  In should be noted that during this test only four sUAS were  available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  Navigation: In the navigation tests, due to the differences  between the through corridor and through apertures tests, two  slightly different cascaded FIS were designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' These can be  found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 6b and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 6d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' The input to the through corri-  dors cFIS includes area (cross section), light level, verticality,  coverage, number of crashes, and duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' The inputs to the  through apertures cFIS include area, light level, number of  10 degrees  10 degrees  5 degrees  5degree  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='2 mCutput_Score  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='5  MC  0  ECOutput_Score  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='2  0:  0  2  200  400  4  600  Area  Light0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='7  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='3  10  10  5  5  Pitch  0  RollOutput_Score  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='5  00  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='5  Crashes  3  CompletionXVariables  Description  MFs: Triangular{Low, Medium, High}  Area  Aperture/Hallway Cross-Section (m2)  [0, 0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='7]  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='6, 3, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='4]  [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='3, 6, 6]  Light  Ambient Light Level (Lux)  [0, 0, 337.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='5]  [75, 375, 675]  [412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='5, 750, 750]  Vert  Verticality (°)  [0, 0, 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='5]  [7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='5, 45, 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='5]  [52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='5, 90, 90]  Crash  Number of Crashes  [0, 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='25]  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='5]  [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='75, 3, 3]  Rollovers  Number of Rollovers  [0, 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='25]  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='5]  [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='75, 3, 3]  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' %  Completion Percentage  [0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='55]  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='92]  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='7, 1, 1]  Yaw/Pitch  Static Yaw/Pitch Angle (°)  [0, 0, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='17]  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='83, 5, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='12]  [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='83, 10, 10]  VR  Static Vertical Obstruction (m)  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='1]  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='7, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='7]  [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='8]  LR  Static Lateral Obstruction (m)  [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='2]  [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='4, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='4]  [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='6, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='6, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='6]  Coverage  Coverage Percentage  [0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='55]  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='92]  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='7, 1, 1]  Cs Detected  Landolt C Depth Detected  [0, 0, 50]  [10, 50, 90]  [50, 100, 100]  Duration  Duration of Test (Minutes)  [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='5, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='25]  [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='05, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='25, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='45]  [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='25, 8, 8]  Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  Number of Obstructions  [0, 0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='5]  [1, 3, 5]  [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='5, 6, 6]  Output Variable  Description  Sugeno MFs: Constant {Very Low to Very High}  Score  Combined Defuzzified Score  [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='75, 1]  TABLE II: Membership Functions (MFs) for each input and output variable used in an FIS in the evaluation of these sUAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 5: Scores for each sUAS as a percentage of the maximum score  possible on the y-axis, with each test on the x-axis  crashes, and completion percentage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' As shown above in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 5,  each sUAS that performed the through apertures test, achieved  a maximum score, besides sUAS G, which performed slightly  worse, due to both issues in correctly traversing the aperture,  as well as being the only sUAS to suffer a crash during the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  Next, for the through corridors test, As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 5, there  is more variance in the performance between each sUAS, with  UAS A performing the best, and UAS E performing the worst,  even though sUAS E tied for best in through apertures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' This  is important, as there is more room for error while traversing  corridors, than there is traversing an aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  Takeoff and Land/Perch: For the takeoff and land/perch  test, the cFIS diagram can be found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 6e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' The inputs  include pitch, yaw, number of crashes, completion percentage,  vertical and lateral obstruction, and number of rollovers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' As  can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 5 and Table III, sUAS A and sUAS G  perform best across both sections of the test and thus provide  the highest level of autonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Likewise, sUAS B performs the  worst in both portions of the test, showcasing a lower level of  autonomy, compared to the other sUAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' This evaluation allows  for the characterization of how an sUAS may perform during  portions of a mission which requires the system to takeoff or  land in a specified spot, of varying difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  Room Clearing: Since the room clearing test is done in  a static environment, we included a time constraint in our  testing making the evaluation to focus on the performance  of the sUAS in regards to the Mission Complexity axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' The  designed cFIS can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 6c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Inputs include light level,  number of obstructions, number of crashes, duration, coverage,  and landolt C depth detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Results are found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 5, and  Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' In this test, the strongest performer was UAS E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  however, most of the UAS performed closely to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  Surprisingly, a strong performance in the runtime endurance  test did not necessarily correlate to a strong performance in  this test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Both of these tests require a system with good maneu-  verability capabilities, but this test also requires a controller  which allows the user to visually identify different landmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Final Results  UAS  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  Takeoff  Land  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
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+page_content='  Predictive  Score  A  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
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+page_content='80  TABLE III: Scores of each sUAS for each test, as well as a weighted  multiple which allows for an overall evaluation of each sUAS  Contextual autonomy evaluations are concerned with the  performance of a system within an environment while per-  forming a specific task with a known level of complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  However, calculating an overall score that represents an av-  erage autonomy for a given system in a spectrum of tests  and environment is desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' To combine the test scores into  a single score, we utilize a weighted product, with equal  weightings for each test, as we did previously in our non-  contextual evaluation [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' The weighted product represented  as  P =  M  �  i  φwi  i ,  (2)  where M is the number of individual tests, φi represents an  individual test score, and wi the weight assigned to that test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='0  T  T  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='9  UASA  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='8 -  UAS B  UAS C  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='7 -  UAS D  UAS E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='6  UAS F  UAS G  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='5  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  Takeoff  Land(a) Runtime Endurance (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=') test FIS  (b) Through Apertures (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=') test FIS  (c) Room Clearing (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=') test FIS  (d) Through Corridors (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=') test FIS  (e) Takeoff and Land/Perch test FIS  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 6: Diagrams of each system of cascaded FIS utilized to  calculate scores for each test  has several benefits including that different test results can  be combined without requiring normalization or scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' The  results for each sUAS are shown in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' It is important to  note that many of the sUAS perform better than the others in  some tests, but worse in others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' One example of this is UAS  A, which has the fourth highest weighted multiple (overall  score), while performing the best in four out of six tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' This  is due to the sUAS’s relatively poor performance in both the  runtime endurance test, as well as in the room clearing test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  The use case of a singular score like this presents itself when  a user would like to know which sUAS is likely to provide  the most overall autonomy, across multiple tests and different  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' CONCLUSIONS AND DISCUSSIONS  In this paper, we proposed a framework for evaluation of  contextual autonomy for robotic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Our framework con-  sists of a cascaded Fuzzy Inference System (cFIS) that com-  bines test results over three axes of evaluation (mission com-  plexity, environment complexity and human independence)  introduced by the ALFUS framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' We have designed  four tests with different mission complexity and environment  complexity levels and performed several experiments with  several sUAS, and we have shown that our modular framework  is adaptable to different tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' For future work, we plan to  extend our framework for performance evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  To achieve this, a desired mission can be decomposed into  base tasks, such as takeoff/landing, traversing through environ-  ments/apertures, clearing rooms, and general maneuverability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  The user then can define a set of weights and calculate a  potential performance score of a sUAS for the target mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  Unlike MPP [5], however, we suggest a method which is based  upon performance in set tasks, rather than a combination of  non-contextual attributes, and environmental factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  ACKNOWLEDGMENT  This work is sponsored by the Department of the Army, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  Army Combat Capabilities Development Command Soldier  Center, award number W911QY-18-2-0006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Approved for  public release #PR2022 88282  REFERENCES  [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Durst and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Gray, “Levels of autonomy and autonomous system  performance assessment for intelligent unmanned systems,” Engineer-  ing research and development center Vicksburg Ms Geotechnical and  Structures lab, Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=', 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  [2] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Clought, “Metrics, schmetrics!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' how the heck do you determine  a uav’s autonomy anyway?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Air Force Research Laboratory, Wright-  Pattterson Air Force Base, OH, Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=', August 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  [3] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Durst, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Gray, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Trentini, “A non-contextual model for  evaluating the autonomy level of intelligent unmanned ground vehicles,”  in Proceedings of the 2011 Ground Vehicle Systems Engineering and  Technology Symposium, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  [4] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Hertel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Donald, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Dumas, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Ahmadzadeh, “Methods  for combining and representing non-contextual autonomy scores for  unmanned aerial systems,” International Conference on Automation,  Robotics, and Applications (ICARA), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 8th, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 145–149, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  [5] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Durst, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Gray, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Nikitenko, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Caetano, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Trentini, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' King,  “A framework for predicting the mission-specific performance of au-  tonomous unmanned systems,” International Conference on Intelligent  Robots and Systems, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 1962–1969, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  [6] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Sheridan, “Automation, authority, and angst – revisited,” vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 35,  September 1991, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 2–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  [7] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Parasuraman, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Sheridan, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Wickens, “A model for types  and levels of human interaction with automation,” IEEE Transactions on  Systems, Man, and Cybernetics–Part A: Systems and Humans, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 30,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 286–297, May 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  [8] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Kendoul, “Towards a unified framework for uas autonomy and  technology readiness assessment (ATRA),” pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 55–71, 2 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  [9] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Huang, “Autonomy levels for unmanned systems framework;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  volume II: Framework models,” NIST Special Publication: Gaithersburg,  MD, USA, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' 30, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Norton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Ahmadzadeh, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Jerath, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Robinette, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Weitzen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Wick-  ramarathne, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Yanco, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Choi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Donald, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' Donoghue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=',  “Decisive test methods handbook: Test methods for evaluating suas in  subterranean and constrained indoor environments, version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='1,” arXiv  preprint arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='01801, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  Crashes  Environmental  Mission  Complexity  Complexity  Light  FIS  FIS  Level  Speed  Output  Test FIS  ScoreArea  Crashes  Environmental  Mission  Complexity  Complexity  Light  FIS  FIS  Level  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' %  Output  Test FIS  ScoreCoverage  Cs  Detected  Crashes  Light  Environmental  Mission  Complexity  Complexity  Duration  FIS  FIS  Ohs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  Output  Test FIS  ScoreArea  Crashes  Environmental  Mission  Complexity  Complexity  Light  FIS  FIS  Level  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' %  Vert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content='  Output  Test FIS  ScoreCrashes  Pitch  Rollovers  Yaw  Environmental  Mission  Complexity  Complexity  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}
+page_content=' %  LR  FIS  FIS  VR  Output  Test FIS  Score' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'}

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+Dowker Complexes and filtrations on self-relations
+Dominic Desjardins Côté
+January 11, 2023
+Abstract
+Given a relation on X × Y , we can construct two abstract simplicial complexes
+called Dowker complexes.
+The geometric realizations of these simplicial complexes
+are homotopically equivalent. We show that if two relations are conjugate, then they
+have homotopically equivalent Dowker complexes. From a self-relation on X, this is a
+directed graph, and we use the Dowker complexes to study their properties. We show
+that if two relations are shift equivalent, then, at some power of the relation, their
+Dowker complexes are homotopically equivalent. Finally, we define a new filtration
+based on Dowker complexes with different powers of a relation.
+Keywords :
+Dowker complex, relation, filtration, graph theory, shift equivalence
+1
+Introduction
+We can use multivalued maps to study dynamical systems [17]. The idea is to use Conley index [7]
+on upper semi-continuous multivalued maps. In applications, it can be hard to study a dynamical
+system. We can use a model that seems to fit data, but it can be a challenge to find it. Another way
+is to discretize the continuous space and use to multivalued maps to approximate the underlying
+dynamical system [8] [30]. Another approach is to use combinatorial structures. To name a few,
+we can use combinatorial vector fields from Forman [13] [18] [9]. Moreover, a generalization was
+proposed by Mrozek called combinatorial multivector fields [23] [25]. Finally, others proposed to
+use the distributive lattices to compute attractors on finite data [19] [20] [21].
+Multivalued maps can be restrictive. In [18], authors generalize them to partial multivalued
+maps. But a partial multivalued map is equivalent to a relation. Some advancements were done
+in [26] by using the Scymczak category of finite sets where objects are sets and morphisms are
+relations. The Szymczak category [29] captures the essence of index pairs and index maps [7] which
+is the core of the theory of Conley index. So one motivation of this paper is to continue to develop
+the theory of relations, and it can be used to study dynamical system with finite data.
+Our main object is a relation which is a subset of the cartesian product of two sets X and
+Y . We can define two different abstract simplicial complexes on a relation. For the first simplicial
+complex, we fixed a value y ∈ Y . For all elements in X, that they are related to y, they will span a
+1
+arXiv:2301.03739v1  [math.CO]  10 Jan 2023
+
+simplex together. For the second one, we reverse the role. We fixed a value x ∈ X. For all elements
+in Y , that they are in relation with x, they will span a simplex together. They are called Dowker
+complexes [10]. An important result is the Dowker’s Theorem. It says that the geometric realization
+of these Dowker complexes are homotopically equivalent. The Dowker’s Theorem is quite useful in
+applications. To name a few examples, we can use Dowker complexes to study signal coverage [14],
+to find errors in relation of programs and files [1], to study the privacy of information [12] and in
+social studies [3]. For the last one, this method is called Q-analysis which is developed by Atkins
+[2]. The idea of Q-analysis is to study the q-connectivity and the q-tunnel of the Dowker complexes.
+Our two main inspirations for definitions come from these articles [24] and [26].
+They are many contributions in this article.
+First, conjugate relations have homotopically
+equivalent Dowker complex. If two relations are shift equivalent with lag l, then at a certain power
+their Dowker complexes are homotopically equivalent. We can define a filtration on relation based
+on the Dowker complex at different powers of a finite self-relation. Moreover, this can be computed
+in finite time. We can also compute the 0th homology of a high enough power with the connected
+components of the graph induced by the relation.
+This article goes as follows. In section 2, we remind some concepts and definitions on finite
+relation, graph, simplicial complex, Dowker complexes and finally the famous Dowker’s Theorem.
+In section 3, we define right morphism and left morphism. If there exists a right or left morphism
+between two relations, then there is an inclusion from one of the Dowker complexes. Moreover,
+we show that if there exists a conjugacy between two relations, then their Dowker complexes are
+homotopically equivalent. In section 4, we generalize the idea of right and left morphism for mul-
+tivalued right and multivalued left morphism. We show two important properties needed for the
+definition of a filtration. The Dowker complex of a certain power of a relation is include in the
+Dowker complex of the same relation with a higher power. For some finite relations at certain a
+power j, every other Dowker complexes of the same relation at power higher than j are the same.
+We call it the stabilization of the Dowker complex. We show that shift equivalence between rela-
+tions have homotopically equivalent Dowker complex at some power. In section 5, we can define a
+filtration on the Dowker complexes of different powers of a relation under some simple conditions by
+using the two properties in section 4. We can use persistent homology on these filtration to extract
+topological features of the Dowker complex. If a relation is acyclic, then we have that the number
+of connected components of the graph associated to the relation up to a certain power is equal to
+the dimension of the 0th homology. It can be generalized to the class of simple relations. We also
+have a similar result for strongly connected relations.
+2
+Preliminaries
+2.1
+Finite Relations
+Let X and Y be finite sets. We define a relation as a subset of X × Y . Let (x, y) ∈ R ⊂ X × Y , we
+denote by xRy or by y ∈ R(x). We define the composition of relations as follows. Let R1 ⊂ X × Y
+and R2 ⊂ Y × Z.
+2
+
+R2 ◦ R1 := {(x, z) ∈ R2 ◦ R1 | ∃y such that xR1y and yR2z}.
+(1)
+We define the inverse relation by swapping the sets of a relation.
+R−1 := {(y, x) ∈ Y × X | y ∈ R(x)}
+(2)
+If a relation is a subset of X × X, then we say it’s a self-relation on X. We define the power of
+a self-relation as follows :
+Rn :=
+�
+�
+�
+�
+�
+R ◦ Rn−1
+n > 0
+IdX
+n = 0
+R−1 ◦ Rn+1
+n < 0
+The domain and the image for a relation R ⊂ X × Y are :
+Dom R := {x ∈ X | ∃y such that (x, y) ∈ R}
+(3)
+Im R := {y ∈ Y | ∃x such that (x, y) ∈ R}.
+(4)
+We can see relations as partial multivalued maps. If Dom R = X, then we say that the relation
+is a multivalued map. A relation is injective, if for all x1, x2 ∈ X, R(x1) = R(x2) implies that
+x1 = x2. A relation is surjective if Im R = Y . Moreover, a map f : X → Y induces a relation
+where (x, f(x)) ∈ R. Without ambiguity, we can compose maps and relations together to obtain a
+new relation.
+Definition 2.1. Let R be a self-relation on X. Let j be the least positive integer such that :
+Rj = Rj+p for some p > 0.
+We say that j is the index and the least p > 0 is the period. If j = 1, then R is periodic. A pair
+(j, p) is the eventual period of R with index j and period p.
+In other words, the period p will eventually be a period for R.
+We sometimes use matrices with values in {0, 1} to represent relations. Let R ⊂ X × Y be a
+relation with card(X) = m and card(Y ) = n. The matrix Mm×n have a value 1 at Mi,j if xiRyj
+otherwise the value is 0. It can be called relation matrix, Boolean relation matrix, binary relation
+matrix, binary Boolean matrix, (0, 1)-Boolean matrix and (0, 1)-matrix. For more information on
+Boolean matrix theory, we refer to the book [22].
+We say that a self-relation R on X has a cycle at x if and only if there exists an n ∈ N such
+that xRnx. We say R as a fixed point at x, if n = 1. If a relation has no cycle at x for all x with
+period n > 1, then the relation is acyclic. A cycle is a sequence x1, x2, . . . , xn such that x1 = xn
+and xiRxi+1. A self-relation R on X is simple if for any two cycles are either disjoints or equals.
+3
+
+2.2
+Graphs
+In this subsection, we remind the definition of a graph and some notations.
+Definition 2.2. A directed graph G is a pair (E, V ) where V is the set of vertices V and E is the
+subset V × V the set of edges.
+A relation can also be seen as a directed graph. If R is a self-relation on X, then X is the set
+of vertices and the set of edges E = R. This graph has at most one directed edge from the vertex
+A to the vertex B, and we also allow a self-loop on vertices. We note GR the graph induced by a
+self-relation R.
+Let x, y ∈ V . There is a (x, y)-path, if there exists a sequence of edges e1, e2, . . . , en ∈ E that
+connect x to y without following the direction of edges. We can define an equivalence relation
+on vertices of G. If there is a path between two vertices x and y, then x and y are in the same
+class of equivalence. For a graph G, we say the number of connected components is the number of
+class equivalences of the relation of paths. We say G is connected if there is only one connected
+component.
+If the sequence of edges of a (x, y)-path follows the direction of edges of the graph, then we
+say it’s a (x, y)-walk. We can also define an equivalence relation with a walk between vertices. If
+there is a walk from x to y and a walk from y to x, then x and y are in the same equivalence class.
+This is the class of strongly connected components. For a graph G, we say the number of strongly
+connected components is the number of class equivalence of the relation of walks. We say G is
+strongly connected if there is only one strongly connected component.
+2.3
+Simplicial Complexes and Dowker Complexes
+In this subsection, we will discuss simplicial complex, Dowker complex and the Dowker’s Theorem.
+For more information about simplicial complex, we suggest to read [27]. We do not present filtration
+and persistent homology, but we refer to [11].
+An abstract simplicial complex is a set K that contains finite non-empty sets such as if A ∈ K,
+then for all subsets of A are also in K. For further examples, we use geometric simplex. A geometric
+n-simplex is the convex hull of a geometrically independent sets of vertices {v0, v1, . . . , vn} ∈ RN.
+This is the set of x ∈ RN such as x = �n
+i=0 tixi and 1 = �n
+i=0 ti where ti ≥ 0 for all i. We denote
+an n-simplex by [v0, v1, . . . , vn] is the simplex spanned by the vertices v0, v1, . . . , vn. Any simplex
+spanned by the subsets of {v0, v1, . . . , vn} are called faces and denote by the symbol ≤. A simplicial
+complex is a collection of simplices for all σ ∈ K, if τ ≤ σ then τ ∈ K and if σ1 ∩ σ2 = τ, then τ
+is either the empty set or τ is a face of σ1 and σ2. We say that L ≤ K if L is a sub-complex of
+K. A simplicial complex is contractible if its homology is equivalent to a point. Given an abstract
+simplicial K, we can define a geometric simplicial complex and |K| call the geometric realization of
+K. We call 0-simplices vertices and 1-simplices edges. The closure of a simplex σ is the set of all
+the faces of the simplex. We denote it by cl(σ). We need one more definition related to simplicial
+complexes. It will be useful in some proofs.
+4
+
+Definition 2.3. A simplicial complex K is edge-connected, if for any two vertices x and y there
+is a sequence of edges e1, e2, . . . , en such that x ∈ e1, y ∈ en and cl(ei) ∩ cl(ei+1) ̸= ∅ for all
+i = 1, 2, . . . , n − 1.
+We have that the simplicial complex is connected if and only if it is edge-connected if and only
+if H0 is dimension 1 [16].
+Now we explain how to construct abstract simplicial complexes from a relation which are called
+Dowker complexes. Let R ⊂ X × Y be a relation and X, Y be two finite sets. There are two ways
+to construct the Dowker complex from a relation.
+Definition 2.4. Let R ⊂ X × Y be a finite relation and KR be the Dowker complex. A simplex
+[x1, x2, . . . , xn] ∈ KR if and only if ∃y ∈ Y such as xiRy for all i = 1, 2, . . . , n.
+We have an analogous construction.
+Definition 2.5. Let R ⊂ X × Y be a finite relation and LR be the Dowker complex. A simplex
+[y1, y2, . . . , ym] ∈ LR if and only if ∃x ∈ X such as xRyi for all i = 1, 2, . . . , n.
+We denote [x1, x2, . . . xn] = σy ∈ KR if and only if xiRy for all i = 1, 2, . . . , n. We use y as an
+index for σy to note that all vertices of σy are in R−1(y). We use the same notation for σx ∈ LR
+but the vertices are in R(x).
+By using the matrix notation, we can use rows and columns to build the simplices. The columns
+are for KR and the rows are for LR.
+Example 2.6. Let R ⊂ X × Y be a finite relation.
+R :=
+�
+���
+1
+0
+0
+0
+1
+0
+0
+1
+1
+0
+1
+0
+0
+0
+1
+1
+1
+0
+0
+0
+�
+���
+(5)
+The first column gives the 2-simplex [x1, x3, x4]. The third and the fourth column give the 0-
+simplex [x2]. The second and the fifth column do not add new simplices. We obtain the simplicial
+complex KR := {[x1, x3, x4], [x2]}.
+The first row adds a 1-simplex [y1, y5] to LR. The second row gives a 1-simplex [y3, y4]. The
+final row adds a 1-simplex [y1, y5]. We obtain the simplicial complex LR = {[y1, y5], [y3, y4], [y1, y2]}.
+We obtain that |KR| and |LR| have two connected components and no higher dimension cycle.
+The next theorem links to the homotopy between |KR| and |LR|.
+Theorem 2.7 (Dowker’s Theorem). Let R ⊂ X × Y be a relation and let KR and LR be the
+associated Dowker complexes. Then, the polyhedra |KR| and |LR| are homotopy equivalent.
+5
+
+(a) Geometric realization of
+the Dowker complex KR.
+(b) Geometric realization of
+the Dowker complex LR.
+Figure 1: These are geometric realizations of the Dowker complexe in Example 2.6. They
+are homotopically equivalent.
+In 1952, Dowker [10] has shown that KR and LR have isomorphic homology groups. In 1995,
+Björner [4] has shown that |KR| and |LR| are homotopically equivalent, which is the more com-
+monly use in the literature. In recent years, Dowker complexes were regained in popularity in the
+community of topology data analysis. We can use them to do a filtration on weighted networks [6].
+In our cases, our filtration will be different and based on different powers of a self-relation.
+3
+Left and Right Morphism
+Let start with the definition of the graph homomorphism and next we define left and right morphisms
+between relations.
+Definition 3.1. Let R be a self-relation on X and R′ be a self-relation on Y . A map f : X → Y
+is a graph homomorphism if the following condition is satisfied :
+For every x1, x2 ∈ X such as x1Rx2 =⇒ f(x1)R′f(x2).
+If f is bijective and its inverse is also a graph homomorphism, then f is a graph isomorphism.
+We obtain that graph homomorphism keeps some information of the Dowker complex coming
+from the first relation.
+Lemma 3.2. Let f : X → Y be a graph homomorphism between R and R′. If f is injective, then
+there exist a map p : KR �→ KR′.
+6
+
+Proof. Consider a n-simplex [x0, x1, . . . , xn] ∈ KR. Then, there exists α ∈ X such as xiRα for
+all i = 0, 1, 2, . . . n. We have that f is a graph homomorphism. This implies that f(xi)R′f(α)
+for all i = 0, 1, 2, . . . , n. Indeed, f is injective implies that [f(x0), f(x1), . . . , f(xn)] is also a n-
+simplex in KR′. So we can construct a map p : KR �→ KR′ by sending a simplex [x0, x1, . . . , xn] to
+[f(x0), f(x1), . . . , f(xn)]. By the previous argument, p is well defined and injective.
+If we have a graph isomorphism between two relations, then the Dowker complexes remain
+unchanged. This holds because graph isomorphisms are relabelling on the vertices of a graph.
+Proposition 3.3. Let R1 be a self-relation on X and R2 be a self-relation on Y . If there exists a
+graph isomorphism f between R1 and R2, then they have the same Dowker complexes up to the label
+of vertices.
+Proof. Graph homomorphisms f and f−1 are injective. By the Lemma 3.2, there exist two injective
+maps p : KR �→ KR′ and p′ : KR′ �→ KR. So we have that KR and KR′ are the same up to the
+label of vertices.
+By similar arguments, we can show it for LR and LR′.
+Graph homomorphisms are nice, but they can drastically change the Dowker complexes. So,
+we defined a left morphism which it changes the source of an edge and a right morphism which it
+changes the target of an edge. In this way, only one of the Dowker complexes will change from the
+right morphism or the left morphism.
+Definition 3.4. A right morphism f : (X, Y, R) → (X, Z, R′) is a map f : Y → Z such that for
+every x ∈ X and y ∈ Y :
+xRy =⇒ xR′f(y).
+We obtain this simple Lemma which is very useful for later proofs.
+Lemma 3.5. If there exists a right morphism f : (X, Y, R) → (X, Z, R′), then KR ≤ KR′. We
+obtain the equality if f is a bijective map.
+Proof. Let f : (X, Y, R) → (X, Z, R′) be a right morphism and [x1, x2, . . . , xn] ∈ KR. This implies
+there exists a y ∈ Y such as xiRy for all i = 1, 2, . . . , n. We obtain that xiR′f(y) for all i. Finally,
+we have [x1, x2, . . . , xn] ∈ KR′.
+Now, we suppose that f is bijective. Let [x1, x2, . . . , xn] ∈ KR′. Then, there exists a z ∈ Z such
+that xiR′z for all i = 1, 2, . . . , n. We have f−1(z) ∈ Y and f−1 is well defined because f is bijective.
+Then, xiRf−1(z) for all i = 1, 2, . . . , n. We obtain that [x1, x2, . . . , xn] ∈ KR.
+The idea of right morphism comes from the article [24]. The author only considered the right
+morphism. But, in our case, we are also interesting of modifying the first set in the cartesian product
+of a relation.
+7
+
+Definition 3.6. A left morphism g : (X, Z, R) → (Y, Z, R′) is a map g : X → Y such that for every
+x ∈ X and z ∈ Z :
+xRz =⇒ g(x)R′z.
+We have an analogous Lemma for left morphism as the Lemma 3.5 for right morphism.
+Lemma 3.7. If there exists a left morphism g : (X, Z, R) → (Y, Z, R), then LR ≤ LR′. We obtain
+the equality if g is a bijective map.
+With the definition of right and left morphism, we can easily show that if two relations are
+conjugate, then there Dowker complexes are homotopically equivalent. We remind the definition of
+conjugacy between relations before showing the proof.
+Definition 3.8. Let R1 be a self-relation on X and R2 be a self-relation on Y . We say that R1 and
+R2 are conjugate if there exists a bijective map ϕ : X → Y such as ϕ ◦ R1 = R2 ◦ ϕ.
+Corollary 3.9. Let R be a self-relation on X and R′ be a self-relation on Y which are conjugate
+by a bijective map ϕ : X → Y . Then, |KR|, |LR|, |LR′| and |KR′| are homotopy equivalent.
+Proof. The map ϕ is bijective. It implies that KR = Kϕ◦R by Lemma 3.5 and LR′ = LR′◦ϕ by
+Lemma 3.7.
+By Dowker’s Theorem, we obtain that |KR| is homotopic equivalent to |LR′|, because Kϕ◦R =
+KR′◦ϕ.
+In [28], the author decides to combine the right and left morphism together. Let R ⊂ X × Y
+and R′ ⊂ X′ × Y ′ be relations and f : X → X′ and g : Y → Y ′ be two maps. A pair (f, g) is
+a morphism between relation R1 and R2 if for all x ∈ X, y ∈ Y such that xR1y it implies that
+f(x)R2g(y). In [28], it is shown that the Dowker complex and (co)sheaf representation have nice
+functoriality properties. In our case, it won’t be useful because we only need right or left morphism.
+But we can see them as a pair (idX, f) where f is a right morphism and idX is the identity function
+on X.
+4
+Multi-right morphism and multi-left morphism
+We want to work with multivalued maps. We generalize left and right morphism to multi-left and
+multi-right morphism.
+Definition 4.1. A multi-right morphism F : (X, Y, R) ⊸ (X, Z, R′) is a multivalued map F : Y ⊸
+Z such as for all x ∈ X, y ∈ Y :
+xRy =⇒ xR′a for all a ∈ F(y).
+8
+
+We also obtain the same Lemma as before.
+Lemma 4.2. Let R ⊂ X × Y and R′ ⊂ X × Z be relations. If there exist a multi-right morphism
+F : (X, Y, R) ⊸ (X, Z, R′), then KR ≤ KR′. We obtain the equality if F is a bijective multivalued
+map.
+Proof. The proof is the same as Lemma 3.5.
+Definition 4.3. A multi-left morphism G : (X, Z, R) ⊸ (Y, Z, R′) is a multivalued map G : X → Y
+such for all x ∈ X, z ∈ Z :
+xRz =⇒ aR′z for all a ∈ G(x).
+Lemma 4.4. Let R ⊂ X × Z and R′ ⊂ Y × Z be relations. If there exist a multi-left morphism
+G : (X, Z, R) ⊸ (Y, Z, R′), then LR ≤ LR′. We obtain the equality if G is a bijective multivalued
+map.
+We denote a multi-right morphism by mr-morphism and a multi-left morphism by ml-morphism.
+Remarks 4.5. We remind that if a relation S ⊂ X×Y satisfies Dom S = X, then S is a well-defined
+multivalued map. Moreover, for any relation R ⊂ Z ×X, we have that S : (Z, X, R) ⊸ (Z, Y, S ◦R)
+is a well-defined mr-morphism. It is also true for ml-morphism.
+The next corollary will be useful to define our filtrations.
+Corollary 4.6. Let R be a self-relation on X. If Dom R = X, then KRn ≤ KRn+1. If Im R = X,
+then LRn ≤ LRn+1.
+Proof. Dom R = X implies that R is a multivalued map. Moreover, R : (X, X, Rn) ⊸ (X, X, Rn+1)
+is well-defined mr-morphism. By Lemma 3.5, we have that KRn ≤ KRn+1.
+In the same way, Im R = X implies that R−1 is a multivalued map.
+We have that R−1 :
+(X, X, Rn) ⊸ (X, X, Rn+1) is well-defined ml-morphism. By Lemma 3.7, we have that LRn ≤
+LRn+1.
+Remark 4.7. The hypothesis Dom R = X in the previous corollary is important. Let us show that
+by an example. If R is a self-relation on X, define by this matrix such as :
+�
+�
+0
+1
+1
+0
+0
+1
+0
+0
+0
+�
+�
+(6)
+Then, we obtain KR = {[x2, x3], [x2], [x3]}, KR2 = {[x3]} and KRn = ∅ for n > 2.
+We don’t have KRn ⊂ KRn+1 for all n ∈ N>0. If the matrix is nilpotent, then KRn is an empty
+set for an n ∈ N>0.
+9
+
+Using Corollary 4.6, we can show that the Dowker complexes of a relation stabilize at some
+power n.
+Corollary 4.8. Let R be a finite self-relation on X with an eventual period (j, p). If Dom R = X,
+then, we have KRj = KRj+i for i ∈ N. If Im R = X, then LRj = LRj+i for i ∈ N.
+Proof. By Corollary 4.6, we have the sequence :
+KRj ≤ KRj+1 ≤ . . . ≤ KRj+p−1 ≤ KRj+p.
+(7)
+But p is the period of R, hence Rj = Rj+p implies that KRj = KRj+p. By (7), we obtain
+KRj = KRj+i for i ∈ N. A similar proof can be done for LRj = LRj+i for i ∈ N.
+We remind the definition of shift equivalence between two relations and we show the assump-
+tion that relations are shift equivalences implies that theirs Dowker complexes are homotopically
+equivalent at some power for each relation.
+Definition 4.9. Let R1 be finite self-relation on X and R2 be finite self-relation on Y . R1 and R2
+are shift equivalent with a lag l, if there exists two relations S ⊂ X × Y and T ⊂ Y × X such as :
+R1 ◦ T = T ◦ R2
+S ◦ R1 = R2 ◦ S
+T ◦ S = Rl
+1
+S ◦ T = Rl
+2
+We say it is a strong shift equivalence if l = 1.
+Corollary 4.10. Let R1 be finite self-relation on X with Dom R1 = X = Im R1 and R2 be finite
+self-relation on Y with Dom R2 = Y = Im R2. Let (jp, p) be an eventual period of R1 and (jq, q)
+be an eventual period of R2. Without loss of generality, we suppose that jp ≥ jq. If R1 and R2 are
+shift equivalent with lag l, then |KR
+jp
+1 |, |KR
+jq
+2 |, |LR
+jp
+1 | and |LR
+jq
+2 | are homotopy equivalent.
+Proof. If R1◦T = T ◦R2 then Rn
+1 ◦T = T ◦Rn
+2 is also true for n ∈ N. Moreover, we have Dom S = X
+and Dom T = Y , because Dom R1 = X, Dom R2 = Y , T ◦ S = Rl
+1 and S ◦ T = Rl
+2. So we have
+that S and R are well defined multivalued maps.
+We want to show that KR
+jp
+1
+= KS◦R
+jp
+1
+and LR
+jq
+2
+= LR
+jq
+2 ◦S. We are going to use T and S as
+mr-morphism with R1 and ml-morphism with R2.
+We have that S : (X, X, Rjp
+1 ) ⊸ (X, Y, S ◦ Rjp
+1 ) and T : (X, Y, S ◦ Rjp
+1 ) ⊸ (X, X, T ◦ S ◦ Rjp
+1 )
+are well-defined mr-morphisms. We have T ◦ S = Rl
+1. It implies that T ◦ S ◦ Rjp
+1 = Rl+jp
+1
+. We
+obtain KR
+jp
+1 ≤ KS◦R
+jp
+1 ≤ KT◦S◦R
+jp
+1 = KR
+l+jp
+1
+. The eventual period of R1 is (jp, p). It implies that
+KR
+jp
+1 = KR
+jp+l
+1
+. We obtain that KR
+jp
+1 = KS◦R
+jp
+1 .
+We can see that S : (Y, Y, Rjp
+2 ) ⊸ (X, Y, Rjp
+2 ◦ S) and T : (X, Y, Rjp
+2 ◦ S) ⊸ (Y, Y, Rjp
+2 ◦ S ◦ T)
+are well-defined ml-morphisms. We have S ◦ T = Rl
+2. It implies that Rjp
+2 ◦ S ◦ T = Rjp+l
+2
+. We
+10
+
+(a) The graph GR1.
+(b) The graph GR2.
+Figure 2: Theses are the graphs from Example 4.11.
+obtain LR
+jp
+2 ≤ LR
+jp
+2 ◦S ≤ LR
+jp
+2 ◦S◦T = LR
+jp+l
+2
+. We have LR
+jq
+2 = LR
+jp+l
+2
+because jp ≥ jq. We obtain
+LR
+jp
+2 = LR
+jp
+2 ◦S.
+Finally, we have KR
+jp
+1 = KS◦R
+jp
+1 = KR
+jp
+2 ◦S and LR
+jp
+2 = LR
+jp
+2 ◦S = LS◦R
+jp
+1 . By Dowker’s Theorem,
+we obtain that |KR
+jp
+1 |, |KR
+jq
+2 |, |LR
+jp
+1 | and |LR
+jq
+2 | are homotopy equivalent.
+Example 4.11. Let X be a finite set with 8 points and Y be a finite set with 3 points. Let R1 be
+a self-relation on X and R2 be a self-relation on X defined by those graphs in Figure 2. R1 has an
+eventual period (3, 3) and R2 has an eventual period (1, 3). We see in Figures 3(a), (b) and (d) that
+the Dowker complexes are not homotopically equivalent. But, in Figures 3 (c) and (d), the Dowker
+complexes with relations at power 3 are homotopically equivalent.
+There is an interesting proposition from [15] that we can use for strongly connected relations.
+We remind that an indecomposable Boolean matrix is a relation which is strongly connected and J
+is a square matrix where all the entries are equals to 1.
+Proposition 4.12 (Proposition 4.3 in [15] ). Every indecomposable Boolean matrix with positive
+trace is strong shift equivalent to J.
+We can easily compute the Dowker complexes of J. It is a (n−1)-simplex where n is the number
+of rows of J for both Dowker complexes. Finally, we obtain that the Dowker complexes of a strongly
+connected self-relation at a power high enough are contractible if the trace is strictly positive.
+11
+
+Y3(a) Dowker complex of |KR1|.
+(b) Dowker complex of |LR1|.
+(c) Dowker complexes of |KR3
+1| and
+|LR3
+1|.
+(d)
+Dowker
+complexes
+|KR2|, |LR2|, |KR3
+2| and |LR3
+2|.
+Figure 3: Theses are the different Dowker complexes from Example 4.11.
+12
+
+5
+Filtrations of Dowker complexes
+For this section, we suppose that a R is a finite self-relation on X. From Corollary 4.6, if Dom R =
+X, then we have KRi ≤ KRi+1 for all i ≥ 0. We have an inclusion and we get this filtration :
+KR �→ KR2 �→ . . . �→ KRi �→ KRi+1 �→ . . .
+(8)
+In the same way with Im R = X, we have LRi ≤ LRi+1. We obtain another filtration :
+LR �→ LR2 �→ . . . �→ LRi �→ LRi+1 �→ . . .
+(9)
+From Corollary 4.8, the Dowker complexes stabilize at a certain power. This means we can
+compute the filtration (8) and (9) in finite time. For our filtrations, we start at i = 1, but we could
+also start with the i = 0. We have that R0 = idX and R : (X, X, idX) ⊸ (X, X, R) is a well-defined
+mr-morphism, if Dom R = X. Then, the filtration (8) becomes :
+KidX = KR0 �→ KR �→ KR2 �→ . . . �→ KRi �→ KRi+1 �→ . . .
+(10)
+And for the filtration (9) by applying similar arguments, we obtain :
+LidX = LR0 �→ LR �→ LR2 �→ . . . �→ LRi �→ LRi+1 �→ . . .
+(11)
+The homology of KidX and Lidx is the homology of n = card(X) points. In some cases, we
+might want to start the filtration at i = 0 or i = 1.
+We remind that, by Dowker’s Theorem, |KRi| and |LRi| are homotopically equivalent for all
+i ∈ N. We obtain the same bar code representation for filtrations (8) and (9).
+Example 5.1. Let R1 be a self-relation on X given by the graph in the Figure 4(a). R1 has 9 nodes
+and is acyclic. The eventual period is (3, 1). We obtain the bar code at Figure 4(b). It has one
+generator of H1 with the interval [1, 2] and we had 3 generators of H0 that die early and 1 generator
+of H0 that survive to infinity.
+Example 5.2. Let R2 be a self-relation on X given by the graph in the Figure 4(c). R2 has 10
+nodes and is simple. The eventual period is (3, 4). We obtain the bar code at the Figure 4(d). It has
+4 generators of H0 that die at time 2 and 2 other generators that survive to infinity.
+For the next results, we compute the 0th homology of the Dowker complexes for different types
+of relation.
+If R is a finite acyclic self-relation, then it has an eventual period p = 1 and j ∈ N>0 such as
+Rj = Ri for all i ≥ j. So we denote this relation Rj by R∞, because it converges to a relation when
+i → ∞.
+13
+
+(a) The graph of the acyclic rela-
+tion from Example 5.1.
+(b) The associated bar code of the rela-
+tion from Example 5.1. We use the fil-
+tration with KRi
+1. The first bar in orange
+is a generator in H1 and the others four
+bars in blue are generators in H0.
+(c) The graph of the relation with
+multiple cycles from Example 5.2.
+(d) The associated bar code of the rela-
+tion from Example 5.2. We use the fil-
+tration with KRi
+2. The six bars in blue
+are generators in H0.
+Figure 4: In these figures, we have the graph on the left and the associated bar code diagram
+on the right for Examples 5.1 and 5.2. Dashed lines in bar code mean it goes to infinity.
+14
+
+4.0
+T
+3.5
+-
+3.0 
+T
+2.5
+2.0 
+1.5
+1.0
+-
+0.5 
+0.0 
+1.0
+1.5 
+2.0
+2.5
+0'E
+3.5 
+4.0
+4.5
+5.0X5
+X105 .
+4 
+3 -
+1
+1.0
+1.5 
+2.0
+2.5
+3.0
+3.5
+4.0
+4.5
+5.0X3Definition 5.3. We say that x ∈ X is a minimum for a self-relation R, if there exists no y ∈ X
+such as x ̸= y and xR∞y. We denote the set Ux := {y ∈ X | yR∞x}.
+We say that x ∈ X is a maximum for a self-relation R, if there exists no y ∈ X such as x ̸= y
+and yR∞x. We denote the set Dx := {y ∈ X | xR∞y}.
+The maximums and minimums of an acyclic relation are important, because they are responsible
+for the maximal simplices of KR∞ and LR∞.
+Lemma 5.4. Let R be a finite acyclic self-relation on X with Dom R = X. Then, the maximal
+simplices of KR∞ are given by the minimums of R∞.
+Proof. We have that Dom R = X it implies that Dom R∞ = X. Then, for all x ∈ X there exists
+z ∈ X such that xR∞z and z is a minimum. Let σy = [x1, x2, . . . , xn] ∈ KR∞ be an arbitrary
+simplex. We have an y ∈ X such as xiR∞y for all i = 1, 2, . . . , n. By the first argument, there
+exists a minimum z ∈ X such as yR∞z. Therefore, we have xiR∞yR∞z. Then, σy ⊂ σz.
+We can do a similar result with LR∞ by using the maximums of R∞, if Im R = X.
+Theorem 5.5. Let R an acyclic finite self-relation on X with Dom R = X.
+number of connected components of GR = dim H0(KR∞) = dim H0(LR∞)
+(12)
+Proof. First, by Dowker’s Theorem, we have dim H0(KR∞) = dim H0(LR∞). We suppose that GR is
+connected and show that dim H0(KR∞) = 1. More precisely, we show that KR∞ is edge-connected.
+We have that Dom R = X implies that for all x ∈ X,[x] ∈ KR∞.
+Let x, x′ ∈ X. There exists y1 ∈ X a minimum such as xR∞y1 and y1R∞y1. This implies
+that e1 = [x, y1] ∈ KR∞. We also have that there exists yn ∈ X a minimum such as x′R∞yn and
+ynR∞yn. This implies that en = [x′, yn] ∈ KR∞.
+Since GR is connected, there exists a (y1, yn)-path of finite length. We denote this sequence by
+y1, z1, z2, z3, . . . , zm, yn. Without loss of generality, we take the shortest path. There exist a i such
+as zi ∈ Uy1 and zi+1 /∈ Uy1. First, we have e2 = [y1, zi] ∈ KR∞ and ziRzi+1. There exists y2 ̸= y1 a
+minimum such as zi+1R∞y2. This implies that ziR∞y2 and we have the edge e3 = [zi, y2] ∈ KR∞.
+We can repeat this process with the (y2, yn)-path until we obtain a sequence of edges that connect
+the vertex [x] and [x′]. We obtain that KR∞ is edge-connected.
+Now suppose that GR is not connected. Let H be a connected component of GR. Then, for all
+x ∈ H and for all y /∈ H, we have that x /∈ R(y) and y /∈ R(x). It implies that for each connected
+component gives a single generator for H0(KR∞).
+We can construct a map j : cc(GR) → H0(KR∞) that sends the connected components of GR to
+the generators of H0(KR∞). By the previous argument, we can make this map j injective . For any
+generator g in H0(KR∞), there exists a x ∈ X such as g is homologous to [x] because Dom R = X.
+This implies there exists a H ∈ cc(GR) such as x ∈ H. We obtain that the map j is bijective and
+the equality (12).
+15
+
+Figure 5: The graph of R4
+2 from Example 5.2. GR4
+2 has 2 connected components.
+We can show a similar proof for simple relations.
+Theorem 5.6. Let R be a finite simple self-relation on X with Dom R = X and (j, p) be an eventual
+period. Assume that GR is connected. There exists a r ∈ N such that :
+dim H0(KRj) = dim H0(LRj) = number of connected components of GRr.
+(13)
+Proof. We can find q big enough so that Rq is acyclic, because R is a simple relation. We choose a i ∈
+N such as iq > j. We fix r = iq. We also have that Rr is also acyclic. By Corollary 4.8, we have that
+KRj = KRr. By Theorem 5.5, we know that dim H0(KRj) = number of connected components of Rr.
+If GR has more than one connected component, we apply this theorem for each connected
+component of GR by using similar arguments as the proof of Theorem 5.5. From preceding results,
+if Dom R ̸= X but Im R = X, we can redo the proofs with LRj. Another approach is to use R−1,
+because Dom R−1 = X.
+Remark 5.7. In Example 5.2, it is a simple relation. We have R4
+2 is acyclic. The graph of R4
+2 is
+shown in Figure 5. It has two connected components and the bar code from Figure 4(b) has 2 bars
+goes to infinity. It is expected from Theorem 5.6.
+We have shown earlier in Corollary 4.10 and Proposition 4.12 that a strongly connected self-
+relation R is shift equivalent to a matrix J, if tr(R) > 0. But we will like to have a result for any
+strongly connected relations. But, first we need some definition and other results from other papers.
+Let gcd(a, b) be the great common divisor of a and b. We define :
+q = gcd(n1, n2, n3, . . .)
+(14)
+where ni is the length of a cycle and i ∈ I is the set of all different cycles from R.
+We obtain this proposition :
+Proposition 5.8 (Proposition 6.12 in [26]). Let R be a strongly connected self-relation on X, (j, p)
+be the eventual period and q defined by (14). We have q|j.
+16
+
+Let’s define a new equivalence relation ∼q for a strongly connected self-relation R. We say that
+x ∼q y, if for each (x, y)-walk has length equal to 0 modulo q. It is an equivalence relation.
+Proposition 5.9 (Proposition 6.16 in [26]). Let R be a strongly connected self-relation on X. Let q
+defined as (14). Then, ∼q is an equivalence relation in X with exactly q distinct equivalence classes.
+We need one more Lemma before showing our final result.
+Lemma 5.10 (Lemma 6.25 in [26]). Let R be a strongly connected self-relation on X and (j, p) an
+eventual period. Then,
+x ∼q x′ =⇒ Rj(x) = Rj(x′).
+(15)
+We are going to show that the number of class equivalence of ∼q is equal to dim(H0(KRj)) for
+a strongly connected self-relation with eventual period (j, p).
+Theorem 5.11. Let R be a finite self-relation on X with an eventual period (j, p), R is strongly
+connected, q defined by (14). Then, we have :
+number of [x]∼q = q = dim(H0(KRj)) = dim(H0(LRj)).
+(16)
+Proof. We show that for any x, y ∈ X, if x ∼q y, then x and y are edge-connected and if x ̸∼q y,
+then x and y are not edge-connected.
+First, we suppose that x ∼q y. By Lemma 5.10, we have that Rj(x) = Rj(y) ̸= ∅. There exists
+a z ∈ Rj(x). It implies that [x, z] and [y, z] are in KRj. So, each vertex in the same equivalence
+class is edge-connected.
+Now, we suppose that, x ̸∼q y. There exists a (x, y)-walk of length n modulo q where n ̸= 0.
+Let show that Rj(x) ∩ Rj(y) = ∅ . Let’s suppose there exists a z ∈ Rj(x) ∩ Rj(y). This implies
+there exists a (x, z)-walk of length j and a (y, z)-walk of length j. But, from Proposition 5.8, q|j.
+This implies x ∼q z and y ∼q z. But ∼q is an equivalence relation. We obtain that x ∼q y which is
+a contradiction. We obtain that x ̸∼q y implies Rj(x) ∩ Rj(y) = ∅. We obtain that if x ∼q y then
+they are edge-connected. But, if x ̸∼q y, then they are not edge-connected. There is q different
+equivalence classes. The proof is complete.
+For the case of R is an arbitrary relation, it is harder to find its homology H0. Also, for higher
+dimensions of the homology groups, it’s hard to tell what happens. Further investigations are needed
+for both cases.
+Now, we return to the filtrations defined earlier. There are two other types of filtrations that we
+can use. If R is a self-relation on X with Dom R = X = Im R, then we can use both filtrations. But
+we obtain the same bar codes for both. That holds because, for each i ∈ N, |KRi| is homotopically
+equivalent to |LRi| for any self-relation. We might need to come with other types of filtration. We
+suggest two other types of filtration.
+It will be interesting to use a zigzag filtration [5] with KRi and LRi by alternating them. It will
+probably depend on the relation. Further investigations are needed.
+17
+
+We will present an interesting bi-filtration with KRm and LRn. We have that, if KRm ⊂ KRm+1
+and LRn ⊂ LRn+1, then KRm ∩LRn ⊂ KRm+1 ∩LRn and KRm ∩LRn ⊂ KRm ∩LRn+1 for all m, n ∈ N.
+We obtain this bi-filtration :
+...
+...
+. . .
+KRm ∩ LRn
+KRm+1 ∩ LRn
+. . .
+. . .
+KRm ∩ LRn+1
+KRm+1 ∩ LRn+1
+. . .
+...
+...
+The computation of the bi-filtration is also finite. Because the relation R is finite and Dom R =
+X = Im R. We obtain an eventual period (j, p). In the bi-filtration, there are, at maximum, j2
+different simplicial complexes to compute.
+One may ask why the intersection is a good idea to consider. Let’s explain it in more details.
+Let R be a self-relation on X with Dom R = X = Im R and m, n ∈ N. Let σ ∈ KRm ∩ LRn where
+σ = [x1, x2, . . . , xd]. Then, there exists xω ∈ X such that xiRmxω and there exists xα ∈ X such
+that xαRnxi for all i. Another way to see this is, for each xi ∈ σ, there exists a (xω, xα)-walk of
+length m+n passing through xi. We can subdivide this (xω, xα)-walk into a (xω, xi)-walk of length
+m and a (xi, xα)-walk of length n. So, by only the existence of a simplex σ in KRm ∩ LRn, each
+vertex of σ, it has a walk with a common starting point and a common ending point of same length
+going through the vertex. It will be interesting to study these Dowker complexes when m tends to
+infinity, n tends to infinity or both.
+6
+Conclusion
+In summary, we used the Dowker complexes to study some properties of self-relation. First, we
+defined the right morphism and left morphism. We also generalized it to the case of multivalued
+maps called multi-right morphism and multi-left morphism. The existence of a right or multi-right
+morphism between R1 and R2 implies that the KR1 is included in KR2. Similarly, the existence of a
+left or a multi-left morphism between R1 and R2 implies that LR1 is included in LR2. We have shown
+that two relations are conjugate implies that they have homotopically equivalent Dowker complexes.
+We have also shown that if two relations are shift equivalent, then their Dowker complexes are
+homotopically equivalent at some power of the relations. We were interested in self-relation which
+is equivalent of a type of directed graph. We have obtained two nice properties. If R is finite and
+Dom R = X = Im R, then we have that KRi ≤ KRi+1 and LRi ≤ LRi+1 for all i ∈ N. Moreover,
+there exists a j ∈ N such as KRj = KRi+j and LRj = LRi+j for all i ∈ N. With these two properties,
+we defined two filtrations with KRi �→ KRi+1 and LRi �→ LRi+1. Also, the filtration ends at some
+finite time. Finally, we proved some results about the 0th homology for some types of self-relation
+at some power. We also proposed the intersection filtration and the zigzag filtration.
+18
+
+We have put some foundations to study directed graph using Dowker complexes. Moreover, we
+think it might be a useful tool to study the dynamics of finite data define by a directed graph of
+Dowker complexes. Let R be a self-relation with an eventual period (j, p). The positive or forward
+invariant is given by the existence of a simplex in KRj when j converges to infinity. The negative
+or backward invariant is given by the existence of a simplex in Lj
+R when j converges to infinity.
+But, by the stabilization of Dowker complexes, we can compute it in finite time. Finally, if we want
+to study the invariant of R, this is given by the existence of a simplex in KRj ∩ LRj which is the
+intersection of the forward and the backward invariant. But further investigations are needed. The
+idea is to use the structure of the Dowker complexes to encode the dynamics of finite data.
+19
+
+References
+[1] K. Ambrose, S. Huntsman, M. Robinson, and M. Yutin. Topological differential testing. CoRR,
+abs/2003.00976, 2020.
+[2] R. Atkin. Mathematical Structure in Human Affairs. London, Heinemann, 1974.
+[3] R. Atkin. Q-analysis: A hard language for the soft sciences. Futures, Heinemann, 10(6), 1978.
+[4] A. Björner. Topological methods. Handb. Comb., 2:1819–1872, 1995.
+[5] G. Carlsson and V. de Silva. Zigzag persistence. Found Comput Math, 10, 2010.
+[6] S. Chowdhurry and F. Mémoli.
+A functorial Dowker theorem and persistent homology of
+asymmetric networks. Journal of Applied and Computational Topology, 2:115–175, 2018.
+[7] C. Conley. Isolated Invariant Sets and the Morse Index. American Mathematical Society, 1978.
+[8] S. Day, O. Junge, and K. Mischaikow. A rigorous numerical method for the global analy-
+sis of infinite-dimensional discrete dynamical systems. SIAM J. Applied Dynamical Systems,
+3(2):117–160, 2004.
+[9] D. Desjardins Côté. From finite vector field data to combinatorial dynamical systems in the
+sense of forman. arXiv, 2021.
+[10] C. Dowker. Homology groups of relations. Annals of Mathematics, pages 84–95, 1952.
+[11] H. Edelsbrunner and J. L. Harer.
+Computational Topology : An Introduction.
+American
+Mathematical Society, 2010.
+[12] M. Erdmann. Topology of privacy: Lattice structures and information bubbles for inference
+and obfuscation. arXiv, 2017.
+[13] R. Forman. Combinatorial vector fields and dynamical systems. Mathematische Zeitschrift,
+228:629–681, 1998.
+[14] R. Ghrist, D. Lipsky, J. Derenick, and A. Speranzon. Topological landmark-based navigation
+and mapping.
+https://www2.math.upenn.edu/~ghrist/preprints/landmarkvisibility.
+pdf. 2012.
+[15] K. Hang Kim and F. W. Roush. On strong shift equivalence over a boolean semiring. Ergod.
+Th. and Dynam. Sys., 6:81–97, 1986.
+[16] T. Kaczynski, K. Mischaikow, and M. Mrozek. Computational Homology. Springer, 2004.
+[17] T. Kaczynski and M. Mrozek. Conley index for discrete multivalued dynamical systems. Topol-
+ogy and Its Appl., 65:83–96, 1995.
+[18] T. Kaczynski, M. Mrozek, and T. Wanner. Towards a formal tie between combinatorial and
+classical vector field dynamics. Journal of Computational Dynamics, 3(1):17–50, 2016.
+[19] W. D. Kalies, K. Mischaukow, and R. C.A.M Vandervorst. Lattice structures for attractors i.
+Journal of Computational Dynamics, 1(2):307–338, 2014.
+20
+
+[20] W. D. Kalies, K. Mischaukow, and R. C.A.M Vandervorst. Lattice structures for attractors ii.
+Foundations of Computational Mathematics, 16:1151–1191, 2016.
+[21] W. D. Kalies, K. Mischaukow, and R. C.A.M Vandervorst. Lattice structures for attractors iii.
+Journal of Dynamics and Differential Equations, 34:1729–1768, 2022.
+[22] K. H. Kim. Boolean Matrix Theory and Applications, volume (Monographs and textbooks in
+pure and applied mathematics, v. 70). New York:Dekker, 1982.
+[23] M. Lipiński, J. Kubica, M. Mrozek, and T. Wanner. Conley-Morse-Forman theory for general-
+ized combinatorial multivector fields on finite topological spaces. arXiv:1911.12698 [math.DS],
+pages 1–44, 2020.
+[24] G. Minian Elias. The geometry of relations. Order, 27:213–224, 2010.
+[25] M. Mrozek, R. Srzednicki, Thorpe J., and Th. Wanner. Combinatorial vs classical dynamics :
+Recurrence. Commun. Nonlinear Sci. Numer. Simul., 108(106226):1–30, 2022.
+[26] Marian Mrozek and Mateusz Przybylski. The szymczak functor on the category of finite sets
+and finite relations. arXiv, 2022.
+[27] J. R. Munkres. Elements of Algebraic Topology. Addison-Weslay, Cambridge, 1984.
+[28] M. Robinson. Cosheaf representations of relations and Dowker complexes. Journal of Applied
+and Computational Topology, 6:27–63, 2022.
+[29] A. Szymczak. The Conley index for disccrete dynamical system. Topology Appl., 66:215–240,
+1995.
+[30] A. Szymczak. Index Pairs : From Dynamics to Combinatorics and Back. Ph.D. thesis, Georgia
+Inst. Tech., 1999.
+21
+

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+Unsupervised Mandarin-Cantonese Machine Translation
+Megan Dare, Valentina Fajardo Diaz, Averie Ho Zoen So, Yifan Wang, Shibingfeng Zhang
+Summer Semester Software Project 2022
+Language Science and Technology, Saarland University
+{mdare,valenfd,averieso,yifwang,[email protected]}
+Abstract
+Advancements in unsupervised machine trans-
+lation have enabled the development of ma-
+chine translation systems that can translate be-
+tween languages for which there is not an
+abundance of parallel data available. We ex-
+plored unsupervised machine translation be-
+tween Mandarin Chinese and Cantonese. De-
+spite the vast number of native speakers of
+Cantonese, there is still no large-scale corpus
+for the language, due to the fact that Can-
+tonese is primarily used for oral communica-
+tion.
+The key contributions of our project
+include: 1.
+The creation of a new corpus
+containing approximately 1 million Cantonese
+sentences, and 2.
+A large-scale compari-
+son across different model architectures, tok-
+enization schemes, and embedding structures.
+Our best model trained with character-based
+tokenization and a Transformer architecture
+achieved a character-level BLEU of 25.1 when
+translating from Mandarin to Cantonese and of
+24.4 when translating from Cantonese to Man-
+darin. In this paper we discuss our research
+process, experiments, and results.
+1
+Introduction
+In recent years, neural machine translation has
+gained massive research interests. Most of these
+studies (e.g. Bahdanau et al. 2014; Luong et al.
+2015; Wu et al. 2016; Vaswani et al. 2017) focus
+on the construction of neural machine translation
+systems leveraging parallel bilingual corpora. Nev-
+ertheless, such an approach is not feasible for many
+language pairs due to the scarcity of resources for
+such pairs, as is the case for Cantonese and Man-
+darin. The study of automatic translation between
+these two languages faces the same problem: to
+the best of our knowledge, despite the vast number
+of native speakers of both languages, there is still
+no large-scale Mandarin-Cantonese parallel corpus.
+In addition, monolingual corpora for Cantonese are
+hard to collect as it is a low-resource language that
+is mainly used for only oral communication.
+Currently, only a few studies have been done
+on Cantonese-Mandarin translation, among which
+some compare various low-resource models for
+this language pair. However, these studies nor-
+mally focus on a comparison between one or two
+model types. Based on our motivation of imple-
+menting and training a Cantonese-Mandarin trans-
+lation model and current state of research, we set
+our goal as building a robust model trained on
+a more diverse dataset, which can help improve
+communication between Cantonese and Mandarin
+speakers. Additionally, we seek to compare vari-
+ous model architectures, tokenization schemes, and
+embedding structures to conduct a comprehensive
+understanding on which settings may lead to the
+best performance for the Cantonese-Mandarin lan-
+guage pair.
+After a close analysis of the current state of re-
+search and the available resources, we propose to
+develop a Cantonese-Mandarin machine translation
+system that is capable of conducting translation in
+both directions. The training of the system involves
+only Mandarin and Cantonese monolingual corpora
+collected from Wikipedia and various websites.
+Our work also makes contributions to the Can-
+tonese language NLP field by collecting Cantonese
+textual data and building a public large-scale mono-
+lingual corpus, which did not exist until now.
+In addition, considering the similarity between
+Cantonese and Mandarin, our translation system
+will provide a foundation for further development
+regarding machine translation tasks that center
+around language pairs composed of two similar
+languages.
+2
+Background
+2.1
+Cantonese and Chinese: an overview
+Cantonese is one of the most widely spoken va-
+rieties of Chinese other than Mandarin Chinese
+(Matthews and Yip, 2013). It is estimated to have
+arXiv:2301.03971v1  [cs.CL]  10 Jan 2023
+
+more than 55 million native speakers, with large
+populations found in southern China provinces
+Guangdong and Guangxi, as well as regions includ-
+ing Hong Kong and Macau, it is also commonly
+spoken in overseas Cantonese communities in Sin-
+gapore, Malaysia, North America and Australia as
+a result of emigration (Matthews and Yip, 2013).
+While numerous NLP applications have been
+developed for Mandarin Chinese, little has been
+developed for Cantonese. One reason for this is the
+limited linguistic resources that have been collected
+for Cantonese. Primarily a spoken language and a
+non-standard variety, written Cantonese is not tra-
+ditionally used or taught in schools. Instead, Can-
+tonese speakers typically learn to read and write
+in standard Chinese through education, so there is
+no language barrier for Cantonese speakers when
+interacting with computer applications designed in
+standard Chinese.
+On the other hand, with the availability of the
+internet and the rise of social media, Cantonese is
+much more commonly used and written online in
+recent years, which can be seen as an indicator for
+a market in Cantonese NLP applications.
+It is important to note that this phenomenon
+might only be applicable to Hong Kong Cantonese,
+and not other variants such as the one in Guang-
+dong province. More recent discussions about Can-
+tonese, such as Bauer (2018), make a point to dis-
+tinguish between the Hong Kong Cantonese variant
+and the others, since the use of Cantonese is on the
+rise in Hong Kong, while declining in provinces
+within mainland China. Not only has this led to
+Hong Kong being named “the Cantonese-speaking
+capital of the world" (Bolton, 2011, p.64), but also
+the rise of written Cantonese locally and subse-
+quently, the Cantonese text data that are available
+online, which are of the Hong Kong variant of Can-
+tonese.
+2.2
+Linguistic Differences between Cantonese
+and Mandarin
+Despite the common misconception that Chinese
+dialects share the same grammar, Cantonese and
+Mandarin are different at phonological, lexical and
+syntactic levels, and are not mutually intelligible
+(Matthews and Yip, 2013). Some suggests it is
+more accurate describe Cantonese as a distinct
+language of the Chinese language family (Snow,
+2004). For the rest of this section, we describe
+some features that differ between Mandarin and
+Hong Kong Cantonese.
+2.2.1
+Writing Systems
+To anyone who can read Chinese, the most notable
+visual variation in written Chinese is the writing
+system - Traditional or Simplified Chinese. The
+two systems are equivalent to each other, and have
+one-to-one correspondence for each character. The
+following is some examples of traditional / sim-
+plified characters: “open" 開/开, “talk" 話/话 and
+“book" 書/书. The usage of either system is primar-
+ily due to regional difference, with mainland China
+using the simplified system, while Hong Kong and
+Taiwan use the traditional system.
+2.2.2
+Lexical and Syntactic comparisons
+Vocabulary difference is the main barrier which
+prevents Mandarin speakers from understanding
+Cantonese (Snow, 2004), it is also the aspect which
+is the most distinguishable between Cantonese and
+Mandarin.
+According to Snow (2004), written
+Cantonese in formal domains can contain around
+10-15% Cantonese-only characters, while this per-
+centage in informal domains can go up to 25-40%.
+Notably, the vocabulary that differ are some of
+the most frequent words, including many func-
+tion words, as seen in Table 1.
+Syntactically,
+Meaning
+Cantonese
+Mandarin
+possessive marker
+ge3
+的de
+perfective marker
+zo2
+了le
+pronoun pluralizer
+dei6
+們mén
+negator
+唔m4
+不bù
+is (copula)
+係hai6
+是shì
+this
+呢ne1
+這zhè
+Table 1: Examples of lexical difference between Can-
+tonese and Mandarin from Snow (2004, p.49). Can-
+tonese romanizations follow the Jyutping system.
+Cantonese and Mandarin are broadly similar but
+with some differences that are often overlooked
+(Matthews and Yip, 2013). Some common differ-
+ences are in terms of word order, including indi-
+rect object and comparative constructions (Snow,
+2004):
+Indirect object construction:
+Cantonese:
+我俾錢佢ngo5 bei2 cin4 keoi5
+(I + give + money + he)
+Mandarin:
+我給他錢wó gˇei t¯a qían
+2
+
+(I + give + he + money)
+‘I give him money’
+Comparative construction:
+Cantonese:
+我高過佢ngo5 gou1 gwo3 keoi5
+(I + tall + more than + he)
+Mandarin:
+我比他高wó bˇı t¯a g¯ao
+(I + compared to + he + tall)
+‘I’m taller than him.’
+2.2.3
+Challenges Unique to Cantonese NLP
+Firstly, there exists a certain degree of variabil-
+ity in written Cantonese since it was never stan-
+dardised. As such, some words can be written
+with completely different characters yet have the
+same meanings and pronunciations. For example,
+“like" can be written as 中意or 鍾意(read: zung1
+ji31), “still" can be written as 仲or 重(read: zung6)
+(Matthews and Yip, 2013). Additionally, when
+some Cantonese words cannot be represented by
+existing Chinese characters, they could be written
+in a romanized form, such as the comparative (eg.
+“-er" in “cheaper") can be written with “D", as well
+as a non-romanized form (read: di1) (Snow, 2004;
+Matthews and Yip, 2013).
+Secondly, code-switching to English is a com-
+mon phenomena in Cantonese, which is not a
+feature in standard Chinese or Mandarin. Code-
+switching in Hong Kong Cantonese is mostly in-
+trasentential (below clause level) (Li, 2000), for
+example:
+我今朝9點有個meeting。
+ngo5 dei6 gam1 ziu1 gau2 dim2 jau5 go3
+MEETING
+‘We have a meeting at 9am today.’
+3
+Related Work
+3.1
+Unsupervised Machine Translation
+Unsupervised machine translation with no parallel
+data is a challenging task that has attracted many
+interests. The presence of cross-lingual embed-
+dings (Mikolov et al., 2013; Artetxe et al., 2016,
+2017a, 2018a,b; Conneau et al., 2017) provides
+prior knowledge for machine translation systems
+and makes it possible to train a machine transla-
+tion model in an unsupervised way. Artetxe et al.
+(2017b) and Lample et al. (2017) are the first at-
+tempts to explore the possibility of constructing
+1romanizations according to the Jyutping system.
+a neural machine translation system using only
+monolingual corpora from both source and target
+languages. The proposed system is based on an
+encoder-decoder architecture with attention mecha-
+nism (Bahdanau et al., 2014), trained with a denois-
+ing auto-encoding task (Vincent et al., 2008) and a
+back-translation task (Sennrich et al., 2015). The
+encoder is shared by both the source and target lan-
+guages, so that sentences from both languages can
+be mapped to a common latent space, while each
+language has its own decoder to reconstruct en-
+coded sentences back into its own language space.
+Cross-lingual embeddings are leveraged as an ini-
+tialization for the system, providing additional lex-
+ical level information. Such a structural property
+allows the translation model to be bi-directional,
+that is, the same model can be employed in both the
+L1-to-L2 translation task and the L2-to-L1 transla-
+tion task.
+This approach is extended in Lample et al. (2018)
+by applying a transformer model and using sub-
+word level tokenization methods. Attention-only
+structures provide higher model capacity, and sub-
+word level tokenization method Byte Pair Encod-
+ing (BPE) reduce the size of vocabulary and helps
+solving <UNK> problems in translation. Addition-
+ally, they re-exploited the potential of statistical
+approaches in unsupervised machine translation
+tasks. A phrase-based machine translation model
+initialized with an automatically populated phrase
+table and language model is trained by iterative
+back-translation. Results of the experiment show
+that a statistical approach can reach similar perfor-
+mance or even outperform neural systems when the
+data is scarce, as the neural model tends to over-
+fit the corpora, and thus does not generalize well.
+Together with Singh and Singh (2020), they show
+that unsupervised approaches can be used to con-
+struct machine translation systems for low-source
+languages (e.g., Urdu, Romanian, Manipuri).
+In recent years, pre-trained language models
+have become popular due to their competitive
+ability of representing and generating natural lan-
+guages learned from transfer learning on large-
+scale self-supervised datasets. Lample and Con-
+neau Lample and Conneau (2019) take their work
+one step further by pre-training both the encoder
+and decoder in their model using a cross-lingual
+language model (XLM). They then fine-tune the
+pre-trained model to an unsupervised neural ma-
+chine translation model following the training pro-
+3
+
+cess described in Lample et al. (2018). The pre-
+training stage results in a sharp BLEU score in-
+crease over previous benchmarks for unsupervised
+machine translation.
+Unsupervised machine translation methods are
+also applied in dialectal machine translation tasks,
+where the similarity and commonality between lan-
+guages can be leveraged. Farhan et al. (2020) uses
+common words between Arabic dialects as anchor
+points to steer projections of surrounding words be-
+tween two dialects, creating a more accurate map-
+ping between source and target words. In this way,
+they construct an unsupervised machine translation
+system with a BLEU score of 32.14, which is re-
+markably high compared with the highest BLEU
+score obtained in the supervised setting (48.25).
+3.2
+Mandarin-Cantonese Machine
+Translation
+Due to the scarcity of available datasets, Cantonese
+language is always under-researched in NLP tasks.
+This issue is even more severe in machine trans-
+lation tasks, which usually requires large amount
+of parallel data. For this reason, many researches
+on Cantonese-Mandarin machine translation are
+intended to collect more data or to fully exploit the
+limited data in a semi-supervised or unsupervised
+way.
+Hei Yi Mak and Tan Lee (2021) construct a
+large-scale Cantonese-Mandarin parallel dataset
+by mining parallel sentences from Mandarin and
+Cantonese Wikipedia. They apply a similarity-
+based sentence alignment approach and use sen-
+tence pairs with high confidence score as parallel
+sentences. In this way, they end up with a paral-
+lel corpus of about 100,000 sentences. They also
+fine-tune a pre-trained language model using the
+collected data and obtain a competitive translation
+system that outperforms Baidu Fanyi, a commonly
+used translator in China.
+Concurrently, some efforts have been made to
+create unsupervised Cantonese-Mandarin transla-
+tion systems. (Wan et al., 2020) handles Cantonese-
+Mandarin translation as a dialect translation prob-
+lem. which attempts to exploit the commonality
+between two language dialects. On the basis of
+(Lample et al., 2018)’s transformer model, they
+make use of pivot-private embeddings and layer
+coordination to better utilize the similarity and dif-
+ference between the two languages. Trained on
+two large monolingual datasets of 20 million collo-
+quial sentences for each Mandarin and Cantonese,
+their model reaches an improvement of up to 12
+BLEU score for Cantonese to Mandarin, and 5
+BLEU from Mandarin to Cantonese compared to
+their baseline transformer model.
+There have been other works relying on pre-
+trained cross-lingual language models (XLM). In
+Wong and Tsai (2022), the authors initialize the
+encoder and decoder with XLM as described in
+(Lample and Conneau, 2019), while using pivot-
+private embeddings rather than cross-lingual em-
+beddings. Using this enriched structure, they are
+able to achieve slight BLEU score improvements
+over previous XLM models.
+4
+Corpus Construction
+While existing Cantonese corpora are scarce, and
+usually collected for linguistic purposes which is
+smaller in scale and of a specific demographic (eg.
+Wong et al. 2017; Luke and Wong 2015), text data
+is available on the internet due to Cantonese being
+the common language used on social media. This
+also led to a rise in Cantonese writing in tradition-
+ally more formal domains such as advertisements,
+online news and subtitles.
+Therefore, we aim for the corpus to span across
+various domains for a comprehensive collection of
+modern Cantonese usage. Secondly, since standard
+Chinese is also commonly used among Cantonese
+speakers in online settings, in the data selection pro-
+cess, we aim to avoid sources which use standard
+Chinese. Lastly, in our pre-processing, we preserve
+some unique features in Cantonese such as code-
+switching in English. Detailed data statistics of the
+corpus is available on the Github repository.
+As we focus on collecting data for Cantonese,
+note that we simply use the Chinese Wikipedia
+for Mandarin data, since there is already a large
+amount of data available just from one source.
+4.1
+Data Collection
+The Cantonese data available from various sources
+on the internet are either readily downloadable (for
+Wikipedia, corpus and dictionary) or are scraped
+by us (for Instagram, subtitles and articles). Due to
+structural differences in the various websites, scrap-
+ing functions are individually written for each of
+the three classes of sources. In general, the script
+moves recursively over the website domain and
+extracts any text in each web page. The scraping
+script is available on our GitHub repository. Fig-
+4
+
+ure 1 shows the distribution in data domain of the
+Cantonese training dataset, which contains only
+monolingual data sources.
+4.1.1
+Monolingual Data
+Cantonese Wikipedia
+The largest source of data
+available was Cantonese Wikipedia, which was
+downloaded from Wikimedia dump2, then pure
+text data is obtained with WikiExtractor (Attardi,
+2015). Cantonese Wikipedia amounts to 690k lines
+of text, making up 70% of the Cantonese corpus
+overall.
+Corpus
+As mentioned, there is a small number
+of open source Cantonese corpora collected for aca-
+demic purposes, mainly transcribed from spoken
+Cantonese. Additionally, there is another corpus
+which contains scraped text data. Existing corpora
+add up to 95k lines of Cantonese text, with the ma-
+jority coming from Openrice restraurant reviews
+(78k).
+• openrice-senti3: scraped restaurant reviews
+from popular Hong Kong website OpenRice
+(https://www.openrice.com/zh/
+hongkong).
+• HK Cantonese Corpus4 (Wong et al., 2017):
+manually
+transcribed
+oral
+conversations
+recorded between 1997-1998, includes spon-
+taneous speech as well as radio programmes.
+• tatoeba5: a website which contains crowd-
+sourced sentences and their translations in
+many languages, including Cantonese.
+Instagram
+Due to its popularity in Hong Kong,
+the domains from Instagram can be varied, ranging
+from blogs, advertisements, news and governmen-
+tal organisations. We scrape posts and comments
+via imginn.org from 14 accounts, 5 of which
+are categorised as news, the others are categorised
+as non-news. Instagram comments make up the
+second largest source of Cantonese data with 108k
+lines (11%), while Instagram news are 58k lines
+and Instagram non-news 30k lines.
+Subtitles
+Cantonese YouTube6
+is a crowd-
+sourced compilation of youtube videos with spo-
+ken Cantonese subtitles. It is a voluntary effort
+2https://dumps.wikimedia.org/zh_yuewiki/20220601
+3https://github.com/toastynews/openrice-senti
+4https://github.com/fcbond/hkcancor
+5https://tatoeba.org/en
+6https://docs.google.com/spreadsheets/d/1CmN8GPalrb4
+5YFIPrWgh7GRYyoUhnizEOImY6kAW82w
+Figure 1: Distribution of data domain in the Cantonese
+training set (monolingual data only).
+from Cantonese learners, and each video is manu-
+ally tagged with “Written Cantonese" or “Standard
+Written Chinese", which allows us to filter for only
+Cantonese videos. We are able to scrape directly
+from Youtube with the help of the Youtube Tran-
+script API7. There are 1,620 lines.
+Articles
+We scrape blog articles written by vari-
+ous authors in Cantonese from the freelancer plat-
+form https://handstopmouthstop.com.
+There are 6,531 lines from the website.
+4.1.2
+Parallel Data
+As the experiments described in the future sections
+are unsupervised, parallel data is not included in
+the training set. They are only used for the test set.
+Corpus
+Cantonese-HK and Chinese-HK Uni-
+versal Dependencies Treebank8(Luke and Wong,
+2015): manually transcribed and annotated film
+subtitles and legislative proceedings of Hong Kong,
+in both Cantonese and Mandarin. There are 1,004
+parallel sentences from this corpus.
+Dictionary
+Kaifangcidian9
+is
+an
+online
+Cantonese-Chinese dictionary which comes with
+parallel sentences for each lexical entry. There are
+13,004 parallel sentences from the dictionary.
+Subtitles
+Kongjisubtitles 10 is a Cantonese sub-
+title team that specialises in “kongji"(meaning
+“Hong Kong words" in romanized Cantonese) and
+focuses on subtitling Thai online series. Since
+7https://github.com/jdepoix/youtube-transcript-api
+8https://github.com/UniversalDependencies/UD_Cantonese-
+HK
+9https://kaifangcidian.com/han/yue/
+10https://sites.google.com/view/lihkg-kongjisubtitles
+5
+
+instagram comments
+restaurantreviews
+11%
+8%
+instagram news
+6%
+instagram non-news
+3%
+2%
+corpus
+1%
+subtitles & articles
+70%
+wikipediasome of the same videos also have Mandarin subti-
+tles, we align them based on the timestamps of the
+videos. This amounts to 77,479 lines of parallel
+data.
+4.2
+Pre-processing
+Our data is scraped from different resources and
+inevitably contains noise. The following tools are
+leveraged for the pre-processing of collected data:
+Sentence Cutter
+Sentence cutter cuts each text
+into sentences. The cutting points are punctuation
+marks such as 。.!? that defines the end of a sen-
+tence.
+Mandarin-Cantonese Filter
+Due to the fact that
+most Cantonese speakers are also native in Man-
+darin, Mandarin text is normally present in Can-
+tonese data scraped from social media. Mandarin-
+Cantonese Filter aims to determine whether a sen-
+tence is written in Mandarin or Cantonese by calcu-
+lating the number of language-specific characters.
+This tool is involved only in the pre-processing of
+Cantonese data.
+Cantonese-specific characters are: , 唔, 係, , 啦,
+, 既, 咁, 佢, , 冇, 仲, , 乜, 噉, 咪, 咩, 俾, 呢, , 黎, ,
+喂, 喇, 喎, 睇
+Mandarin-specific characters are: 是, 的, 他, 她,
+沒, 也, 看, 說, 在,说
+Foreign Text Filter
+Text written in foreign lan-
+guages such as Russian, Japanese and Korean
+abounds in collected data.
+Foreign Text Filter
+serves to filter out all sentences that are not writ-
+ten in Chinese characters. If the Chinese charac-
+ters contributes to less than 5% of sentence’s total
+length, the sentence is removed.
+url, emoji, hashtag Remover
+This tool serves
+to remove url, emoji, and hashtag from sentence
+using regular expression.
+Jieba Tokenizer
+Jieba 11 is a Mandarin NLP li-
+brary. In our project, we used Jieba tokenizer to
+pre-process our Mandarin data.
+PyCantonese Tokenizer
+PyCantonese 12 is a
+Cantonese NLP library. In our project, we used Py-
+Cantonese tokenizer to pre-process our Cantonese
+data.
+We did not include any Mandarin data from so-
+cial media in our dataset, considering that data
+11https://github.com/fxsjy/jieba
+12https://pycantonese.org/
+(a) Mandarin corpus
+(b) Cantonese corpus
+Figure 2: Distribution of sentence length.
+scraped from social media is always full of noises
+and Mandarin data from Wikipedia is already abun-
+dant for our task. We included Cantonese data
+scraped from social media since Cantonese data
+from Wikipedia is not sufficient.
+4.2.1
+Overall Data Statistics
+After pre-processing, there are 912,258 lines of
+monolingual Cantonese data and 16M lines of
+monolingual Mandarin data. In terms of domains,
+the Cantonese corpus has 70% data from Wikipedia
+while the Mandarin corpus is 100% Wikipedia. Fig-
+ure 2 shows that the distribution of sentence length
+in Cantonese and Mandarin are broadly similar af-
+ter pre-processing.
+5
+Methodology
+As shown in Figure 3, we follow a standard un-
+supervised machine translation architecture with
+a shared encoder and language-specific decoders
+in our experiment. Models are trained on a de-
+6
+
+1e6
+1.4
+1.2
+1.0
+frequency
+0.8
+0.6
+0.4
+0.2
+0.0
+0
+5
+10
+15
+20
+25
+30
+sentencelength(punctuationincluded)60000
+50000
+40000
+frequency
+30000
+20000
+10000
+0
+0
+5
+10
+15
+20
+25
+30
+sentencelength(punctuationincluded)Figure 3: General architecture of the unsupervised machine translation systems in this experiment. A shared
+encoder maps sentences from L1/L2 to a common latent space, then a language-specific decoder reconstructs the
+encoded sentence back into its own language space. The model is trained by a denoising auto-encoding task and a
+back-translation task.
+noising auto-encoding task and an on-the-fly back-
+translation task. To have an overall study of how
+different setups affect the model performance, we
+make three sets of comparisons:
+1. Model architectures.
+2. Cross-lingual embeddings.
+3. Tokenization methods.
+5.1
+Model Architectures
+In this experiment, we compare an RNN-based
+attention model and a transformer model.
+• RNN-based model: We adopt the architecture
+from (Artetxe et al., 2017b): Both encoder
+and decoder have 2-layer bidirectional GRU
+(Cho et al., 2014), Luong’s attention (Luong
+et al., 2015) is applied to align the source sen-
+tence and translation. Input sentences are con-
+verted to 512-dimensional cross-lingual em-
+beddings. Considering the relatively lower ca-
+pacity, the cross-lingual embeddings are fixed
+during training.
+• Transformer model: Following (Lample et al.,
+2018), we use 4-layer encoder and decoder
+with 3-layer sharing parameters for both Can-
+tonese and Mandarin sides.
+When gener-
+ating translations, the decoder starts with a
+language-specific <BOS> token, specifying
+the language it is operating with. The embed-
+ding matrices are trainable during the training
+process.
+5.2
+Cross-lingual Embeddings
+Cross-lingual embeddings can be learned in various
+different ways. In our experiments we compare the
+following three approaches:
+• Mapping: It has been extensively studied how
+to map monolingual word embeddings into
+a cross-lingual space.(Mikolov et al., 2013;
+Artetxe et al., 2016, 2017a, 2018a,b; Conneau
+et al., 2017) In this project, we use Vecmap
+13 by Artexte to obtain cross-lingual embed-
+dings from monolingual ones. In particular,
+we adopt the “identical” setting, where the
+shared vocabulary in two languages can be
+used as anchors to learn the mapping. This
+approach is applied to RNN-based models.
+• Learning from concatenated data: Another
+setup is to learn embeddings on the concatena-
+tion of source and target corpora in a monolin-
+gual way. As embeddings are learned in the
+context of both languages, the resultant em-
+beddings can be seen as cross-lingual. This
+approach is applied on both RNN-based mod-
+els and transformer models.
+• Pivot-private embeddings: We also experi-
+ment with 512-dimensional pivot-private em-
+beddings which consists of a 256-dimensional
+cross-lingual embedding learned on the con-
+catenated dataset and a 256-dimensional pri-
+vate embedding, which is learned on two
+monolingual datasets separately.
+This ap-
+proach is assumed to be able to capture the
+commonality between both languages and pre-
+serve language-specific characteristics as well
+(Wan et al., 2020). We adopt this approach on
+transformer models.
+5.3
+Tokenization Methods
+We are also interested whether byte-pair encod-
+ing helps training Cantonese-Mandarin translation
+systems, so we compare it to a character-level tok-
+enization method.
+13https://github.com/artetxem/vecmap
+7
+
+L1 decoder
+L1output
+Sharedencoder(L1/L2)
+L2 decoder
+L2output
+Cross-lingual
+embeddings
+L1/L2 input• Word-level tokenization: As a baseline, we do
+no further tokenization on the collected data
+which is separated by words using Jieba and
+PyCantonese. In this setting, a total number
+of 80K/1M unique words are present in the
+Cantonese/Mandarin corpora respectively.
+• Character-level tokenization: Since Mandarin
+and Cantonese are both analytic languages,
+character-level tokenization is a valid option
+to tokenize sentences. This results in 8K/14K
+unique tokens in Cantonese/Mandarin training
+data respectively.
+• Byte-pair encoding: We also use byte-pair
+encoding to obtain a vocabulary of 50K sub-
+words on word-tokenized datasets. The em-
+beddings of sub-words are learned using meth-
+ods described above.
+6
+Experiments and Results
+In this section, we describe the experiments we
+conducted and the results of both automatic and hu-
+man evaluation. Our code and relevant repositories
+are publicly available online 14.
+6.1
+Task Setup
+6.1.1
+Baseline Model
+Due to the large overlap in vocabulary between
+Mandarin and Cantonese and the lack of compli-
+cated morphology in both languages, for our base-
+line model we take advantage of these character-
+istics by evaluating Mandarin sentences as if they
+were a translation into Cantonese, and visa-versa.
+This method is carried out by simply converting
+both Mandarin and Cantonese evaluation datasets
+to the same character set using OpenCC 15 (our
+experiments used the Traditional Chinese (Hong
+Kong variant) character set) and evaluating the
+BLEU score directly.
+6.1.2
+RNN-based Experiments
+In order to improve upon the baseline model perfor-
+mance, we train several models using Artetxe’s
+RNN+Attention-based architecture for unsuper-
+vised machine translation 16. The primary objec-
+tive, aside from improving BLEU scores over the
+baseline, is to identify which settings (e.g. tok-
+enization scheme and embedding training method)
+14https://github.com/meganndare/cantonese-nlp
+15https://github.com/BYVoid/OpenCC
+16https://github.com/artetxem/undreamt
+lead to the best model performance. As detailed in
+the methodology section we experiment with word,
+character, and byte-pair encoding (BPE) tokeniza-
+tion, as well as cross-lingual embeddings obtained
+by learning a mapping into cross-lingual space, and
+by concatenation and training a skip-gram model.
+Additionally, for the BPE-tokenized models we
+have experimented with learning the BPE tokens
+separately for each language, or jointly.
+6.1.3
+Balanced Dataset Experiments
+One characteristic of our full training dataset is that
+it is imbalanced (1 million Cantonese sentences
+versus 16 million Mandarin sentences). This is
+due to the abundance of Mandarin text data and
+the scarcity of Cantonese text data available. As
+a result, we were curious to understand whether
+having an imbalanced dataset negatively affects
+our training results. To this end we conducted an
+experiment using what we refer to as our ’Balanced
+Dataset’. To create the set, Mandarin sentences are
+chosen at random to be removed from the training
+set until a downsampled version of approximately
+the same size as the Cantonese training set was ob-
+tained, that also preserves the sentence length dis-
+tribution of the original Mandarin training set. We
+then compare the performance of models trained
+using the balanced dataset to those trained using the
+full set, utilizing some simple baseline settings for
+comparison, namely word and character-tokenized
+models.
+6.1.4
+Transformer Experiments
+Guided by advancements in neural network model
+architectures over the past several years, we are
+interested in how using a transformer architecture
+would impact our results. For the transformer ex-
+periments we leveraged Facebook Research’s Un-
+supervised Neural Machine Translation Model 17
+for training. Using the results from our RNN-based
+models, we primarily focused on character and
+BPE tokenization schemes, and have also experi-
+mented with a more complex cross-lingual embed-
+ding type called pivot-private embeddings. Due to
+differences in implementation between the RNN
+and Transformer-based models, we were unable
+to train Vecmap embeddings for this set of experi-
+ments.
+17https://github.com/facebookresearch/UnsupervisedMT
+8
+
+Model Name
+Can>Man Char BLEU
+Man>Can Char BLEU
+Baseline (Character Conversion) Model
+13.3
+13.2
+RNN (Word Tok + Vecmap Embed)
+13.1
+14.9
+RNN (Char Tok + Vecmap Embed)
+19.8
+22.5
+RNN (Char Tok + Concat Embed)
+19.4
+20.3
+RNN (BPE Tok learned separately + Vecmap Embed)
+18.0
+18.8
+RNN (BPE Tok learned jointly + Vecmap Embed)
+19.3
+19.5
+RNN (Balanced Dataset + Word Tok + Vecmap Embed)
+6.2
+11.5
+RNN (Balanced Dataset + Char Tok + Vecmap Embed)
+17.1
+20.4
+Transformer (Char Tok + Concat Embed)**
+24.4
+25.1
+Transformer (Char Tok + Pivot-Private Embed)
+21.2
+20.5
+Transformer (BPE Tok learned jointly + Concat Embed)
+20.2
+17.4
+Table 2: Overview of all automatic evaluation results. All BLEU (Bilingual Evaluation Understudy) metric
+scores are calculated at the character-level. Best-performing model indicated by **.
+6.2
+Results
+6.2.1
+Automatic Evaluation
+Model Architectures
+The first metric that our
+study sought to investigate was the varying per-
+formances of Mandarin-Cantonese unsupervised
+machine translation based on the underlying neu-
+ral network architecture, namely an RNN-based
+architecture versus a Transformer architecture. We
+observed that the transformer model led to higher
+BLEU scores when other factors are constant. This
+can be observed in the RNN (Char Tok + Con-
+cat Embed) versus Transformer (Char Tok + Con-
+cat Embed) models, where Cantonese-to-Mandarin
+translation yielded 19.4 versus 24.4, respectively;
+and Mandarin-to-Cantonese yielded 20.3 versus
+25.1, respectively. In fact, our highest performing
+model from the study was trained on a Transformer
+architecture.
+Cross-lingual
+Embeddings
+The
+study
+also
+makes comparisons between different types of
+cross-lingual embeddings.
+Of primary interest
+are training monolingual embeddings and map-
+ping them to a shared cross-lingual space using
+Vecmap (as detailed in the Methodology section),
+and learning embeddings from the concatenated
+data. In a comparison between RNN (Char Tok
++ Vecmap Embed) and RNN (Char Tok + Con-
+cat Embed) models, we can see that the mapping-
+based cross-lingual embeddings have outperformed
+the concatenation-based technique, yielding a
+Cantonese-to-Mandarin BLEU of 19.8 and 19.4,
+respectively; and a Mandarin-to-Cantonese BLEU
+of 22.5 and 20.3, respectively.
+In addition to mapping-based and concatenation-
+based cross-lingual embeddings, we also had time
+to run one experiment on pivot-private embeddings
+(as detailed in the Methodology section). By com-
+paring the Transformer (Char Tok + Concat Em-
+bed) and Transformer (Char Tok + Pivot-Private
+Embed) models, we observe that concatenation-
+based embeddings outperform pivot-private em-
+beddings, with a Cantonese-to-Mandarin BLEU
+of 24.4 versus 21.2, and a Mandarin-to-Cantonese
+BLEU of 25.1 to 20.5, respectively.
+Tokenization Methods
+Our study additionally
+makes a comparison between different types of
+tokenization methods: word, character, and BPE-
+tokenized models. Word-tokenization always per-
+forms the worst, in all cases aside from one (see
+RNN (Word Tok + Vecmap Embed) Mandarin-to-
+Cantonese results in Table 2), models trained with
+word-tokenized training data did not outperform
+even the Baseline (Character Conversion) Model
+in which no neural network was trained.
+While BPE-tokenized data tends to perform very
+well for languages with an alphabet system, such
+as French or English, we did not observe a such
+a strong result in the models trained using BPE-
+tokenized data for the Mandarin-Cantonese lan-
+guage pair. We experimented by learning BPE
+token vocabularies both separately and jointly, ob-
+serving a slight performance improvement when
+learned jointly. However, neither BPE setting could
+outperform our character-tokenized models (see Ta-
+ble 2 for two results that lead to this conclusion:
+RNN (Char Tok + Vecmap Embed) versus RNN
+9
+
+(BPE Tok learned jointly + Vecmap Embed), as well
+as Transformer (Char Tok + Concat Embed) versus
+Transformer (BPE Tok learned jointly + Concat
+Embed)).
+Balanced Dataset
+We conclude that neither
+word nor character-tokenized models trained on
+the balanced dataset outperformed models trained
+using the full training dataset. Thus, it is advanta-
+geous to use as much data as possible for model
+training, even if the two languages have an uneven
+amount of sentences.
+6.2.2
+Human Evaluation
+We conduct human evaluation on the Transformer
+(Char Tok + Concat Embed) model output in order
+to assess the extent to which our translation system
+would be useful to Cantonese and Mandarin speak-
+ers respectively. Considering that Cantonese speak-
+ers can understand Standard Chinese, a translation
+system from Mandarin to Cantonese should aim
+for localisation and fluency in Cantonese, while not
+losing the original meaning of the sentence. On the
+other hand, the primary purpose of a Cantonese-
+to-Mandarin translation system is to facilitate Can-
+tonese comprehension for Mandarin speakers. For
+these diverging purposes in our translation direc-
+tions, we manually evaluate each translation direc-
+tion with separate criteria, which is explained in
+the following sections.
+Procedure
+100 lines from the test set are selected
+for evaluation, identical for both translation di-
+rections. One native speaker of each target lan-
+guage evaluates for that direction only (i.e. Can-
+tonese speaker evaluates Mandarin to Cantonese
+sentences, and visa-versa). During evaluation, the
+evaluator has access to the original input and the
+target output. The evaluation decision is binary for
+both criteria, the evaluator can only choose either
+YES or NO. In the example sentences below, Man-
+darin features are highlighted in orange, Cantonese
+features are highlighted in teal and ungrammatical
+features are highlighted in red.
+Cantonese to Mandarin
+System outputs are
+evaluated against the criteria concerning whether
+the output helps Mandarin speakers understand
+Cantonese text. 34% were found helpful for un-
+derstanding Cantonese text, 61% were found not
+helpful, 5% sentences are discarded because the
+original text in Cantonese is already perfectly com-
+prehensible for Mandarin speaker.
+Mandarin to Cantonese
+System outputs are
+evaluated against the criteria “Does the system out-
+put contribute to Cantonese fluency / localisation?".
+It is found to be the case for 47% of the sentences,
+false for 52% of the sentences with 1%sentences
+discarded since the input and target were identical.
+(1)-(4) are examples of the system output for the
+Mandarin to Cantonese direction. In (1), the out-
+put is evaluated as helpful even though it has not
+completely transformed all Mandarin features into
+Cantonese ones, however, the components with the
+highest semantic value (拍拖dating and 散break
+up) are in Cantonese where it was originally in
+Mandarin. Compared to (3), where the output still
+retains mostly Mandarin and has no Cantonese fea-
+tures. Comparing (2) and (4), they both have some
+grammatical errors (in red), but the impact of such
+error in (2) is less significant to the overall meaning
+of the sentence, while in (4) the overall sentence is
+incomprehensible.
+Examples of output that is helpful:
+(1)
+Mandarin reference (source):
+身邊有兩位好朋友，交往了三年，
+就那樣分手了。
+Cantonese reference (target):
+身邊有兩位好友，拍三年拖，就噉散
+。
+System output:
+身邊有兩位好友，拍了三年拖，就這
+樣散了。
+Sentence meaning: I have two friends
+who had been dating for three years,
+and they broke up just like that.
+(2)
+Mandarin reference (source):
+別這麼犟，快點向媽認錯。
+Cantonese reference (target):
+咪咁硬頸，快同亞媽認錯。
+System output:
+否“硬頸，快些和亞媽認錯。
+Sentence meaning: Don’t be so stubborn,
+apologize to your mother at once.
+Examples of output that is not helpful:
+(3)
+Mandarin reference (source):
+別小看他，他已經有了三項發明。
+Cantonese reference (target):
+10
+
+咪睇小佢，佢已經有三項發明。
+System output:
+否看小她，她已經有了三項發明。
+Sentence meaning: Don’t underestimate
+him, he already has three inventions.
+(4)
+Mandarin reference (source):
+���海關沒收了那些東西。
+Cantonese reference (target):
+畀海關執。
+System output:
+給海關執了那麼。
+Sentence meaning: The things that were
+confiscated by customs.
+7
+Discussion
+Our Mandarin-Cantonese machine translation
+project displays the differences between two to-
+kenization methods (character-level and byte pair
+encoding), with an outcome different than expected
+regarding byte pair encoding. A possible reason
+for this may be that such a big vocabulary size can
+lead to worse embeddings, taking into account the
+size of our corpus.
+One of our approaches was down-sampling the
+full dataset into a balanced one, from which we
+expected a higher BLEU score compared to when
+using the full dataset. However, this had the op-
+posite effect on the BLEU score and it ended up
+being lower than in the previous occasions. This
+is perhaps due to the fact that 1 million sentences
+is just simply not enough data for a machine to
+become ’fluent’ in a language.
+As further work, we propose that this project
+can be extended by combining out best architec-
+ture, best tokenization and best embedding training
+method (transformer + character + mapping), by de-
+veloping a cross-lingual mapping for embeddings
+that is compatible with a transformer network in
+order to confirm whether it does lead to higher
+results.
+In addition, other options worth exploring would
+be the grammatical similarity between Cantonese
+and Mandarin and developing an statistical ma-
+chine translation model.
+8
+Summary and conclusion
+The aim of implementing a Cantonese-Mandarin
+MT-model was accomplished by:
+• Creating a large-scale corpus out of several
+online sources such as Wikipedia, scraped
+Instagram comments, YouTube subtitles and
+restaurant reviews.
+• Implementing and training several Cantonese-
+Mandarin translation models while studying
+the effects of different tokenization strategies,
+such as character-level and byte-pair encod-
+ing. While BPE was expected to outperform
+character-level tokenization, this was not the
+case in our experiments.
+The outcomes of this project showed that overall,
+in 61% of the cases, the outcome translation was
+not useful to help Mandarin speakers understand
+Cantonese text. As far as what fluency concerns,
+in 52 out of 100 cases, the system’s output did not
+show any contribution.
+Further work and research is essential in order to
+reach good percentages of performance and fluency
+in such a machine translation model. This project
+has contributed a large Cantonese dataset that was
+not available before as it is now.
+We hope that with this project we moved one
+step forward into a direction that has been studied
+for some years now, contributing to further devel-
+opments and advancement.
+References
+Mikel Artetxe, Gorka Labaka, and Eneko Agirre. 2016.
+Learning principled bilingual mappings of word em-
+beddings while preserving monolingual invariance.
+In Proceedings of the 2016 conference on empiri-
+cal methods in natural language processing, pages
+2289–2294.
+Mikel Artetxe, Gorka Labaka, and Eneko Agirre.
+2017a. Learning bilingual word embeddings with
+(almost) no bilingual data.
+In Proceedings of the
+55th Annual Meeting of the Association for Compu-
+tational Linguistics (Volume 1: Long Papers), pages
+451–462.
+Mikel Artetxe, Gorka Labaka, and Eneko Agirre.
+2018a. Generalizing and improving bilingual word
+embedding mappings with a multi-step framework
+of linear transformations.
+In Proceedings of the
+AAAI Conference on Artificial Intelligence, vol-
+ume 32.
+Mikel Artetxe, Gorka Labaka, and Eneko Agirre.
+2018b. A robust self-learning method for fully un-
+supervised cross-lingual mappings of word embed-
+dings. arXiv preprint arXiv:1805.06297.
+11
+
+Mikel Artetxe, Gorka Labaka, Eneko Agirre, and
+Kyunghyun Cho. 2017b. Unsupervised neural ma-
+chine translation. arXiv preprint arXiv:1710.11041.
+Giusepppe Attardi. 2015. Wikiextractor. https://
+github.com/attardi/wikiextractor.
+Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Ben-
+gio. 2014.
+Neural machine translation by jointly
+learning to align and translate.
+arXiv preprint
+arXiv:1409.0473.
+Robert S Bauer. 2018. Cantonese as written language
+in hong kong. Global Chinese, 4(1):103–142.
+Kingsley Bolton. 2011. Language policy and planning
+in hong kong: Colonial and post-colonial perspec-
+tives. Applied linguistics review, 2(1):51–71.
+Kyunghyun Cho, Bart Van Merriënboer, Dzmitry Bah-
+danau, and Yoshua Bengio. 2014. On the properties
+of neural machine translation: Encoder-decoder ap-
+proaches. arXiv preprint arXiv:1409.1259.
+Alexis Conneau, Guillaume Lample, Marc’Aurelio
+Ranzato, Ludovic Denoyer, and Hervé Jégou. 2017.
+Word translation without parallel data.
+arXiv
+preprint arXiv:1710.04087.
+Wael Farhan, Bashar Talafha, Analle Abuammar,
+Ruba Jaikat, Mahmoud Al-Ayyoub, Ahmad Bisher
+Tarakji, and Anas Toma. 2020. Unsupervised dialec-
+tal neural machine translation. Information Process-
+ing & Management, 57(3):102181.
+Guillaume Lample and Alexis Conneau. 2019. Cross-
+lingual language model pretraining. arXiv preprint
+arXiv:1901.07291.
+Guillaume Lample, Alexis Conneau, Ludovic Denoyer,
+and Marc’Aurelio Ranzato. 2017. Unsupervised ma-
+chine translation using monolingual corpora only.
+arXiv preprint arXiv:1711.00043.
+Guillaume Lample, Myle Ott, Alexis Conneau, Lu-
+dovic Denoyer, and Marc’Aurelio Ranzato. 2018.
+Phrase-based & neural unsupervised machine trans-
+lation. arXiv preprint arXiv:1804.07755.
+David CS Li. 2000. Cantonese-english code-switching
+research in hong kong: A y2k review. World En-
+glishes, 19(3):305–322.
+Kang Kwong Luke and May LY Wong. 2015.
+The
+hong kong cantonese corpus: design and uses. Jour-
+nal of Chinese Linguistics Monograph Series, pages
+312–333.
+Minh-Thang Luong, Hieu Pham, and Christopher D
+Manning. 2015. Effective approaches to attention-
+based neural machine translation.
+arXiv preprint
+arXiv:1508.04025.
+Stephen Matthews and Virginia Yip. 2013. Cantonese:
+A comprehensive grammar. Routledge.
+Tomas Mikolov, Quoc V Le, and Ilya Sutskever. 2013.
+Exploiting similarities among languages for ma-
+chine translation. arXiv preprint arXiv:1309.4168.
+Rico Sennrich, Barry Haddow, and Alexandra Birch.
+2015.
+Improving neural machine translation
+models with monolingual data.
+arXiv preprint
+arXiv:1511.06709.
+Salam Michael Singh and Thoudam Doren Singh. 2020.
+Unsupervised neural machine translation for english
+and manipuri. In Proceedings of the 3rd Workshop
+on Technologies for MT of Low Resource Languages,
+pages 69–78.
+Don Snow. 2004. Cantonese as written language: The
+growth of a written Chinese vernacular, volume 1.
+Hong Kong University Press.
+Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob
+Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz
+Kaiser, and Illia Polosukhin. 2017. Attention is all
+you need. Advances in neural information process-
+ing systems, 30.
+Pascal Vincent, Hugo Larochelle, Yoshua Bengio, and
+Pierre-Antoine Manzagol. 2008.
+Extracting and
+composing robust features with denoising autoen-
+coders. In Proceedings of the 25th international con-
+ference on Machine learning, pages 1096–1103.
+Yu Wan, Baosong Yang, Derek F Wong, Lidia S Chao,
+Haihua Du, and Ben CH Ao. 2020. Unsupervised
+neural dialect translation with commonality and di-
+versity modeling. In Proceedings of the AAAI Con-
+ference on Artificial Intelligence, volume 34, pages
+9130–9137.
+Ka
+Ming
+Wong
+and
+Richard
+Tzong-Han
+Tsai.
+2022. Mixed embedding of xlm for unsupervised
+cantonese-chinese neural machine translation (stu-
+dent abstract).
+Tak-sum Wong, Kim Gerdes, Herman Leung, and
+John SY Lee. 2017.
+Quantitative comparative
+syntax on the cantonese-mandarin parallel depen-
+dency treebank.
+In Proceedings of the fourth in-
+ternational conference on Dependency Linguistics
+(Depling 2017), pages 266–275.
+Yonghui Wu, Mike Schuster, Zhifeng Chen, Quoc V
+Le,
+Mohammad Norouzi,
+Wolfgang Macherey,
+Maxim Krikun,
+Yuan Cao,
+Qin Gao,
+Klaus
+Macherey, et al. 2016.
+Google’s neural machine
+translation system: Bridging the gap between hu-
+man and machine translation.
+arXiv preprint
+arXiv:1609.08144.
+Hei Yi Mak and Tan Lee. 2021.
+Low-resource nmt:
+A case study on the written and spoken languages in
+hong kong. In 2021 5th International Conference on
+Natural Language Processing and Information Re-
+trieval (NLPIR), pages 81–87.
+12
+

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+Time-domain observation of ballistic orbital-angular-momentum currents with giant relaxation 
+length in tungsten 
+Tom S. Seifert1,2, Dongwook Go3, Hiroki Hayashi4, Reza Rouzegar1,2,  
+Frank Freimuth3,5, Kazuya Ando4,6-7, Yuriy Mokrousov3,5, Tobias Kampfrath1,2 
+1Freie Universität Berlin, 14195 Berlin, Germany 
+2Fritz Haber Institut der Max-Planck-Gesellschaft, 14195 Berlin, Germany 
+3Forschungszentrum Jülich, 52425 Jülich, Germany 
+4Department of Applied Physics and Physico-Informatics, Keio University, Yokohama 223-8522, Japan 
+5Johannes Gutenberg-Universität Mainz, 55099 Mainz, Germany 
+6Keio Institute of Pure and Applied Sciences, Keio University, Yokohama 223-8522, Japan 
+7Center for Spintronics Research Network, Keio University, Yokohama 223-8522, Japan 
+ 
+Abstract 
+The emerging field of orbitronics exploits the electron orbital momentum 𝐿, which may allow magnetic-
+information transfer with significantly higher density over longer distances in more materials than possible 
+with spin-polarized electrons. However, direct experimental observation of 𝐿  currents, their extended 
+propagation lengths and their conversion into charge currents has remained challenging. Here, we optically 
+trigger ultrafast angular-momentum transport in Ni|W|SiO2 thin-film stacks. The resulting terahertz charge-
+current bursts exhibit a marked delay and width that grow linearly with W thickness. We consistently ascribe 
+these observations to a ballistic 𝐿 current from Ni through W with giant decay length (∼ 80 nm) and slow 
+velocity (∼ 0.1 nm/fs). At the W/SiO2 interface, the 𝐿 flow is converted into a charge current by the inverse 
+orbital Rashba-Edelstein effect. Our findings establish orbitronic materials with long-distance ballistic 𝐿 
+transport as possible candidates for future ultrafast devices and an approach to discriminate Hall- and 
+Rashba-Edelstein-like conversion processes. 
+ 
+Introduction 
+Spintronics research aims at utilizing the flow of spin angular momentum carried by electrons to transport 
+information and eventually manipulate magnetic order [1]. Actually, electrons have two distinct channels of 
+angular momentum: the electron spin 𝑆 and orbital angular momentum 𝐿. While 𝑆 is successfully exploited 
+in the field of spintronics to transport information by spin currents and to convert the latter into detectable 
+charge currents by spin-to-charge conversion (S2C) [2], 𝐿 has only recently gained attention in the field of 
+orbitronics. To make this fascinating concept compatible and competitive with conventional electronics [3, 
+4], the speed of spin-orbitronic functionalities needs to be scalable to terahertz (THz) rates [5].  
+A first key advantage of 𝐿 over 𝑆 is that it can assume arbitrarily high values for one electron, which is 
+interesting for efficient manipulation of future orbitronic devices [1, 6, 7]. Second, 𝐿-to-charge conversion 
+(L2C) does not rely on spin-orbit interaction (SOI), which opens the orbitronic workbench to abundant light 
+metals [8]. Third, 𝐿-currents are predicted to propagate over increased lengths reaching almost 100 nm [9]. 
+Finally, 𝐿-induced torques should have a starkly different behavior compared to 𝑆-induced torques [10-14].  
+Recent studies provided strong indications of 𝐿 transport and charge-to-𝐿-current conversion by the orbital 
+Hall effect (OHE) in a thin layer of a paramagnetic material (PM). The 𝑆 or 𝐿 accumulation resulting from an 
+in-plane charge current was interrogated by magnetooptic imaging [8] or by measuring the torque it exerted 
+on the magnetization of an adjacent thin-film ferromagnetic material (FM) [1, 9-24]. The FM was chosen to 
+be either susceptible to 𝑆 (e.g., Ni81Fe19, CoFeB) or 𝐿 accumulation (e.g., Ni).  
+
+Unfortunately, it remains experimentally challenging to measure 𝐿 curents by L2C. First, it is difficult to 
+distinguish L2C by the OHE from L2C by an orbital Rashba-Edelstein effect (OREE) because both phenomena 
+obey identical macroscopic symmetries. Second and for the same reason, OHE and OREE are difficult to 
+separate from their S2C counterparts, i.e., from the spin Hall effect (SHE) and the spin-based Rashba-
+Edelstein effect (SREE) [25]. Previous work, however, indicates different spatial propagation and relaxation 
+dynamics of 𝑆  and 𝐿  currents [9-11]. Therefore, an experimental approach such as THz emission 
+spectroscopy [26, 27], which monitors currents with femtosecond resolution, is perfectly suited to access the 
+possibly different ultrafast 𝐿/𝑆 propagation and conversion dynamics.  
+Here, we study ultrafast signatures of 𝑆 and 𝐿 transport from a FM into a PM that is launched by exciting 
+FM|PM stacks with a femtosecond laser pulse. L2C and S2C in the PM is measured by monitoring the emitted 
+THz pulse. Upon changing the FM from Ni to Ni81Fe19 (Py) and interfacing them with the PMs Pt, Ti and W, 
+we find the same characteristic sign changes in the emitted THz pulse as in previous magnetotransport 
+studies [11]. Consequently, we interpret our observations as signatures of ultrafast L2C and S2C. Remarkably, 
+the emitted THz field from Ni|W is strongly time-delayed and low-pass-filtered compared to that from Ni|Pt. 
+The bandwidth and amplitude of the underlying charge-current burst decreases with W thickness, whereas 
+its delay increases linearly. We assign this observation to long-distance ballistic 𝐿 transport in W, which has 
+a more than 10 times larger relaxation length than 𝑆 transport. Specifically, our data suggest a dominant 
+contribution to L2C through the inverse OREE (IOREE) at the W/SiO2. Interestingly, this effect is absent in 
+Ni|Ti and attributed to a dominant bulk L2C by the inverse OHE (IOHE). Our results may help establish an 
+ultrafast experimental and theoretical methodology to extract the propagation dynamics of 𝐿 currents. 
+ 
+ 
+FIGURE 1: Launching and detecting terahertz 𝑺 and 𝑳 currents. Upon ultrafast laser excitation of the FM, 
+the FM magnetization 𝐌 is quenched, leading to 𝑆 accumulation 𝜇�, 𝐿 accumulation 𝜇� and the injection of 
+spin and orbital currents 𝑗� and 𝑗�, respectively, into the PM. Various bulk and interfacial L2C and S2C 
+processes generate an ultrafast in-plane charge current 𝑗� that radiates a THz pulse with electric field 𝐸 vs 
+time 𝑡 directly behind the sample. 
+Conceptual background. Our approach is guided by the idea that 𝐿 currents obey the same phenomenology 
+as 𝑆 currents, whereas 𝐿 transport is expected to have comparatively different spatiotemporal dynamics on 
+ultrashort time and length scales [1, 9-11]. As schematically depicted in Fig. 1, a femtosecond optical pump 
+pulse excites a FM|PM stack and triggers ultrafast 𝑆 and 𝐿 currents with density 𝑗� and 𝑗�, respectively, from 
+FM to PM. S2C and L2C result in ultrafast in-plane charge currents acting as a sources of a THz 
+electromagnetic pulse [28]. The resulting THz electric-field amplitude 𝐸(𝑡) directly behind the sample is 
+proportional to the sheet charge current 𝐼�(𝑡), which reads 
+
+Femtosecond
+THz pulse
+heating pulse
+js
+E
+S2C
+us
+ee
+UL
+L2C
+iL
+M
+jc
+FM
+PM
+Z𝐸(𝑡) ∝ 𝐼�(𝑡) =
+�
+d𝑧 [θ���(𝑧)𝑗�(𝑧, 𝑡) + θ���(𝑧)𝑗�(𝑧, 𝑡)]
+�������
+�
+. 
+(1) 
+Here, θ���(𝑧) and θ���(𝑧) describe the local efficiency of instantaneous L2C and S2C, respectively. They 
+include microscopic mechanisms like the inverse SHE (ISHE) or IOHE [27, 29], which occur in the bulk, or the 
+inverse SREE and IOREE, which require regions of broken inversion symmetry such as interfaces [30, 31]. 
+To understand the emergence of 𝑗� and 𝑗�, we note that sudden laser heating of the FM induces 𝑆 and 𝐿 
+accumulations, 𝜇�  and 𝜇� , respectively. The spin accumulation 𝜇�  is proportional to the excess 
+magnetization, i.e., the difference between the instantaneous magnetization and the equilibrium 
+magnetization that would be attained at the instantaneous electron temperature [32-35]. Consequently, the 
+FM releases 𝑆 at a rate proportional to 𝜇�, by transferring 𝑆 to both the crystal lattice and the PM. 
+Recent studies on single-element FMs showed that the 𝑆- and 𝐿-type magnetizations exhibit very similar 
+ultrafast time evolution following laser excitation [36-38]. Therefore, we expect a very similar time evolution 
+of 𝜇� and 𝜇�, i.e., 𝜇�(𝑡) ∝ 𝜇�(𝑡), where their amplitudes depend on details of the electronic structure [14]. 
+Despite this common origin of 𝑆 and 𝐿 currents, the relation between 𝑗�(𝑧, 𝑡) and 𝑗�(𝑧, 𝑡) (Fig. 1) can be 
+highly nontrivial as 𝑆 and 𝐿 may propagate differently through the FM/NM interface and the NM bulk. 
+Eq. ( 1 ) does not account for contributions due to magnetic dipole radiation of the time-dependent 
+magnetization and of photocurrents even in magnetic order, because both components can be discriminated 
+experimentally [32, 39]. 
+Experiment details. We study thin film FM|PM samples, where the two FMs Py and Ni are chosen for their 
+high efficiency in generating 𝑆 and 𝐿 currents, respectively [11]. The PMs are chosen to have a strong ISHE 
+(Pt, W) and IOHE (W, Ti) response. The reported signs for the ISHE are opposite for Pt vs W with a vanishing 
+ISHE in Ti, but the expected IOHE signs are the same for all three PMs [40]. The studied FM|PM stacks have 
+thicknesses of a few nanometers deposited onto 500 μm thick glass substrates or 625 μm thick thermally 
+oxidized Si substrates (see Fig. S1 and Methods). The samples are characterized by optical and THz 
+transmission spectroscopy [41], yielding the pump absorptance, DC conductivity and Drude relaxation rate 
+(Fig. S2). 
+In our experiment (Fig. 1), ultrashort laser pulses (15 fs duration, 800 nm center wavelength, 80 MHz 
+repetition rate, 1.9 nJ pulse energy, 0.2 mJ/cm2 incident fluence) derived from a Ti:sapphire oscillator excite 
+the FM|PM samples. We record the emitted THz radiation by electrooptic sampling in a 1 mm or 10 µm thick 
+ZnTe(110) or a 250 μm thick GaP(110) electro-optic crystal [42]. The resulting THz emission signal 𝑆(𝐌, 𝑡) vs 
+time 𝑡 is proportional to the THz electric-field waveform 𝐸 (Fig. 1) convoluted with a setup-response function 
+[43]. The presented data is low-pass filtered by convolution with a Gaussian function with a full width at half 
+maximum of about 80 fs for better visibility unless noted otherwise. 
+All experiments are performed under ambient conditions unless stated otherwise. We apply an in-plane 
+magnetic field of about 10 mT to the sample and monitor the THz field component perpendicular to the 
+sample magnetization 𝐌. The component parallel to 𝐌 is found to be minor (Fig. S3). Measurements with 
+linearly and circularly polarized pump pulses reveal a negligible impact of the pump polarization on the THz 
+emission (Fig. S4).  
+To isolate magnetic signals, we reverse 𝐌  and focus on the odd-in- 𝐌  THz signal 𝑆(𝑡) =
+[𝑆(+𝐌, t) − 𝑆(−𝐌, t)] 2
+⁄ . Even-in-𝐌 signal components are minor. As expected from a transport scenario, 
+further experiments, in which the samples are reversed, reveal a dominant structural-inversion-asymmetry 
+(SIA) character of the emitted THz signals compared to minor contributions unrelated to SIA, which most 
+likely arise from magnetic-dipole radiation due to ultrafast demagnetization (Fig. S5) [32].  
+
+ 
+ 
+FIGURE 2: Terahertz raw data. THz 
+emission signals 𝑆(𝑡) from FM|PM stacks 
+with a FM=Py and b FM=Ni capped with 
+PM=Pt, W or Ti. Note the rescaling of the 
+Pt-based sample signals. Film thicknesses 
+in nanometers are given as numerals in 
+parenthesis. As THz detector, a 1 mm 
+ZnTe(110) crystal was used. 
+Results 
+FM=Py. Figure 2a shows THz emission signals 𝑆 from Py|PM samples with PM=Pt, W, Ti, where the time-axis 
+origin is the same for all signals. All three waveforms have identical shapes. Minor differences in the shape 
+of 𝑆��|�� vs 𝑆��|�� are attributed to contributions unrelated to SIA (see above and Fig. S6).  
+The relative signal magnitudes as well as the opposite polarities for PM=Pt and W are consistent with 
+previous reports of ISHE-dominated THz emitters [28]. The polarity of the signal from Py|Ti is the same as 
+from Py|Pt and consistent with the calculations and measurements that found the same sign of the ISHE in 
+Pt and the IOHE in Ti [8, 27, 40].  However, the Py|Ti signal has a significantly smaller amplitude than the 
+Py|Pt signal even though Ti has a sizeable L2C efficiency. We ascribe this observation to a small amplitude of 
+the 𝐿 current injected into Ti, consistent with the small 𝐿 component of the Py magnetization [11].  
+To summarize, for Py|PM, our THz signals are consistent with the notion that we predominantly observe 
+transport of 𝑆 and 𝐿 into the PM bulk and its conversion into a charge current through the ISHE and the IOHE. 
+A possible Rashba-type L2C or S2C, or skew-scattering at the FM/PM interface [44] may make an additional 
+yet relatively small contribution.  
+FM=Ni. When the FM=Py is replaced by Ni, the signal polarity remains the same for Pt and Ti, and the two 
+waveforms exhibit identical dynamics (Fig. 2b and Fig. S7). In stark contrast, however, the signal polarity for 
+Ni|W reverses, the waveform is less symmetric, and its maximum is time-shifted relative to Py|W. This 
+striking observation indicates that Py|W and Ni|W show competing THz-generation mechanisms, the 
+a 
+b 
+
+X10-6
+3
+Ni(5)IPt(3) /3
+Ni(5)ITi(3)
+2
+Ni(5)/W(3)
+Terahertz signal
+-2
+-3
+0
+1
+2
+Time (ps)3
+Py(5)/Pt(3) /3
+Py(5)/Ti(3)
+2
+Py(5)/W(3)
+Terahertz signal
+-2
+.3
+0
+2
+1
+Time (ps)dominance of which depends sensitively on the FM material. To gain more insight into the different dynamics 
+in Ni|W, we next vary the W thickness.        
+ 
+ 
+FIGURE 3: Impact of W thickness in 
+Ni|W. THz emission signals for Ni|W 
+samples with varying W thickness 
+normalized to the absorbed pump-
+pulse fraction in the Ni layer and to the 
+sample impedance (see Methods and 
+Table S1). Note the rescaling of the 
+reference signal from Ni|Pt. Film 
+thicknesses in nanometers are given as 
+numerals in parenthesis. A 250 µm 
+GaP(110) crystal was used as THz 
+detector. 
+Impact of W thickness. Figure 3 shows THz emission signals from Ni|W(𝑑�) for various 𝑑� and from a Ni|Pt 
+reference sample. Consistent with Fig. 2b, we see a clear trend with increasing W thickness relative to Ni|Pt: 
+The THz signal amplitude has a reversed sign, reduces with increasing 𝑑� and undergoes a significant 
+reshaping from asymmetric (Ni|Pt) to more symmetric (Ni|W) around the signal maximum. Interestingly,  
+𝑑� = 2 nm is already sufficient to induce a shift of the maximum of the THz signal by about 100 fs.  
+We emphasize that the changes in THz-signal dynamics solely originate from changing the PM thickness. 
+Therefore, the FM is not primarily responsible for the signal-dynamics changes and, thus, considered as an 
+PM-independent 𝑆 and 𝐿 injector in the following.  
+ 
+FIGURE 4: Ultrafast charge currents in Ni|W. a Charge sheet currents in Ni|W for various W thicknesses 𝑑� 
+as extracted from the data of Fig. 3. The feature at 0.9 ps is a remainder of a THz-field reflection echo in the 
+10 µm ZnTe electro-optic detection crystal (see Methods). Film thicknesses in nanometers are given as 
+numerals in parenthesis. Note the rescaling of the Pt-based sample signal. The apparent signal delays and 
+amplitudes are highlighted by a circular marker. b Extracted time delay with a straight line  as a guide to the 
+eye, c relative amplitude at the delay marked in panel a, d temporal width at half maximum, and e integrated 
+charge current between 0.2 to 0.9 ps vs 𝑑� from the data in panel a. Error bars are estimated for panels c 
+a 
+b 
+c 
+d 
+e 
+
+X10-9
+5
+Ni(5)/Pt(3) /6
+Ni(5)/W(2)
+4
+Norm. terahertz signal
+Ni(5)/W(3)
+Ni(5)/W(5)
+3
+Ni(5)/W(10)
+Ni(5)/W(15)
+2
+Ni(5)/W(20)
+0
+7
+-2
+-3
+-0.5
+0
+0.5
+Time (ps)Ni(5)/Pt(3) /6
+14
+Ni(5)/W(2)
+12
+Ni(5)/W(3)
+Ni(5)/W(5)
+2
+per abs. fluence in Ni (J/m
+10
+Ni(5)/W(10)
+Ni(5)/W(15)
+8
+Ni(5)IW(20)
+delay
+6
+4
+2
+0
+-2
+-4
+-6
+0
+0.5
+Time (ps)80
+Ampl. (norm.
+Delay (fs)
+60
+40
+0.5
+20
+0
+0
+0
+10
+20
+10
+20
+0
+340
+320
+Area (norm.)
+(fs)
+300
+280
+Width
+260
+0.5
+240
+220
+0
+0
+10
+20
+10
+20
+0
+d... (nm)
+d... (nm)and e from the signal-to-noise ratio in panel a, for panels b and d as 10% of the delay and width, respectively, 
+and in all panels b-e as ± 1 nm for 𝑑�. 
+Current dynamics in Ni|W. To obtain a sample-intrinsic measurement of the L2C dynamics, we extract the 
+sheet charge current 𝐼�(𝑡) flowing in Ni|W (Eq. (1)) normalized to the absorbed laser fluence in the Ni layer 
+(see Methods). This procedure eliminates any impact of sample exchange on pump-pulse absorption 
+efficiency, sample impedance or setup response function (see Methods).  
+Figure 4a presents 𝐼�(𝑡) in Ni|W with a resolution of 50 fs for various W thicknesses 𝑑�. The 𝐼�(𝑡) traces 
+have striking features. (i) They have opposite polarity relative to Py|W. (ii) Their maximum shifts by delays 
+Δ𝑡��� ∝ 𝑑� at a rate Δ𝑡��� 𝑑�
+⁄
+≈ 4 fs/nm (Fig. 4b), implying a velocity of 0.25 nm/fs. (iii) The 𝐼� peak 
+value decreases approximately linearly with 𝑑� to about 50% after 20 nm (Fig. 4c), indicating attenuation 
+and dispersion upon propagation. (iv) The 𝐼� width increases linearly at a rate of ≈ 8 fs/nm (Fig. 4d). (v) The 
+time-integrated current ∫ d𝑡 𝐼�(𝑡)  is only weakly dependent on 𝑑�  with a decreasing trend, thereby 
+indicating an extremely large relaxation length >20 nm (Fig. 4e). 
+Features (i) and (iii) imply that 𝐼�(𝑡) cannot arise from 𝑆 transport. Otherwise, an opposite signal polarity 
+would result because S2C in W is dominated by the ISHE [28]. In addition, 𝑆 currents in W relax over distances 
+much smaller than 20 nm [41]. Our data, therefore, strongly indicate that 𝐿 transport plus L2C is the 
+dominant origin of the THz charge current in Ni|W. Second, features (ii) and (iv) are a hallmark of a signal 
+arising from ballistic transport of a pulse that is detected in an arrival layer. In this picture, the increase of 
+the 𝐼�(𝑡) width with 𝑑� arises from velocity dispersion along the 𝑧-direction of the particles that make up 
+the pulse (Fig. 5a). Feature (v) implies a minor L2C in the W bulk because it would otherwise result in an 
+integrated charge current ∫ d𝑡 𝐼�(𝑡) that increases monotonically with 𝑑�.  
+Model: 𝑳 current and IOREE in Ni|W. The preceding discussion suggests the following transport scenario in 
+Ni|W. Upon excitation of the Ni layer, a transient 𝑆 and 𝐿 accumulation is induced (Fig. 1). Their dynamics 
+are expected to be very similar (see above) [36-38] and monitored well by the ISHE charge current in Ni|Pt 
+(Fig. 4a). Finally, L2C is dominated in regions close to the W/SiO2 interface (Fig. 5a). Such interfacial L2C can 
+be very efficient, as argued in previous works [7, 24, 45-49], which, however, lacked the required 
+femtosecond resolution. 
+This scenario can explain all charge-current features (i)-(v) (Fig. 4) and is consistent the above experimental 
+findings. As the 𝑗� pulse propagates predominantly ballistically, its arrival in the W/SiO2 L2C region is delayed 
+by a time Δ𝑡��� ∝ 𝑑�. The velocity of the 𝑗�-pulse peak (∼ 0.1 nm/fs) is consistent with that of 𝐿-carrying 
+d-band states of W [50]. During propagation through PM=W, the 𝑗� pulse disperses due to different electron 
+velocities along the surface normal (Fig. 5a) and attenuation with a typical relaxation length >20 nm.  
+To quantitatively model the charge-current dynamics in Ni|W (Fig. 3), we assume ballistic 𝐿 transport with a 
+characteristic decay length 𝜆� in W. The ��� arriving at the W/SiO2 interface is obtained by summing over all 
+Fermi-surface states with positive group velocity along the 𝑧-axis (see Fig. 5a and Methods). The resulting 𝑗� 
+at the W/SiO2 interface induced by a fictitious 𝛿(𝑡)-like 𝐿 accumulation in Ni reads 
+𝑟(𝑡) ∝ 𝑑�
+𝑡� Θ(𝑣�𝑡 − 𝑑�)e���� ��
+⁄
+, 
+(2) 
+where Θ is the Heaviside step function, and 𝑣� is the Fermi velocity of the 𝐿-polarized electrons in W. We 
+convolute 𝑟(𝑡) with 𝜇�(𝑡) ∝ 𝜇�(𝑡), which is given by the charge current measured in Ni|Pt (Fig. 4a). Our 
+modeled 𝐼�(𝑡) curves (Fig. 5c) reproduce the measured charge currents in Ni|W (Fig. 4a) semiquantitatively 
+for the choice 𝜆� = 80 nm and 𝑣� = 0.14 nm/fs. These values are in good agreement with the estimates 
+obtained above (Figs. 4b-e). 
+ 
+
+ 
+ 
+FIGURE 5: Simulated ultrafast inverse 
+orbital Rashba-Edelstein effect in W. 
+a Schematic of the suggested scenario for 
+𝐿 transport and L2C by the IOREE in Ni|W 
+showing the different wave vector 
+contributions of the 𝐿 currents inside the 
+W layer driven by the magnetization 
+quenching in the Ni layer. Upon reaching 
+the W back surface, the orbital currents 𝐣� 
+are converted into a transverse charge 
+current 𝐣� by the inverse orbital Rashba-
+Edelstein 
+effect 
+(IOREE). 
+In 
+the 
+experiment, many of the point-like 
+sources 
+of 
+orbital 
+currents 
+are 
+superimposed 
+along 
+the 
+FM/PM 
+interface. 
+b Qualitative 
+response 
+functions 𝑟(𝑡) to a fictitious delta-like 𝑗� 
+pulse injected at the Ni/W interface for 
+different W layer thicknesses 𝑑�, where 
+𝑑�� < 𝑑�� < 𝑑�� . c Simulated IOREE 
+charge 
+currents 𝐼�(𝑡)  obtained 
+by 
+convoluting 𝑟(𝑡) [Eq. (2), panel b] with 
+the 𝐼�(𝑡) of the Ni|Pt reference sample. 
+Inputs for the simulation are a ballistic 𝐿 
+velocity of 0.14 fs/nm, an 𝐿 decay length 
+of 𝜆� = 80 nm and a global scaling factor.   
+To summarize, the THz charge currents in Ni|W (Fig. 4) can be considered as signatures of 𝐿 currents injected 
+into W. The charge-current generation [see Eq. (1)] is dominated by an extremely long-range 𝑗� and L2C at 
+the W/SiO2 interface, i.e., by θ��� at 𝑧 = 𝑑�� + 𝑑��. Such long-range 𝐿 transport is a unique feature of 
+orbitronic materials, and first indications for it were found previously in Ti [10, 11]. Note that within our 
+interpretation, the sign of 𝑆��|� agrees coincidentally with the calculated sign of θ��� for the IOHE in W [40].   
+Discussion 
+Our interpretation neglects other possible contributions to the THz charge current. First, the inverse Faraday 
+effect as a source of 𝑆 and 𝐿 currents can be ruled out by the pump-polarization independence (see Fig. S4). 
+Second, for the 𝑆  channel, a dominant Seebeck-type contribution due to an electronic temperature 
+difference Δ𝑇��/�� across the Ni/PM interface is neglected as found in previous studies [32]. For the 𝐿 
+channel, we estimate Δ𝑇��/�� right after pump pulse absorption (see Methods) and find Δ𝑇��/�� ∼ +400 K 
+and Δ𝑇��/�� ∼ −100 K in Ni|Ti and Ni|W. The observed THz-emission signals, in contrast, show the same 
+sign  from all three samples (Fig. 2b). Therefore, interfacial electronic temperature differences are a minor 
+a 
+b 
+c 
+
+14
+Ni(5)/Pt(3) /6
+12
+Ni(5)/W(2)
+Ni(5)/W(3)
+10
+Ni(5)/W(5)
+Ni(5)/W(10)
+8
+Ni(5)/W(15)
+Ni(5)/W(20)
+6
+4
+2
+per abs.
+0
+-2
+4
+-6
+0
+0.5
+1
+Time (ps)W
+Ni
+eee
+μL
+IOREE
+GG
+jL
+dw
+Z
+r(t)
+r (0)
+0
+t
+t(dw1) t(dw2) t(dw3driving force. Additional pump-propagation simulations show that, even for the thickest samples, pump-
+intensity gradients in the FM and PM bulk are relatively small (Fig. S8).  
+Third, regarding transport in W, we consider dominant angular-momentum transport by magnons unlikely 
+because W is not magnetically ordered. An outstandingly long propagation of 𝑆 transport is ruled out, too, 
+because the Drude scattering times for all studied samples are substantially shorter (<50 fs, Figs. S2) than the 
+peak delays of 𝐼�(𝑡) (Fig. 4a).  
+Fourth, even though our data imply a dominant IOREE contribution to charge-current generation (see above), 
+the positive shoulder-like feature at around 0.1 ps for 𝑑� ≤ 3 nm in Fig. 4a may indicate a small contribution 
+of bulk L2C, i.e., the IOHE. A 𝐿-to-𝑆 conversion plus ISHE in the PM [23] might contribute but is considered 
+negligible here given the good agreement of our experimental data (Fig. 4) and the IOREE scenario (Fig. 5).  
+The dominance of an 𝐿-type angular momentum current in Ni|W highlights the role of Ni as an 𝐿 source and 
+indicates that the Ni/W interface may transmit 𝐿 currents more efficiently than 𝑆 currents. 
+We finally turn to other interesting aspects of our study. A more detailed comparison of Fig. 2a and 2b reveals 
+further changes in amplitude between Ni- and Py-based samples. The pronounced amplitude changes for 
+PM=W or Pt when changing FM=Py to Ni are related to the intricate interplay of all parameters in Eq. (1) in 
+addition to changes in the relative amplitudes of 𝜇� and 𝜇� and interface transmission coefficients for 𝑗� and 
+𝑗�. Therefore, further experiments for a robust separation of 𝑆 and 𝐿 transport are needed.  
+We further find that the THz signals from the Ni-based samples increase linearly with pump fluence. Slight 
+sublinearities at the highest fluences do not alter the THz emission dynamics (Fig. S9). We emphasize that 
+samples deposited on Si rather than glass substrates show very similar THz emission characteristics (Fig. S1). 
+These observations demonstrate the robustness of the observed effects. 
+When adding a Cu layer on top of the Ni|W sample, we find almost no change in the THz emission signal 
+(Fig. S10). Future studies are needed to elaborate the exact character of the IOREE for different interfaces. 
+Interestingly, a Cu intermediate layer in Ni|Cu|W slightly modifies the amplitude and dynamics of the THz 
+signal, suggesting that Cu does not block 𝐿 transport strongly. 
+Regarding earlier reports of different THz emission dynamics in Fe|Au and Fe|Ru samples [29], we note that 
+a possible IOREE in Ru can, in hindsight, not be excluded. A dominant IOREE might also explain the seemingly 
+strong dependence of the Fe|Ru THz emission dynamics on the exact growth details [29, 51, 52].  
+In conclusion, we observe THz-emission signals from optically excited Ni|W stacks that are consistent with 
+an ultrafast injection of 𝐿 currents into W and long-distance ballistic transport through W. Remarkably, we 
+find strong indications for the occurrence of the IOREE. This result can be considered as time-domain 
+evidence of the long-range nature of orbital currents and IOREE in typical metals such as W.  
+Our study highlights the power of broadband THz emission spectroscopy in disentangling of spin and orbital 
+transport and Hall- and Rashba-Edelstein-like angular-momentum conversion processes through ultrafast 
+time-of-flight experiments. Future studies may aim at exploiting L2C physics in materials with nontrivial 
+topology, in which the OHE is predicted to be drastically enhanced close to Weyl points [53]. Our study opens 
+the field of THz orbitronics whose distinct dynamical features allow for tailoring ultrafast spin and orbital 
+currents on femtosecond time scales by a targeted material, thickness and interface engineering of 
+multilayers.  
+ 
+ 
+
+Methods 
+Current extraction. To extract the in-plane sheet current flowing inside the sample from the measured THz 
+signal 𝑆, we first measure our setup response function ℎ by having a reference electro-optic emitter (50 μm 
+GaP on a 500 μm glass substrate) at the same position as the sample, which yields a reference THz signal 𝑆���. 
+By calculating the emitted THz electric field from that reference emitter 𝐸���, ℎ is determined by solving the 
+convolution 𝑆��� = (ℎ ∗ 𝐸���)(𝑡) for ℎ [43]. Further measured inputs for this calculation are the excitation 
+spot size with a full width at half maximum of 22 μm, the excitation pulse energy of 1.9 nJ and a transform 
+limited pump pulse with a spectrum centered at 800 nm and 110 nm full width at half maximum. We perform 
+the deconvolution directly in the time domain by recasting it as a matrix equation [54].  
+Next, the electric field 𝐸 directly behind the sample is obtained from the recorded THz signal 𝑆 with the help 
+of the derived function ℎ by solving again the similar equation 𝑆 = (ℎ ∗ 𝐸)(𝑡) for 𝐸. Finally, the sheet charge 
+current (see Table S1) as shown in Fig. 3 is derived from a generalized Ohm’s law [28] that reads 𝐸(𝜔) =
+𝑒𝑍(𝜔)𝐼�(𝜔) , where −𝑒  is the electron charge and the sample impedance is given by 𝑍(𝜔) =
+𝑍� [1 + 𝑛��� + 𝑍�𝑑𝜎(𝜔)]
+⁄
+ with the free space impedance 𝑍�, the metal-stack thickness 𝑑 and the measured 
+mean sample conductivity 𝜎 (see Table S1) that we assume to be frequency independent due to the large 
+Drude scattering rate (see Fig. S2). To enable comparison of THz currents from different samples, we 
+normalize 𝐼� by the absorbed fluence in the FM layer. The data shown in Fig. 3 was obtained in a dry-air 
+atmosphere.    
+Sample preparation. The FM|PM samples (FM = Ni and Py, PM = Pt, Ti, Cu and W) were fabricated on glass 
+substrates of 500 μm thickness or thermally oxidized Si substrates of 625 μm thickness by radio frequency 
+(RF) magnetron sputtering under 6N-purity-Ar atmosphere. The sample structure and thickness are described 
+in Table S1. For the sputtering, the base pressure in the chamber was better than 5.0 × 10-7 Pa. To avoid 
+oxidation, 4-nm-thick SiO2 was sputtered on the surface of the films. All sputtering processes were performed 
+at room temperature. 
+Estimate of electronic temperatures. We calculate the electronic temperatures increase upon pump-pulse 
+absorption by 
+Δ𝑇� = �𝑇�
+� + 2𝑓
+𝑑𝛾 − 𝑇�. 
+(3) 
+Here, 𝑇� = 300 K is the ambient temperature, 𝑓  is the absorbed fluence in the respective layer (see 
+Table S1), 𝑑 is the layer thickness and 𝛾 is the specific electronic heat capacity that is 300 J/m3 K2 for W, 320 
+J/m3 K2 for Ni, 330 J/m3 K2 for Ti and 90 J/m3 K2 for Pt [55]. To obtain the absorbed fluences in each layer, we 
+note that the pump electric field is almost constant throughout the sample (see Fig S8). Therefore, local pump 
+absorption scales solely with the imaginary part of the dielectric function at a wavelength of 800 nm, which 
+equals 22.07 for Ni, 9.31 for Pt, 19.41 for Ti and 19.71 for W [56]. Consequently, the absorbed fluence is 
+determined by 
+𝑓��/�� = 𝑓���
+𝑑��/�� Im𝜀��/��
+𝑑�� Im𝜀�� + 𝑑�� Im𝜀��
+ 
+(4) 
+with the total absorbed fluence 𝑓��� that is obtained from the absorbed pump power (see Table S1) and the 
+beam size on the sample (see above). 
+Model of orbital transport. To model the ballistic current in the PM, we assume that a 𝛿(𝑡)-like transient 
+spin accumulation in the FM causes a change Δ𝑛𝐤� in the occupation of each electronic wavepacket in the 
+PM with mean wavevector 𝐤 right behind the FM/PM interface at 𝑧 = 0� (Fig. 5a). Subsequently, this 
+occupation change propagates into the PM bulk according to Δ𝑛𝐤(𝑧, 𝑡) = Δ𝑛𝐤�𝛿(𝑧 − 𝑣𝐤𝑡), where 𝑣𝐤 is the 
+𝑧 component of the group velocity of the electronic wavepacket. Note that we here restrict ourselves to 𝐤 
+
+with nonnegative 𝑣𝐤. The occupation change Δ𝑛𝐤(𝑧, 𝑡) is accompanied by a particle current density Δ𝑗𝐤 =
+𝑣𝐤Δ𝑛𝐤. The total pump-induced current density flowing into the depth of the PM layer is given by the sum 
+Δ𝑗(𝑧, 𝑡) =
+�
+Δ𝑛𝐤�𝑣𝐤𝛿(𝑧 − 𝑣𝐤𝑡)
+𝐤, �𝐤�� 
+. 
+(5) 
+Because Δ𝑛𝐤�  predominantly affects states close to the Fermi energy, the summation of Eq. ( 5 ) is 
+approximately proportional to an integration over the Fermi-surface parts with 𝑣𝐤 ≥ 0. One obtains 
+Δ𝑗(𝑧, 𝑡) = � d𝑣 𝑤(𝑣)𝑣
+�
+�
+𝛿(𝑧 − 𝑣𝑡), 
+(6) 
+where 𝑤(𝑣) is the weight of the group velocity 𝑣 along the 𝑧 axis. We assume that the Fermi surface is a 
+sphere with isotropic occupation change Δ𝑛𝐤� and radius 𝑘�. Consequently, the integrand Δ𝑛𝐤�𝑣𝐤 d�𝐤 
+becomes Δ𝑛𝐤�𝑣� cos 𝜃 𝑘�d𝑘d𝜑d cos𝜃 ∝ 𝑣 d𝑣d𝑘d𝜑, where 𝑣� is the Fermi velocity. In other words, all 
+velocities from 0 to the Fermi velocity 𝑣� have equal weight, and we find 
+Δ𝑗(𝑧, 𝑡) ∝ �
+d𝑣 𝑣
+��
+�
+𝛿(𝑧 − 𝑣𝑡) = 𝑧
+𝑡� Θ(𝑣�𝑡 − 𝑧), 
+(7) 
+for 𝑧 > 0. Finally, we phenomenologically account for relaxation of the ballistic current with time constant 𝜏 
+by multiplying Δ𝑗(𝑧, 𝑡) with e��/�, which directly takes us to Eq. (2).  
+ 
+Acknowledgements 
+We thank G. Sala for fruitful discussions. TSS, RR and TK acknowledge funding by the German Research 
+Foundation (DFG) through the collaborative research center SFB TRR 227 “Ultrafast spin dynamics” (project 
+ID 328545488, projects A05 and B02) and financial support from the Horizon 2020 Framework Programme 
+of the European Commission under FET-Open Grant No. 863155 (s-Nebula). FF  and YM  acknowledge DFG 
+collaborative research center SFB TRR 173/2 “Spin+X”(project ID 268565370, project A11). KA and HH 
+acknowledge funding by JSPS (Grant Number 22H04964 and 20J20663) and Spintronics Research Network of 
+Japan. 
+ 
+
+References 
+[1] Go, D., D. Jo, H.-W. Lee, M. Kläui, and Y. Mokrousov Orbitronics: orbital currents in solids. EPL (Europhysics 
+Letters), 2021. 135: p. 37001. 
+[2] Miron, I.M., K. Garello, G. Gaudin, P.-J. Zermatten, M.V. Costache, S. Auffret, S. Bandiera, B. Rodmacq, A. 
+Schuhl, and P. Gambardella Perpendicular switching of a single ferromagnetic layer induced by in-plane 
+current injection. Nature, 2011. 476: p. 189. 
+[3] Ji, B., Y. Han, S. Liu, F. Tao, G. Zhang, Z. Fu, and C. Li Several Key Technologies for 6G: Challenges and 
+Opportunities. IEEE Communications Standards Magazine, 2021. 5: p. 44. 
+[4] Schwierz, F. and J.J. Liou RF transistors: Recent developments and roadmap toward terahertz applications. 
+Solid-State Electronics, 2007. 51: p. 1079. 
+[5] Vedmedenko, E.Y., R.K. Kawakami, D.D. Sheka, P. Gambardella, A. Kirilyuk, A. Hirohata, C. Binek, O. 
+Chubykalo-Fesenko, S. Sanvito, and B.J. Kirby The 2020 magnetism roadmap. Journal of Physics D: Applied 
+Physics, 2020. 53: p. 453001. 
+[6] Salemi, L., M. Berritta, A.K. Nandy, and P.M. Oppeneer Orbitally dominated Rashba-Edelstein effect in 
+noncentrosymmetric antiferromagnets. Nature communications, 2019. 10: p. 1. 
+[7] Johansson, A., B. Göbel, J. Henk, M. Bibes, and I. Mertig Spin and orbital Edelstein effects in a two-
+dimensional electron gas: Theory and application to SrTiO 3 interfaces. Physical Review Research, 2021. 3: p. 
+013275. 
+[8] Choi, Y.-G., D. Jo, K.-H. Ko, D. Go, K.-H. Kim, H.G. Park, C. Kim, B.-C. Min, G.-M. Choi, and H.-W. Lee 
+Observation of the orbital Hall effect in a light metal Ti. arXiv preprint arXiv:2109.14847, 2021. 
+[9] Go, D., D. Jo, K.-W. Kim, S. Lee, M.-G. Kang, B.-G. Park, S. Blügel, H.-W. Lee, and Y. Mokrousov Long-Range 
+Orbital Transport in Ferromagnets. arXiv preprint arXiv:2106.07928, 2021. 
+[10] Bose, A., F. Kammerbauer, D. Go, Y. Mokrousov, G. Jakob, and M. Klaeui Detection of long-range orbital-
+Hall torques. arXiv preprint arXiv:2210.02283, 2022. 
+[11] Hayashi, H., D. Jo, D. Go, Y. Mokrousov, H.-W. Lee, and K. Ando Observation of long-range orbital 
+transport and giant orbital torque. arXiv preprint arXiv:2202.13896, 2022. 
+[12] Zheng, Z., Q. Guo, D. Jo, D. Go, L. Wang, H. Chen, W. Yin, X. Wang, G. Yu, and W. He Magnetization 
+switching driven by current-induced torque from weakly spin-orbit coupled Zr. Physical Review Research, 
+2020. 2: p. 013127. 
+[13] Kim, J., D. Go, H. Tsai, D. Jo, K. Kondou, H.-W. Lee, and Y. Otani Nontrivial torque generation by orbital 
+angular momentum injection in ferromagnetic-metal/Cu/Al 2 O 3 trilayers. Physical Review B, 2021. 103: p. 
+L020407. 
+[14] Lee, D., D. Go, H.-J. Park, W. Jeong, H.-W. Ko, D. Yun, D. Jo, S. Lee, G. Go, and J.H. Oh Orbital torque in 
+magnetic bilayers. Nature communications, 2021. 12: p. 1. 
+[15] An, H., Y. Kageyama, Y. Kanno, N. Enishi, and K. Ando Spin–torque generator engineered by natural 
+oxidation of Cu. Nature communications, 2016. 7: p. 1. 
+[16] Go, D., D. Jo, C. Kim, and H.-W. Lee Intrinsic spin and orbital Hall effects from orbital texture. Physical 
+Review Letters, 2018. 121: p. 086602. 
+[17] Go, D., F. Freimuth, J.-P. Hanke, F. Xue, O. Gomonay, K.-J. Lee, S. Blügel, P.M. Haney, H.-W. Lee, and Y. 
+Mokrousov Theory of current-induced angular momentum transfer dynamics in spin-orbit coupled systems. 
+Physical review research, 2020. 2: p. 033401. 
+[18] Go, D. and H.-W. Lee Orbital torque: Torque generation by orbital current injection. Physical review 
+research, 2020. 2: p. 013177. 
+[19] Tazaki, Y., Y. Kageyama, H. Hayashi, T. Harumoto, T. Gao, J. Shi, and K. Ando Current-induced torque 
+originating from orbital current. arXiv preprint arXiv:2004.09165, 2020. 
+[20] Lee, S., M.-G. Kang, D. Go, D. Kim, J.-H. Kang, T. Lee, G.-H. Lee, J. Kang, N.J. Lee, and Y. Mokrousov Efficient 
+conversion of orbital Hall current to spin current for spin-orbit torque switching. Communications Physics, 
+2021. 4: p. 1. 
+[21] Liao, L., F. Xue, L. Han, J. Kim, R. Zhang, L. Li, J. Liu, X. Kou, C. Song, and F. Pan Efficient orbital torque in 
+polycrystalline ferromagnetic− metal/Ru/Al 2 O 3 stacks: Theory and experiment. Physical Review B, 2022. 
+105: p. 104434. 
+
+[22] Xiao, Z.-Y., Y.-J. Li, W. Zhang, Y.-J. Han, D. Li, Q. Chen, Z.-M. Zeng, Z.-Y. Quan, and X.-H. Xu Enhancement 
+of torque efficiency and spin Hall angle driven collaboratively by orbital torque and spin–orbit torque. Applied 
+Physics Letters, 2022. 121: p. 072404. 
+[23] Sala, G. and P. Gambardella Giant orbital Hall effect and orbital-to-spin conversion in 3 d, 5 d, and 4 f 
+metallic heterostructures. Physical Review Research, 2022. 4: p. 033037. 
+[24] Ding, S., A. Ross, D. Go, L. Baldrati, Z. Ren, F. Freimuth, S. Becker, F. Kammerbauer, J. Yang, and G. Jakob 
+Harnessing orbital-to-spin conversion of interfacial orbital currents for efficient spin-orbit torques. Physical 
+review letters, 2020. 125: p. 177201. 
+[25] Manchon, A., J. Železný, I.M. Miron, T. Jungwirth, J. Sinova, A. Thiaville, K. Garello, and P. Gambardella 
+Current-induced spin-orbit torques in ferromagnetic and antiferromagnetic systems. Reviews of Modern 
+Physics, 2019. 91: p. 035004. 
+[26] Seifert, T.S., L. Cheng, Z. Wei, T. Kampfrath, and J. Qi, Spintronic sources of ultrashort terahertz 
+electromagnetic pulses. 2022, AIP Publishing LLC. p. 180401. 
+[27] Xu, Y., F. Zhang, Y. Liu, R. Xu, Y. Jiang, H. Cheng, A. Fert, and W. Zhao Inverse Orbital Hall Effect Discovered 
+from Light-Induced Terahertz Emission. arXiv preprint arXiv:2208.01866, 2022. 
+[28] Seifert, T., S. Jaiswal, U. Martens, J. Hannegan, L. Braun, P. Maldonado, F. Freimuth, A. Kronenberg, J. 
+Henrizi, I. Radu, E. Beaurepaire, Y. Mokrousov, P.M. Oppeneer, M. Jourdan, G. Jakob, D. Turchinovich, L.M. 
+Hayden, M. Wolf, M. Munzenberg, M. Klaui, and T. Kampfrath Efficient metallic spintronic emitters of 
+ultrabroadband terahertz radiation. Nature Photonics, 2016. 10: p. 483. 
+[29] Kampfrath, T., M. Battiato, P. Maldonado, G. Eilers, J. Notzold, S. Mahrlein, V. Zbarsky, F. Freimuth, Y. 
+Mokrousov, S. Blugel, M. Wolf, I. Radu, P.M. Oppeneer, and M. Munzenberg Terahertz spin current pulses 
+controlled by magnetic heterostructures. Nat Nanotechnol, 2013. 8: p. 256. 
+[30] Jungfleisch, M.B., Q. Zhang, W. Zhang, J.E. Pearson, R.D. Schaller, H. Wen, and A. Hoffmann Control of 
+Terahertz Emission by Ultrafast Spin-Charge Current Conversion at Rashba Interfaces. Phys Rev Lett, 2018. 
+120: p. 207207. 
+[31] Zhou, C., Y.P. Liu, Z. Wang, S.J. Ma, M.W. Jia, R.Q. Wu, L. Zhou, W. Zhang, M.K. Liu, Y.Z. Wu, and J. Qi 
+Broadband Terahertz Generation via the Interface Inverse Rashba-Edelstein Effect. Phys Rev Lett, 2018. 121: 
+p. 086801. 
+[32] Rouzegar, R., L. Brandt, L. Nádvorník, D.A. Reiss, A.L. Chekhov, O. Gueckstock, C. In, M. Wolf, T.S. Seifert, 
+and P.W. Brouwer Laser-induced terahertz spin transport in magnetic nanostructures arises from the same 
+force as ultrafast demagnetization. Physical Review B, 2022. 106: p. 144427. 
+[33] Lichtenberg, T., M. Beens, M.H. Jansen, B. Koopmans, and R.A. Duine Probing optically induced spin 
+currents using terahertz spin waves in noncollinear magnetic bilayers. Physical Review B, 2022. 105: p. 
+144416. 
+[34] Mueller, B.Y. and B. Rethfeld ThermodynamicμTmodel of ultrafast magnetization dynamics. Physical 
+Review B, 2014. 90: p. 144420. 
+[35] Choi, G.M., B.C. Min, K.J. Lee, and D.G. Cahill Spin current generated by thermally driven ultrafast 
+demagnetization. Nat Commun, 2014. 5: p. 4334. 
+[36] Boeglin, C., E. Beaurepaire, V. Halté, V. López-Flores, C. Stamm, N. Pontius, H. Dürr, and J.-Y. Bigot 
+Distinguishing the ultrafast dynamics of spin and orbital moments in solids. Nature, 2010. 465: p. 458. 
+[37] Stamm, C., N. Pontius, T. Kachel, M. Wietstruk, and H. Dürr Femtosecond x-ray absorption spectroscopy 
+of spin and orbital angular momentum in photoexcited Ni films during ultrafast demagnetization. Physical 
+Review B, 2010. 81: p. 104425. 
+[38] Hennecke, M., I. Radu, R. Abrudan, T. Kachel, K. Holldack, R. Mitzner, A. Tsukamoto, and S. Eisebitt 
+Angular Momentum Flow During Ultrafast Demagnetization of a Ferrimagnet. Physical Review Letters, 2019. 
+122: p. 157202. 
+[39] Zhang, W., P. Maldonado, Z. Jin, T.S. Seifert, J. Arabski, G. Schmerber, E. Beaurepaire, M. Bonn, T. 
+Kampfrath, P.M. Oppeneer, and D. Turchinovich Ultrafast terahertz magnetometry. Nat Commun, 2020. 11: 
+p. 4247. 
+[40] Salemi, L. and P.M. Oppeneer First-principles theory of intrinsic spin and orbital Hall and Nernst effects 
+in metallic monoatomic crystals. Physical Review Materials, 2022. 6: p. 095001. 
+[41] Seifert, T.S., N.M. Tran, O. Gueckstock, S.M. Rouzegar, L. Nadvornik, S. Jaiswal, G. Jakob, V.V. Temnov, 
+M. Münzenberg, M. Wolf, M. Kläui, and T. Kampfrath Terahertz spectroscopy for all-optical spintronic 
+
+characterization of the spin-Hall-effect metals Pt, W and Cu80Ir20. Journal of Physics D: Applied Physics, 
+2018. 51: p. 364003. 
+[42] Huber, R., A. Brodschelm, F. Tauser, and A. Leitenstorfer Generation and field-resolved detection of 
+femtosecond electromagnetic pulses tunable up to 41 THz. Applied Physics Letters, 2000. 76: p. 3191. 
+[43] Braun, L., G. Mussler, A. Hruban, M. Konczykowski, T. Schumann, M. Wolf, M. Münzenberg, L. Perfetti, 
+and T. Kampfrath Ultrafast photocurrents at the surface of the three-dimensional topological insulator Bi 2 
+Se 3. Nature communications, 2016. 7: p. 1. 
+[44] Gueckstock, O., L. Nadvornik, M. Gradhand, T.S. Seifert, G. Bierhance, R. Rouzegar, M. Wolf, M. Vafaee, 
+J. Cramer, M.A. Syskaki, G. Woltersdorf, I. Mertig, G. Jakob, M. Klaui, and T. Kampfrath Terahertz Spin-to-
+Charge Conversion by Interfacial Skew Scattering in Metallic Bilayers. Adv Mater, 2021. 33: p. e2006281. 
+[45] Ding, S., Z. Liang, D. Go, C. Yun, M. Xue, Z. Liu, S. Becker, W. Yang, H. Du, and C. Wang Observation of the 
+orbital Rashba-Edelstein magnetoresistance. Physical review letters, 2022. 128: p. 067201. 
+[46] Santos, E., J. Abrão, D. Go, L. de Assis, Y. Mokrousov, J. Mendes, and A. Azevedo Inverse Orbital Torque 
+via Spin-Orbital Entangled States. arXiv preprint arXiv:2204.01825, 2022. 
+[47] Okano, G., M. Matsuo, Y. Ohnuma, S. Maekawa, and Y. Nozaki Nonreciprocal spin current generation in 
+surface-oxidized copper films. Physical review letters, 2019. 122: p. 217701. 
+[48] Yoda, T., T. Yokoyama, and S. Murakami Orbital Edelstein effect as a condensed-matter analog of 
+solenoids. Nano letters, 2018. 18: p. 916. 
+[49] Go, D., J.-P. Hanke, P.M. Buhl, F. Freimuth, G. Bihlmayer, H.-W. Lee, Y. Mokrousov, and S. Blügel Toward 
+surface orbitronics: giant orbital magnetism from the orbital Rashba effect at the surface of sp-metals. 
+Scientific reports, 2017. 7: p. 1. 
+[50] Chattaraj, A., S. Joulie, V. Serin, A. Claverie, V. Kumar, and A. Kanjilal Crucial role of oxygen on the bulk 
+and surface electronic properties of stable β phase of tungsten. Scientific reports, 2022. 12: p. 1. 
+[51] Wu, Y., M. Elyasi, X. Qiu, M. Chen, Y. Liu, L. Ke, and H. Yang High-Performance THz Emitters Based on 
+Ferromagnetic/Nonmagnetic Heterostructures. Adv Mater, 2017. 29: p. 1603031. 
+[52] Zhang, S., Z. Jin, Z. Zhu, W. Zhu, Z. Zhang, G. Ma, and J. Yao Bursts of efficient terahertz radiation with 
+saturation effect from metal-based ferromagnetic heterostructures. Journal of Physics D: Applied Physics, 
+2017. 51: p. 034001. 
+[53] Niu, C., J.-P. Hanke, P.M. Buhl, H. Zhang, L. Plucinski, D. Wortmann, S. Blügel, G. Bihlmayer, and Y. 
+Mokrousov Mixed topological semimetals driven by orbital complexity in two-dimensional ferromagnets. 
+Nature communications, 2019. 10: p. 1. 
+[54] Seifert, T.S., S. Jaiswal, J. Barker, S.T. Weber, I. Razdolski, J. Cramer, O. Gueckstock, S.F. Maehrlein, L. 
+Nadvornik, S. Watanabe, C. Ciccarelli, A. Melnikov, G. Jakob, M. Munzenberg, S.T.B. Goennenwein, G. 
+Woltersdorf, B. Rethfeld, P.W. Brouwer, M. Wolf, M. Klaui, and T. Kampfrath Femtosecond formation 
+dynamics of the spin Seebeck effect revealed by terahertz spectroscopy. Nat Commun, 2018. 9: p. 2899. 
+[55] Lin, Z., L.V. Zhigilei, and V. Celli Electron-phonon coupling and electron heat capacity of metals under 
+conditions of strong electron-phonon nonequilibrium. Physical Review B, 2008. 77: p. 075133. 
+[56] Ordal, M.A., L.L. Long, R.J. Bell, S.E. Bell, R.R. Bell, R.W. Alexander, and C.A. Ward Optical properties of 
+the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared. Applied Optics, 1983. 
+22: p. 1099. 
+[57] Zak, J., E. Moog, C. Liu, and S. Bader Universal approach to magneto-optics. Journal of Magnetism and 
+Magnetic Materials, 1990. 89: p. 107. 
+ 
+ 
+ 
+
+Supplementary Materials 
+First, we summarize briefly the content of the Supplementary Materials before showing the 
+corresponding data: 
+ 
+ 
+Samples on Si show qualitatively the same THz emission waveforms for Ni with Pt, W and Ti. 
+Most importantly, the strong change in W dynamics is also observed on Si (Fig. S1). However, the 
+THz waveforms of Si vs glass differ in the details, which might be related to slightly changed 
+transport times. 
+ 
+Drude scattering times are estimated to be <50 fs for all studied samples (Figs. S2). None of the 
+samples showed any indication of a drastically different Drude scattering time compared to all 
+other samples. 
+ 
+Emitted THz signals are found to be linearly polarized and perpendicular to the sample 
+magnetization (Fig. S3). 
+ 
+Pump-polarization dependent studies (pump helicity and linear polarization direction) show a 
+minor impact on the measured THz emission signal (Figs. S4). 
+ 
+We perform THz emission measurements upon reversing the sample. Only the pure Ni film shows 
+a dominant contribution even in sample rotation, which we ascribe to SIA or magnetic dipole 
+radiation (Fig. S5) [32, 39].    
+ 
+For all Py-based bilayer samples, we find almost identical THz emission waveform shapes even 
+for PM thicknesses of 20 nm (Fig. S6). 
+ 
+For Ni|Ti samples, we find almost identical THz emission dynamics to Ni|Pt (Fig. S7).  
+ 
+Currents driven by pump light gradients in thick films of Ni|W and Ni|Ti can be neglected (Fig. 
+S8). 
+ 
+All fluence dependencies are to a good approximation linear (Fig S9) with minor sublinearities 
+overserved for Ni|Ti and Ni|W samples. Related to that, only minor changes in the THz waveform 
+dynamics can be observed for different pumping fluences (Fig S9) 
+ 
+Cu has only minor impact on the emitted THz waveforms (Fig. S10), either as a spacer layer or as 
+a capping layer as confirmed by comparison to the same sample without Cu. 
+ 
+All data in the Supplementary Materials was measured with a 1 mm ZnTe(110) detection crystal. 
+ 
+ 
+ 
+
+ 
+FIGURE S1: Si vs glass substrate. a Terahertz-emission waveforms from Ni|PM stacks on Si substrates. THz 
+waveforms for Si based samples are multiplied by -1 to account for the reversed sample orientation due 
+to the intransparency of the Si substrate for the pump pulse. a Terahertz-emission waveforms from Ni|PM 
+stacks on glass substrates. Film thicknesses in nanometers are given as numerals in parenthesis. Note the 
+rescaling of the Ni|Pt sample THz waveforms. 
+ 
+ 
+ 
+b 
+a 
+
+Glass substrate
+X10-7
+5
+4
+3
+Terahertz signal
+2
+-2
+-3
+-1.5
+-1
+-0.5
+0
+0.5
+Time
+e (ps)Si substrate
+X10-7
+2.5
+Ni(5)/Pt(3)/4
+2
+Ni(5)ITi(3)
+Ni(5)/Ti(20)
+1.5
+Ni(5)IW(3)
+Terahertz signal
+Ni(5)/W(20)
+1
+0.5
+O
+-0.5
+-1
+-1.5
+-2
+-0.5
+0
+0.5
+1
+1
+Time
+(ps)FIGURE S2: Terahertz conductivities for samples on glass. Mean complex-valued terahertz conductivities 
+obtained from terahertz transmission measurements for a Ni, b Ni|Ti, c Ni|Pt and d Ni|W samples. For the 
+extraction, a thin film formula is applied [41] and a terahertz refractive index of 2.1 for glass is assumed. 
+Film thicknesses in nanometers are given as numerals in parenthesis. 
+ 
+ 
+ 
+c 
+d 
+a 
+b 
+
+Ni(5)
+X106
+7
+6
+real
+imag
+5
+Conductivity (S/m)
+4
+3
+2
+1
+0
+-1
+1.5
+2
+1
+2.5
+3
+3.5
+Freguency (THz)Ni(5)[Pt(3)
+X106
+7
+real
+6
+imag
+5
+Conductivity (S/m)
+4
+3
+2
+1
+0
+-1
+1
+1.5
+2
+2.5
+3
+3.5
+Freguency (THz)Ni(5)/W(3)
+X106
+7
+real
+6
+imag
+5
+Conductivity (S/m)
+4
+3
+2
+1
+0
+-1
+1
+1.5
+2
+2.5
+3
+3.5
+Freguency (THz)Ni(5)/Ti(3)
+X106
+7
+real
+6
+imag
+5
+Conductivity (S/m)
+4
+3
+2
+1
+0
+-1
+1
+1.5
+2
+2.5
+3
+3.5
+Freguency (THz) 
+FIGURE S3: Polarization state of the THz emission signal. Samples are magnetized along the s-direction 
+and pump pulses are polarized along the p-direction. Film thicknesses in nanometers are given as numerals 
+in parenthesis. 
+ 
+-2
+-1
+0
+1
+2
+Time (ps)
+-2
+0
+2
+4
+6
+8
+10
+Terahertz signal
+10-7
+Ni(5)|Pt(3),p-pol
+Ni(5)|Ti(3),p-pol
+Ni(5)|W(3),p-pol
+Ni(5)|Pt(3),s-pol
+Ni(5)|Ti(3),s-pol
+Ni(5)|W(3),s-pol
+
+ 
+FIGURE S4: Impact of pump polarization. a and b: Circular pump polarization. Terahertz emission signals 
+even and odd in change of pump helicity for terahertz emission polarized along the p-direction (a) and s-
+direction (b). c and d: Linear pump polarization. Terahertz emission signals even and odd in change of 
+pump polarization from s to p polarized for terahertz emission polarized along the p-direction (c) and s-
+direction (d). Samples are magnetized along the s-direction. Film thicknesses in nanometers are given as 
+numerals in parenthesis. 
+ 
+-2
+-1
+0
+1
+2
+Time (ps)
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+Terahertz signal
+10-7
+helicity,THz p-pol
+Ni(5)|Pt(3),even
+Ni(5)|Ti(3),even
+Ni(5)|W(3),even
+Ni(5)|Pt(3),odd
+Ni(5)|Ti(3),odd
+Ni(5)|W(3),odd
+-2
+-1
+0
+1
+2
+Time (ps)
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+Terahertz signal
+10-7
+helicity,THzs-pol
+-2
+-1
+0
+1
+2
+Time (ps)
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+Terahertz signal
+10-7s vs p pump,THz p-pol
+Ni(5)|Pt(3),even
+Ni(5)|Ti(3),even
+Ni(5)|W(3),even
+Ni(5)|Pt(3),odd
+Ni(5)|Ti(3),odd
+Ni(5)|W(3),odd
+-2
+-1
+0
+1
+2
+Time (ps)
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+Terahertz signal
+10-7 s vs p pump,THz s-pol
+a 
+b 
+c 
+d 
+
+ 
+FIGURE S5: Front-side vs back-side pump geometry. a Samples pumped from the front side. b Samples 
+pumped from the back side. The back-side pumping is defined as the direction where the pump pulse first 
+traverses the substrate before exciting the sample and is the standard direction used for all measurements 
+throughout this work. Film thicknesses in nanometers are given as numerals in parenthesis. 
+ 
+ 
+FIGURE S6: Terahertz emission signals for Py based samples as shown in Fig. 2a in addition to terahertz 
+emission signals from thicker Ti and W layers on Py. Film thicknesses in nanometers are given as numerals 
+in parenthesis. 
+ 
+ 
+b 
+a 
+b 
+a 
+
+front-side
+back-side
+X10-6
+X10-6
+Ni(5)
+6
+6
+Ni(5)/Pt(3)/5
+5
+5
+Ni(5)/Ti(3)
+4
+4
+Ni(5)IPt(3)/5
+Terahertz signal
+3
+Y
+erahertz
+2
+Ni(5)/W(3)
+Ni(5)IPt(3)/5
+0
+0
+-2
+.1
+0
+2
+-2
+-1
+0
+1
+2
+Time (ps)
+Time (ps)X10-6
+3
+Py(5)/Pt(3) /3
+Py(5)/Ti(3)
+2
+Py(5)/W(3)
+Py(5)ITi(20)
+Terahertz signal
+Py(5)IW(20)
+0
+-2
+-3
+0
+2
+1
+Time (ps)Py(5)IPt(3)
+Py(5)/Ti(3)
+Norm. terahertz signal
+Py(5)/W(3)
+0.5
+-0.5
+0
+2
+1
+Time
+(ps) 
+FIGURE S7: Ni|Pt vs Ni|Ti. Film thicknesses in nanometers are given as numerals in parenthesis. Note the 
+rescaling of the Ni|Pt sample waveform. 
+ 
+ 
+FIGURE S8: Calculated pump-light gradient in Ni for Ni(5)|Ti(20) and Ni(5)|W(20) samples, which are the 
+thickest samples measured. However, even in these thickest samples, the pump-light gradient is minor. 
+The calculation is based on a transfer matrix formalism [57]. Film thicknesses in nanometers are given as 
+numerals in parenthesis. 
+ 
+0
+1
+2
+3
+4
+Time (ps)
+-3
+-2
+-1
+0
+1
+2
+3
+4
+Terahertz signal
+10-7
+Ni(5)|Pt(3)/5
+Ni(5)|Ti(3)
+Ni(5)|Ti(20)
+0
+2
+4
+Thickness (nm)
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+Rel. pump light intensity
+Ni(5)|Ti(20)
+Ni(5)|W(20)
+
+ 
+FIGURE S9: Pump fluence dependencies. a Fluence dependencies of Ni capped with Pt, W or Ti. The data 
+was contracted by taking the root mean square (RMS) of the time-domain traces. b-f Normalized THz 
+emission signals for different pump fluences. Film thicknesses in nanometers are given as numerals in 
+parenthesis. 
+ 
+ 
+b 
+a 
+d 
+c 
+f 
+e 
+
+X10-6
+10
+RMS of terahertz signal
+Ni(5)IPt(3)/3
+Ni(5)ITi(3)
+8
+Ni(5)IW(3)
+6
+4
+2
+0
+0.05
+0.1
+Incident fluence
+(mJ/cm²Ni(5)[Pt(3)
+0.25
+0.5
+0.75
+Norm. terahertz signal
+0.5
+1
+-0.5
+-0.5
+0
+0.5
+Time
+(psNi(5)/Ti(3)
+Norm. terahertz signal
+0.5
+0
+-0.5
+-1.5
+-0.5
+0
+0.5
+7
+Time
+(psNi(5)/Ti(20)
+0.5
+terahertz signal
+-0.5
+Norm. t
+-1.5
+-0.5
+0
+0.5
+Time
+e (ps)Ni(5)/W(3)
+0.5
+Norm. terahertz signal
+-0.5
+-1.5
+-0.5
+0
+0.5
+Time
+(psNi(5)/W(20)
+0.5
+Norm. terahertz signal
+-0.5
+-1.5
+-0.5
+0
+0.5
+Time
+e (ps)FIGURE S10: Impact of cupper inter- and capping layers. a Reference samples without Cu b Samples with 
+Cu intermediate layer c Samples with Cu capping layer. Film thicknesses in nanometers are given as 
+numerals in parenthesis. 
+ 
+ 
+ 
+a 
+b 
+c 
+
+X10-6
+2
+Ni(5)/W(3)/Cu(2
+Ni(5)IPt(3)ICu(2)
+1.5
+Ni(5)/Ti(3)ICu(2)
+Ni(5)ICu(2)
+Terahertz signal
+0.5
+0
+-0.5
+-2
+0
+2
+Time (ps)X10-6
+1.5
+Ni(5)/Cu(2)W(3)
+Ni(5)ICu(2)IPt(3)
+Ni(5)ICu(2)/Ti(3)
+Ni(5)ICu(2)
+Terahertz signal
+0.5
+0
+-0.5
+-2
+0
+2
+Time (ps)X10-6
+Ni(5)/W(3)
+Ni(5)IPt(3)
+3
+Ni(5)ITi(3)
+Terahertz signal
+2
+-2
+-2
+-1
+0
+2
+Time (ps) 
+Sample 
+Absorptance 
+Absorbed fluence 
+in the FM layer 
+(mJ/cm2) 
+Absorbed 
+fluence in the 
+PM layer 
+(mJ/cm2) 
+Conductivity (1e6 
+S/m) 
+Glass| Ti(50) 
+- 
+- 
+- 
+1.6 
+Glass| Ni(5)|W(20) 
+0.52 
+0.06 
+0.20 
+5.1 
+Glass| Ni(5)|Pt(3) 
+0.63 
+0.25 
+0.06 
+3.6 
+Glass| Ni(5)|Ti(3) 
+0.58 
+0.19 
+0.10 
+2.2 
+Glass| Ni(5)|W(3) 
+0.58 
+0.19 
+0.10 
+2.1 
+Glass| Ni(5)|Ti(20) 
+0.51 
+0.06 
+0.20 
+1.6 
+Glass| Ni(5) 
+0.51 
+0.25 
+- 
+1.7 
+Glass| Py(5)|W(3) 
+- 
+- 
+- 
+2.2 
+Glass| Py(5)|Ti(3) 
+- 
+- 
+- 
+1.5 
+Glass| Py(5)|Pt(3) 
+- 
+- 
+- 
+2.5 
+Glass| Py(5)|W(20) 
+- 
+- 
+- 
+5.3 
+Glass| Py(5) 
+- 
+- 
+- 
+2.4 
+Glass| Py(5)|Ti(20) 
+- 
+- 
+- 
+1.2 
+Glass| Ni(5)|Ti(3)|Cu(2) 
+0.53 
+- 
+- 
+3.1 
+Glass| Ni(5)|Pt(3)|Cu(2) 
+0.54 
+- 
+- 
+4.2 
+Glass| Ni(5)|W(3)|Cu(2) 
+0.58 
+- 
+- 
+4.0 
+Glass| Ni(5)|Cu(2) 
+0.52 
+- 
+- 
+4.5 
+Glass| Ni(5)|Cu(2)|Ti(3) 
+0.56 
+- 
+- 
+3.4 
+Glass| Ni(5)|Cu(2)|Pt(3) 
+0.54 
+- 
+- 
+3.7 
+Glass| Ni(5)|Cu(2)|W(3) 
+0.57 
+- 
+- 
+3.4 
+Glass| Ni(5)|W(15) 
+0.54 
+0.07 
+0.19 
+4.7 
+Glass| Ni(5)|W(10) 
+0.57 
+0.10 
+0.18 
+4.2 
+Glass| Ni(5)|W(5) 
+0.63 
+0.16 
+0.15 
+3.6 
+Glass| Ni(5)|W(2) 
+0.60 
+0.22 
+0.08 
+2.9 
+Si| Ni(5)|W(3) 
+- 
+- 
+- 
+2.9 
+Si| Ni(5)|Ti(20) 
+- 
+- 
+- 
+1.6 
+Si| Ti(50) 
+- 
+- 
+- 
+1.5 
+Si| Ni(5)|Pt(3) 
+- 
+- 
+- 
+3.4 
+Si| Ni(5)|W(20) 
+- 
+- 
+- 
+4.3 
+Si| Ni(5) 
+- 
+- 
+- 
+3.3 
+Si| Ni(5)|Ti(3) 
+- 
+- 
+- 
+2.3 
+Table S1. Optical properties of all studied samples. To obtain the absorbed fluence in the FM and PM layer, we 
+assume imaginary parts of the dielectric constants at 800 nm of 22.07 for Ni, 9.31 for Pt, 19.41 for Ti and 19.71 for W 
+[56]. Note that all films are additionally capped with 4 nm SiO2. 
+ 
+ 
+ 
+

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+A Bayesian treatment of the German tank problem
+Cory M. Simon
+School of Chemical, Biological, and Environmental Engineering. Oregon State
+University. Corvallis, OR. USA.
+[email protected]
+Abstract
+The German tank problem has an interesting historical background and is an engaging
+problem in statistical estimation for the classroom. The objective is to estimate the size
+of a population of tanks inscribed with sequential serial numbers, from a random sample.
+In this tutorial article, we outline the Bayesian approach to the German tank problem,
+(i) whose solution assigns a probability to each tank population size, thereby quantifying
+uncertainty, and (ii) which provides an opportunity to incorporate prior information and/or
+beliefs about the tank population size into the solution. We illustrate with an example.
+Finally, we survey other research problems that bear resemblance to the German tank
+problem.
+s1=15
+s2=14
+s3=3
+serial numbers
+1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
+size of tank population, n
+0
+10
+20
+30
+40
+probability
+0.00
+0.05
+0.10
+0.15
+0.20
+prior
+likelihood
+posterior
+1
+arXiv:2301.00046v1  [stat.OT]  30 Dec 2022
+
+1
+Background
+1.1
+History
+To inform their military strategy during World War II (1939-1945), the Allies sought to es-
+timate the rate of production of various military equipment (tanks, tires, rockets, etc.) by
+Germany. Conventional methods to estimate armament production—including (i) extrapo-
+lating data on prewar manufacturing capabilities, (ii) obtaining reports from secret sources,
+and (iii) interrogating prisoners of war—were unreliable and/or contradictory.
+In 1943, British and American economic intelligence agencies exploited a German man-
+ufacturing practice in order to statistically estimate their armament production. Germany
+marked their military equipment with serial numbers and codes for the date and/or place of
+manufacture to handle spare parts and trace faulty/defective equipment/parts back to the
+manufacturer for quality control. However, these markings on a captured sample of German
+equipment provided the Allies information about Germany’s production of it.
+To estimate Germany’s production of tanks, the Allies collected serial numbers on the
+chassis, engines, gearboxes, and bogie wheels of samples of tanks by inspecting captured
+tanks and examining captured records1. Despite lacking an exhaustive sample, the sequential
+nature of2 and patterns in these samples of serial numbers enabled the Allies to estimate
+Germany’s tank production—postwar, we know, much more accurately than conventional
+American and British intelligence (Tab. 1).
+See Ruggles and Brodie [1] for the detailed historical account of serial number analysis to
+estimate German armament production during World War II.
+Table 1: Monthly production of tanks by Germany. [1]
+estimates
+date
+conventional
+American
+& British Intelligence
+serial number analysis
+German
+records
+June, 1940
+1000
+169
+122
+June, 1941
+1550
+244
+271
+August, 1942
+1550
+327
+342
+1Eg., captured records from tank repair depots listed serial numbers of the chassis and engine of repaired
+tanks, and records from divisional headquarters listed chassis serial numbers of tanks held by a specific unit.
+2Gearboxes on captured tanks, for example, were inscribed with serial numbers belonging to an unbroken
+sequence. Chassis serial numbers, on the other hand, were broken into blocks to distinguish models/designs,
+leaving gaps between the serial numbers assigned to them.
+2
+
+1.2
+The German tank problem
+Simplification of the historical context to estimate German tank production via serial number
+analysis [1] motivated the formulation of the textbook-friendly German tank problem [2]:
+Problem statement
+In the backdrop of World War II, the German military has n tanks.
+Each tank is
+inscribed with a unique serial number in {1, ..., n}.
+As the Allies, we do not know n, but we captured (without replacement, of course) a
+sample of k German tanks with inscribed serial numbers (s1, ..., sk).
+s1
+s2
+· · ·
+sk
+Assuming all tanks in the population were equally likely to be captured, our objective
+is to estimate n in consideration of the data (s1, ..., sk).
+In 1942, Alan Turing and Andrew Gleason discussed a variant of the German tank prob-
+lem, “how to best to estimate the total number of taxicabs in a town, having seen a random
+selection of their license numbers”, in a crowded restaurant in Washington DC [3,4]. Today,
+with its interesting historical background [1], the German tank problem is still a suitable con-
+versation topic for dinners and serves as an intellectually engaging, challenging, and enjoyable
+problem to illustrate combinatorics and statistical estimation in the classroom [5–8].
+Uncertainty quantification.
+Any estimate of the tank population size n from the data
+(s1, ..., sk) is subject to uncertainty, since we (presumably) have not captured all of the tanks
+(ie., k ̸= n, probably). Quantifying uncertainty in our estimate of the tank population size n
+is important because high-stakes military decisions may be made on its basis.
+Our contribution.
+In this pedagogical article, we outline the Bayesian approach to the
+German tank problem, (i) whose solution assigns a probability to each tank population size,
+thereby quantifying uncertainty, and (ii) which provides an opportunity to incorporate prior
+information and/or beliefs about the tank population size into the solution.
+1.3
+Survey of previous work on the German tank problem
+The frequentist approach.
+Border [9] calls the German tank problem a ”weird case” in
+frequentist estimation. The maximum likelihood estimator of the tank population size n is
+3
+
+the maximum serial number observed among the k captured tanks, m(k) := maxi∈{1,...,k} si.
+This is a biased estimator, as certainly m(k) ≤ n.
+Goodman [2, 10] derives the minimum-variance, unbiased point estimator of the tank
+population size
+ˆn = m(k) +
+�
+m(k)
+k
+− 1
+�
+.
+(1)
+To intuit this estimator, note (i) n must be greater than or equal to m(k) and (ii) if we observe
+large (small) gaps between the serial numbers (s1, ..., sk) after sorting them (incl. the gap
+preceding the smallest serial number), then n is likely (unlikely) to be much greater than m(k).
+The estimator of n in eqn. 1 quantifies how far beyond m(k) we should estimate the tank
+population size, based on the gaps; m(k)/k − 1 is the average size of the gaps. Goodman
+also derives a frequentist confidence interval for n.
+Clark, Gonye, and Miller explore using simulations and linear regression to discover the
+estimator in eqn. 1 [11].
+For pedagogy.
+Champkin highlights the historical context of the German tank problem
+as a ”great moment in statistics” [12]. Johnson lists and evaluates several intuitive point
+estimators for the size of the tank population [5]. Scheaffer, Watkins, Gnanadesikan, and
+Witmer [13] propose a hands-on learning activity to illustrate the German tank problem by
+sampling chips, labeled with numbers from 1 to n, from a bowl. Berg [6] uses the German
+tank problem as a competition in the classroom.
+The Bayesian approach.
+Closely related to our paper, Roberts [14], H¨ohle and Held [15],
+and Linden, Dose, and Toussaint [16], and Cocco, Monasson, and Zamponi [17] provide a
+Bayesian analysis of the German tank problem. They derive an analytical formula for the
+mean of the posterior distribution of the tank population size under an improper, uniform
+prior distribution. Andrews [18] outlines the Bayesian approach to the German tank problem
+in a blog post containing code in the R language.
+Generalizations/variants.
+Goodman [2, 10] poses a variant of the German tank problem
+where the initial serial number is not known; ie., where the n tanks are inscribed with serial
+numbers {b + 1, ..., n + b} with b and n unknown. Lee and Miller generalize the German
+tank problem to the settings where the serial numbers are continuous and/or lie in two
+dimensions [19].
+1.4
+Overview of the Bayesian approach to the German tank problem
+Under a Bayesian perspective [8,20,21], we treat the (unknown) total number of tanks as a
+discrete random variable N (hence, capitalization) to model our uncertainty in it. A proba-
+4
+
+bility mass function of N assigns a probability to each possible tank population size n. This
+probability is a measure of our degree of belief, perhaps with some basis in knowledge/data,
+that the tank population size is n [22].
+Because the observed serial numbers (s1, ..., sk) provide information about the tank pop-
+ulation size, the probability mass function of N differs before and after they are collected and
+considered. Hence, N has a prior and posterior probability mass function.
+The three inputs to a Bayesian treatment of the German tank problem are:
+• the prior mass function of N, which expresses a combination of our subjective beliefs
+and objective knowledge about the tank population size before we collect and consider
+the sample of serial numbers.
+• the data, the observed serial numbers (s1, ..., sk), viewed as realizations of random
+variables owing to the stochasticity of tank-capturing.
+• the likelihood function, giving the probability of the data (s1, ..., sk) under each tank
+population size N = n, based on a probabilistic model of the tank-capturing process.
+The output of a Bayesian treatment of the German tank problem is the posterior mass
+function of the tank population size N, conditioned on the data (s1, ..., sk). The posterior
+follows from Bayes’ theorem and can be viewed as an update to the prior in light of the
+data. The posterior mass function of N assigns each possible tank population size n with a
+probability according to a compromise between its (i) likelihood, which quantifies the support
+the observed serial numbers (s1, ..., sk) lend to the tank population size being n according to
+our probabilistic tank-capturing model, and (ii) prior probability, which quantifies how likely
+we thought the tank population size might be n before the serial numbers (s1, ..., sk) were
+collected and considered. [21]
+The posterior mass function of N is the raw, uncertainty-quantifying, Bayesian solution
+to the German tank problem. We may summarize the posterior by reporting its median and
+the high-mass subset of the natural numbers that credibly contains the tank population size.
+Also, we can use the posterior to answer questions such as, what is the probability that N
+exceeds some threshold quantity n′ that would alter military strategy?
+2
+A Bayesian approach to the German tank problem
+We now tackle the German tank problem from a Bayesian standpoint.
+For reference, the variables are listed in Tab. 2. We use upper- and lower-case letters to
+represent random variables and realizations of them, respectively. Throughout, we employ
+the indicator function IA(x) which maps its input x to 1 if x belongs to the set A and to 0
+otherwise (if x /∈ A).
+5
+
+Table 2: List of parameters/variables.
+parameter/variable
+∈
+description
+n
+N≥0
+size of population of tanks
+k
+N>0
+number of captured tanks
+si
+N>0
+serial number on captured tank i
+s(k)
+Nk
+>0
+vector listing the serial numbers on the k captured tanks
+m(k)
+N>0
+maximum serial number among the k captured tanks
+2.1
+The data, data-generating process, and likelihood function
+The data.
+The data we obtain in the German tank problem is the vector of serial numbers
+inscribed on the k captured tanks
+s(k) := (s1, ..., sk).
+(2)
+We view the data s(k) as a realization of the discrete random vector S(k) := (S1, ..., Sk).
+Note, at this point, we are entertaining the possibility that the order in which tanks are
+captured matters.
+The data-generating process.
+The stochastic data-generating process constitutes sequen-
+tial capture of k tanks from a population of n tanks, without replacement, then inspecting
+their serial numbers to construct s(k). We assume that each tank in the population is equally
+likely to be captured at each step. Then, mathematically, the stochastic data-generating
+process is sequential, uniform random selection of k integers, without replacement, from the
+set {1, ..., n}.
+The likelihood function.
+The likelihood function specifies the probability of the data S(k) =
+s(k) given each tank population size N = n. Each outcome s(k) in the sample space Ω(k)
+n
+is
+equally likely, where
+Ω(k)
+n
+:= {(s1, ..., sk)̸= : si ∈ {1, ..., n} for all i ∈ {1, ..., k}},
+(3)
+with (· · · )̸= meaning the elements of the vector (· · · ) are unique. The number of outcomes in
+the sample space, |Ω(k)
+n |, is the number of distinct ordered arrangements of k distinct integers
+from the set {1, ..., n}, given by the falling factorial:
+(n)k := n(n − 1) · · · (n − k + 1) = n!/(n − k)!.
+(4)
+Under the data-generating process, then, the probability of observing data S(k) = s(k) given
+the tank population size N = n is the uniform distribution:
+πlikelihood(S(k) = s(k) | N = n) =
+1
+(n)k
+IΩ(k)
+n
+�
+s(k)�
+.
+(5)
+6
+
+Interpretation.
+We view πlikelihood(S(k) = s(k) | N = n) as a function of n, since in
+practice we possess the data s(k) but not n. The likelihood quantifies the support the serial
+numbers on the k captured tanks in s(k) lend for any particular tank population size n [21].
+The likelihood as a sequence of events.
+Alternatively, we may arrive at eqn. 5 from
+a perspective of sequential events S1 = s1, S2 = s2, ..., Sk = sk. First, the probability of a
+given serial number on the ith captured tank, conditioned on the tank population size and
+the outcomes of the previous serial numbers, is the uniform distribution
+π(Si = si | N = n, S1 = s1, ..., Si−1 = si−1) =
+1
+n − i + 1I{1,...,n}\{s1,...,si−1}(si)
+(6)
+since there are n − i + 1 tanks to choose from at uniform random. By the chain rule, the
+joint probability
+πlikelihood(S1 = s1, ..., Sk = sk | N = n) =
+k�
+i=1
+π(Si = si | N = n, S1 = s1, ..., Si−1 = si−1)
+(7)
+giving eqn. 5 after simplifying the product of indicator functions.
+The likelihood function in terms of the maximum observed serial number.
+We will
+find in Sec. 2.3 that only two independent features of the data (s1, ..., sk) provide information
+about the tank population size, N: its (i) size, k, and (ii) maximum observed serial number
+m(k) =
+max
+i∈{1,...,k} si.
+(8)
+Thus, we also write a different likelihood: the probability of observing a maximum serial
+number m(k) given the tank population size N = n, πlikelihood(M(k) = m(k) | N = n).
+Because each outcome s(k) ∈ Ω(k)
+n
+is equally likely, πlikelihood(M(k) = m(k) | N = n) is the
+fraction of sample space under population size n where the maximum serial number is m(k).
+To count the outcomes (s1, ..., sk) ∈ Ω(k)
+n
+where the maximum serial number is m(k), consider
+(i) one of the k captured tanks has serial number m(k) and (ii) the remaining k −1 tanks have
+a serial number in {1, ..., m(k) − 1}. For each of the k possible positions of the maximum
+serial number in the vector s(k), there are (m(k) − 1)k−1 distinct outcomes specifying the
+other k − 1 entries. Thus:
+πlikelihood(M(k) = m(k) | N = n) = k(m(k) − 1)k−1
+(n)k
+I{k,...,n}(m(k)).
+(9)
+2.2
+The prior distribution
+The prior probability mass function πprior(N = n) expresses a combination of our subjective
+beliefs and objective knowledge about the total number of tanks N before the data (s1, ..., sk)
+7
+
+are collected and considered. Context-dependent, the prior mass function we impose on N
+can vary in the amount of uncertainty it admits about the tank population size (measured by
+eg. entropy [23]).
+Prior distributions can be loosely classified as informative, weakly informative, or diffuse
+[21]. If we do not possess prior information about the tank population size, we adopt the
+principle of indifference and impose a diffuse prior to ”let the data speak for itself” [8], eg.
+a uniform distribution over a set of feasible tank population sizes. On the other hand, an
+informative prior might concentrate its mass around some estimate of the total number of
+tanks obtained through other means. An informative prior will have a larger impact on the
+posterior mass function of N than a diffuse one [21].
+Generally, as the number of captured tanks k increases (decreases), we expect the prior
+mass function we impose to have a lesser (greater) influence on the posterior distribution [8].
+2.3
+The posterior distribution
+The posterior probability mass function of N assigns a probability to each possible tank
+population size n in consideration of its consistency with (1) the data (s1, ..., sk), according
+to the likelihood in eqn. 5, and (2) our prior beliefs/knowledge encoded in πprior(N = n).
+The posterior distribution is a conditional distribution related to the likelihood and prior
+mass functions by Bayes’ theorem:
+πposterior(N = n | S(k) = s(k)) = πlikelihood(S(k) = s(k) | N = n)πprior(N = n)
+πdata(S(k) = s(k))
+,
+(10)
+where the denominator is the probability of the data s(k):
+πdata(S(k) = s(k)) =
+∞
+�
+n′=0
+πlikelihood(S(k) = s(k) | N = n′)πprior(N = n′).
+(11)
+We view πposterior(N = n | S(k) = s(k)) as a probability mass function of N, since in practice
+we have s(k). Then, πdata(S(k) = s(k)) is just a normalizing factor for the numerator in
+eqn. 10.
+Interpreting eqn. 10, the prior mass function of N is updated, in light of the data
+(s1, ..., sk), to yield the posterior mass function of N. The posterior probability of N = n is
+proportional to the product of the likelihood at and prior probability of N = n—a compromise
+between the likelihood and prior.
+We simplify the posterior mass function of N in eqn. 10 by (i) substituting eqn. 5, (ii)
+restricting the sum in eqn. 11 to tank population sizes where the likelihood is nonzero, and
+(iii) noting the only two features of the data (s1, ..., sk) that appear are (a) its size k and (b)
+8
+
+the maximum serial number m(k):
+πposterior(N = n | M(k) = m(k)) =
+(n)−1
+k πprior(N = n)
+∞
+�
+n′=m(k)
+(n′)−1
+k πprior(N = n′)
+I{m(k),m(k)+1,...}(n)
+(12)
+Note, we may arrive at eqn. 12 through eqn. 9 as well.
+Interpretation.
+The posterior probability mass function of N in eqn. 12 is our raw, uncertainty-
+quantifying solution to the German tank problem. It assigns a probability to each tank popula-
+tion size n in consideration of the serial numbers (s1, ..., sk) observed on the captured tanks,
+our probabilistic model of the tank-capturing process, and our prior beliefs and knowledge
+about the tank population size expressed in the prior mass function.
+A remark on ”uncertainty”.
+The spread of the posterior mass function of N in eqn. 12
+reflects epistemic [24] uncertainty about the tank population size, attributed to a lack of
+complete data. Accounting for the data (s1, ..., sk) (probably) does not eliminate uncertainty
+about the tank population size because we (presumably) have not captured all of the tanks
+(ie. k < n) and observed their serial numbers. In practice, posterior uncertainty about the
+tank population size also has a contribution from the possible inadequacy of the model of the
+tank-capturing process (uniform sampling) in eqn. 5, which our analysis here neglects.
+Summarizing the posterior mass function of N.
+We may summarize the posterior mass
+function of N with a point estimate of the tank population size and a credible subset of the
+natural numbers that likely3 contains it. A suitable point estimate of the tank population
+size is a median of the posterior mass function of N; by definition, the posterior probability
+that the tank population size is greater (less) than or equal to a median is at least 0.5. A
+suitable credible subset, which entertains multiple tank population sizes, is the α-high-mass
+subset [25]
+Hα := {n′ : πposterior(N = n′ | M(k) = m(k)) ≥ πα}
+(13)
+where πα is the largest mass to satisfy
+πposterior(N ∈ Hα | M(k) = m(k)) ≥ 1 − α.
+(14)
+In words, the α-high-mass subset Hα is the smallest to (i) contain at least a fraction 1 − α
+of the posterior mass of N and (ii) ensure every tank population size belonging to the subset
+is more probable than all outside of it.
+3Well, ”likely”, under our assumptions embedded in the likelihood and prior mass functions.
+9
+
+Querying the posterior distribution.
+We may find the posterior probability that the tank
+population size belongs to any set of interest by summing the posterior mass over it. Eg.,
+the probability the tank population size exceeds some number n′ is:
+πposterior(N > n′ | M(k) = m(k)) =
+∞
+�
+n=n′+1
+πposterior(N = n | M(k) = m(k)).
+(15)
+2.3.1
+Posterior predictive checking
+We may check the consistency of the data s(k) with the posterior mass function of N.
+Conceptually, we can simulate new data ˜s(k) using the model of the tank-capturing process
+under a sample of the tank population size from the posterior, then compare the simulated
+data ˜s(k) to the real data s(k) [21,26]. More appropriately, we can compare the serial numbers
+in the real data (s1, ..., sk) with the mass function giving the probability that the tank with
+serial number ˜s would be captured under this process:
+π(˜s ∈ ˜S(k)) =
+∞
+�
+n′=0
+k
+n′ πposterior(N = n′ | S(k) = s(k))I{1,...,n′}(˜s),
+(16)
+since k/n′ is the probability any given viable serial number ˜s will be observed given the tank
+population size N = n′.
+3
+Example
+We illustrate the Bayesian approach to the German tank problem through an example.
+The prior probability mass function of N.
+Suppose we have an upper bound nmax for the
+possible number of tanks but no other information. Then, we may impose a diffuse prior, a
+uniform prior probability mass function:
+πprior(N = n) =
+1
+nmax + 1I{0,...,nmax}(n).
+(17)
+This prior mass function expresses: in the absence of any data (s1, ..., sk) (ie., no serial
+numbers, not k either), we believe the total number of tanks N is equally likely to be a value
+in {0, ..., nmax}. Particularly, suppose nmax = 35. Fig. 1a visualizes πprior(N = n).
+The data (s1, ..., sk).
+Now suppose we capture k = 3 tanks, with serial numbers s(3) =
+(15, 14, 3). See Fig. 1b. So, the maximum observed serial number is m(3) = 15.
+10
+
+size of tank population, n
+0
+10
+20
+30
+40
+πprior(N=n)
+0.00
+0.01
+0.02
+(a) prior mass function of N
+s1=15
+s2=14
+s3=3
+serial numbers
+1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
+(b) the data s(k=3)
+tank population size, n
+0
+10
+20
+30
+40
+πlikelihood(M(k=3)=15 | N=n)
+0.00
+0.05
+0.10
+0.15
+0.20
+(c) the likelihood function
+size of tank population, n
+0
+10
+20
+30
+40
+πposterior(N=n | M(k = 3)=15)
+0.00
+0.05
+0.10
+0.15
+nmax=35
+(d) posterior mass function of N
+Figure 1: A Bayesian approach to the German tank problem.
+(a, prior) The prior mass
+function. (b, data) The data s(3), with maximum observed serial number m(3) = 15. (c,
+likelihood) The likelihood function associated with the data s(3). (d, posterior) The posterior
+mass function of N. H0.2 highlighted; median marked with vertical, dashed line.
+11
+
+The posterior probability mass function of N.
+Under the uniform prior in eqn. 17, the
+posterior probability mass function of N in eqn. 12 becomes:
+πposterior(N = n | M(k) = m(k)) =
+(n)−1
+k
+nmax
+�
+n′=m(k)
+(n′)−1
+k
+I{m(k),m(k)+1,...,nmax}(n).
+(18)
+Fig. 1d visualizes the posterior probability mass function of N for the data s(3) in Fig. 1b and
+the prior in eqn. 17 (nmax = 35).
+Summarizing the posterior.
+Summarizing the posterior mass function of N, its median
+is 19 and its high-mass credible subset H0.2 = {15, ..., 25} (highlighted in Fig. 1d). For what
+it’s worth, the data in Fig. 1b was generated from a tank population size of n = 20 (explaining
+the choice of scale in Fig. 1b).
+Querying the posterior.
+Suppose our military strategy would change if the size of the
+tank population exceeds 30. From the posterior distribution of N, we calculate πposterior(N >
+30 | M(3) = 15) ≈ 0.066.
+Posterior predictive checking.
+As a posterior predictive check, Fig. 2a shows how the
+observed serial numbers in the data s(3) compare with the probability of observing each serial
+number under the posterior mass function of N, according to eqn. 16.
+Sensitivity of the posterior to the prior.
+Because of the subjectivity involved in construct-
+ing the prior, checking the sensitivity of the posterior to the prior is good practice [21]. Fig. 2b
+shows how the posterior mass function of N changes as we increase the upper-bound on the
+tank population nmax we impose via the prior mass function of N in eqn. 17. The median of
+the posterior under nmax ∈ {60, 70} is 20 (an increase of one compared to nmax = 35). The
+maximum of the high-mass subset H0.2 increases to 29 for nmax = 70.
+Capturing more tanks.
+Suppose we capture an additional 9 tanks and re-run the Bayesian
+analysis. Fig. 3 shows the updated posterior mass function of N. The high-mass credible
+subset H0.2 shrinks considerably, to {19, 20}. This shows how more data—increasing the
+number of tanks captured, k—generally reduces our uncertainty about the tank population
+size.
+4
+Discussion
+Selection bias.
+A strict assumption in the textbook-friendly German tank problem, which
+enables us to estimate the size of the population of tanks from a random sample of their
+12
+
+serial number, s̃
+0
+10
+20
+30
+40
+probability
+0.00
+0.05
+0.10
+0.15
+nmax=35
+data, s(k=3)
+(a) posterior predictive check
+size of tank population, n
+0
+20
+40
+60
+πposterior(N=n | M(k = 3)=15)
+0.00
+0.05
+0.10
+0.15
+nmax=50
+size of tank population, n
+0
+20
+40
+60
+nmax=60
+size of tank population, n
+0
+20
+40
+60
+nmax=70
+prior
+posterior
+(b) sensitivity of the posterior to the prior
+Figure 2: Checking (a) the consistency of the data s(3) with the probability of the serial
+numbers under the posterior mass function of N and (b) the sensitivity of the posterior mass
+function of N to the upper bound nmax imposed by the prior mass function of N.
+(sequential) serial numbers, is that sampling is uniform. To check consistency of the sample
+with this assumption, Goodman [10] demonstrates a test of the hypothesis that the sample of
+serial numbers is from a uniform distribution. Interesting extensions of the textbook German
+tank problem could involve modeling selection bias in the tank-capturing process. Such bias
+could arise eg. hypothetically, if older tanks with smaller serial numbers were more likely to be
+deployed in the fronts opened earlier in the war, where capturing tanks is more difficult than
+at less fortified fronts opened more recently.
+The German tank problem in other contexts.
+The Bayesian probability theory to solve
+the German tank problem applies (perhaps, with modification) to many other contexts where
+we wish to estimate the size of some finite, hidden set [27], eg.: the number of taxicabs in a
+city [12], the number of accounts at a bank [15], the number of furniture pieces purchased
+13
+
+s1=15
+s2=14
+s3=3
+s4=6
+s5=2
+s6=10
+s7=5
+s8=16
+s9=8
+s10=1
+s11=4
+s12=19
+serial numbers
+1
+2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
+(a) the updated data s(k=12)
+size of tank population, n
+0
+10
+20
+30
+40
+πposterior(N=n | M(k = 12)=19)
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+nmax=35
+(b) the updated posterior mass function of N
+Figure 3: The updated posterior mass function of N (b) after we capture an additional 9
+tanks with serial numbers in (a).
+by a university [10], the number of aircraft operations at an airport [28], the extent of leaked
+classified government communications [29], the time needed to complete a project deadline
+[30], the time-coverage of historical records of extreme events like floods [31], the length of
+a short-tandem repeat allele [32], the size of a social network [33], the number of cases in
+14
+
+court [34], the lifetime of a flower of a plant [35], or the duration of existence of a species [36].
+Mark and recapture methods in ecology to estimate the size of an animal population [37,38]
+are tangentially related to the German tank problem.
+The practice of inscribing sequential serial numbers on military equipment.
+Germany
+adopted the practice of marking their military equipment with serial numbers and codes to
+trace the equipment/parts/components back to the manufacturer. However, the sequential
+nature of these serial numbers was exploited by the Allies to estimate their armament pro-
+duction. To reduce vulnerability to serial number analysis for estimating production while
+maintaining advantages of tracing equipment back to the manufacturer, serial numbers and
+codes could instead be obfuscated by eg. chaffing [39].
+Data and code availability
+The Julia [40] code to reproduce all of our visualizations drawn using Makie.jl [41] is available
+on Github at github.com/SimonEnsemble/the˙German˙tank˙problem.
+Acknowledgements
+Thanks to Bernhard Konrad for providing detailed feedback on the first draft and to my
+students Gbenga Fabusola, Adrian Henle, and Paul Morris for feedback on the introduction.
+References
+[1] Richard Ruggles and Henry Brodie. An empirical approach to economic intelligence in
+World War II. Journal of the American Statistical Association, 42(237):72–91, 1947.
+[2] Leo A Goodman. Serial number analysis. Journal of the American Statistical Association,
+47(260):622–634, 1952.
+[3] Andrew Hodges. Alan Turing: the enigma. In Alan Turing: The Enigma. Princeton
+University Press, 2014.
+[4] Marshall Hall. Alan Turing, Marshall Hall, and the Alignment of WW2 Japanese Naval
+Intercepts. Notices of the AMS, 61(3), 2014.
+[5] Roger W Johnson. Estimating the size of a population. Teaching Statistics, 16(2):50–52,
+1994.
+[6] Arthur Berg. Bayesian modeling competitions for the classroom. Revista Colombiana
+de Estad´ıstica, 44(2):243–252, 2021.
+15
+
+[7] Frederick Mosteller. Fifty challenging problems in probability with solutions. Courier
+Corporation, 1987.
+[8] Allen B Downey. Think Bayes 2. https://allendowney.github.io/ThinkBayes2/
+index.html, 2021.
+[9] Kim. C Border. Lecture 18: Estimation. https://healy.econ.ohio-state.edu/kcb/
+Ma103/ (2021 version), 2017.
+[10] Leo A Goodman. Some practical techniques in serial number analysis. Journal of the
+American Statistical Association, 49(265):97–112, 1954.
+[11] George Clark, Alex Gonye, and Steven J Miller. Lessons from the german tank problem.
+The Mathematical Intelligencer, 43(4):19–28, 2021.
+[12] Carlos G´omez Grajalez, Eileen Magnello, Robert Woods, and Julian Champkin. Great
+moments in statistics. Significance, 10(6):21–28, 2013.
+[13] Richard L Scheaffer, Ann Watkins, Mrudulla Gnanadesikan, and Jeffrey Witmer. Activity-
+based statistics: student guide. Springer Science & Business Media, 2013.
+[14] Harry V Roberts. Informative stopping rules and inferences about population size. Journal
+of the American Statistical Association, 62(319):763–775, 1967.
+[15] Michael H¨ohle and Leonhard Held.
+Bayesian estimation of the size of a population.
+Technical Report 499, LMU Munich, Discussion Paper, 2006.
+[16] Wolfgang Von der Linden, Volker Dose, and Udo Von Toussaint. Bayesian probability
+theory: applications in the physical sciences. Cambridge University Press, 2014.
+[17] Simona Cocco, R´emi Monasson, and Francesco Zamponi. From Statistical Physics to
+Data-Driven Modelling: with Applications to Quantitative Biology. Oxford University
+Press, 2022.
+[18] Mark Andrews. German tank problem: A bayesian analysis. https://www.mjandrews.
+org/blog/germantank. Accessed: 2022-12-03.
+[19] Anthony Lee and Steven J Miller. Generalizing the german tank problem. arXiv preprint
+arXiv:2210.15339, 2022.
+[20] William M Bolstad and James M Curran. Introduction to Bayesian statistics. John Wiley
+& Sons, 2016.
+[21] Rens van de Schoot, Sarah Depaoli, Ruth King, Bianca Kramer, Kaspar M¨artens,
+Mahlet G Tadesse, Marina Vannucci, Andrew Gelman, Duco Veen, Joukje Willem-
+sen, and Christopher Yau. Bayesian statistics and modelling. Nature Reviews Methods
+Primers, 1(1):1–26, 2021.
+16
+
+[22] Jayanta K Ghosh, Mohan Delampady, and Tapas Samanta. An introduction to Bayesian
+analysis: theory and methods, volume 725. Springer, 2006.
+[23] Kevin P Murphy. Probabilistic machine learning: an introduction. MIT press, 2022.
+[24] Craig R Fox and G¨ulden ¨Ulk¨umen. Chapter 1: Distinguishing two dimensions of uncer-
+tainty. Perspectives on Thinking, Judging, and Decision Making, 2011.
+[25] Rob J Hyndman.
+Computing and graphing highest density regions.
+The American
+Statistician, 50(2):120–126, 1996.
+[26] Jonah Gabry, Daniel Simpson, Aki Vehtari, Michael Betancourt, and Andrew Gelman.
+Visualization in Bayesian workflow. Journal of the Royal Statistical Society: Series A
+(Statistics in Society), 182(2):389–402, 2019.
+[27] Si Cheng, Daniel J Eck, and Forrest W Crawford. Estimating the size of a hidden finite
+set: Large-sample behavior of estimators. Statistics Surveys, 14:1–31, 2020.
+[28] John H Mott, Margaret L McNamara, and Darcy M Bullock. Estimation of aircraft op-
+erations at airports using nontraditional statistical approaches. In 2016 IEEE Aerospace
+Conference, pages 1–11. IEEE, 2016.
+[29] Michael Gill and Arthur Spirling. Estimating the severity of the WikiLeaks US diplomatic
+cables disclosure. Political Analysis, 23(2):299–305, 2015.
+[30] Thomas M Fehlmann and Eberhard Kranich. A new approach for continuously monitoring
+project deadlines in software development.
+In Proceedings of the 27th International
+Workshop on Software Measurement and 12th International Conference on Software
+Process and Product Measurement, pages 161–169, 2017.
+[31] Ilaria Prosdocimi. German tanks and historical records: the estimation of the time cover-
+age of ungauged extreme events. Stochastic environmental research and risk assessment,
+32(3):607–622, 2018.
+[32] Haibao Tang, Ewen F Kirkness, Christoph Lippert, William H Biggs, Martin Fabani,
+Ernesto Guzman, Smriti Ramakrishnan, Victor Lavrenko, Boyko Kakaradov, Claire Hou,
+Barry Hicks, David Heckerman, Franz J. Och, C. Thomas Caskey, J. Craig Venter, and
+Amalio Telenti. Profiling of short-tandem-repeat disease alleles in 12,632 human whole
+genomes. The American Journal of Human Genetics, 101(5):700–715, 2017.
+[33] Liran Katzir, Edo Liberty, and Oren Somekh. Estimating sizes of social networks via
+biased sampling. In Proceedings of the 20th international conference on World wide
+web, pages 597–606, 2011.
+17
+
+[34] Xiaohan Wu, Margaret E Roberts, Rachel E Stern, Benjamin L Liebman, Amarnath
+Gupta, and Luke Sanford.
+Augmenting Serialized Bureaucratic Data: The Case of
+Chinese Courts. 21st Century China Center Research, (11), 2022.
+[35] William D Pearse, Charles C Davis, David W Inouye, Richard B Primack, and T Jonathan
+Davies. A statistical estimator for determining the limits of contemporary and historic
+phenology. Nature Ecology & Evolution, 1(12):1876–1882, 2017.
+[36] David L Roberts and Andrew R Solow. When did the dodo become extinct? Nature,
+426(6964):245–245, 2003.
+[37] James D Nichols. Capture-recapture models. BioScience, 42(2):94–102, 1992.
+[38] Anne Chao. An overview of closed capture-recapture models. Journal of Agricultural,
+Biological, and Environmental Statistics, 6(2):158–175, 2001.
+[39] Ronald L Rivest et al.
+Chaffing and winnowing: Confidentiality without encryption.
+CryptoBytes (RSA laboratories), 4(1):12–17, 1998.
+[40] Jeff Bezanson, Stefan Karpinski, Viral B Shah, and Alan Edelman. Julia: A fast dynamic
+language for technical computing. arXiv preprint arXiv:1209.5145, 2012.
+[41] Simon Danisch and Julius Krumbiegel. Makie.jl: Flexible high-performance data visual-
+ization for julia. Journal of Open Source Software, 6(65):3349, 2021.
+18
+

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+Error Tolerant Multi-Robot System for Roadside
+Trash Collection
+1st Lee Milburn
+College of Engineering
+Northeastern University
+Boston, Massachusetts
+[email protected]
+2nd John Chiaramonte
+College of Engineering
+Northeastern University
+Boston, Massachusetts
+[email protected]
+3rd Jack Fenton
+College of Engineering
+Northeastern University
+Boston, Massachusetts
+[email protected]
+Abstract—In this paper, we present the first iteration of an
+error-tolerant, autonomous, multi-robot system that monitors
+highway road verges and identifies and collects roadside litter. It
+is designed to use an aerial vehicle that can rapidly cover a vast
+area and collect data on the road verge. This data is then passed
+to a ground vehicle that constructs a map of the road verge and
+uses a trained Convolutional Neural Network (CNN) to identify
+pieces of litter. After the pieces of litter are identified on the
+map of the road verge, the ground robot navigates to each piece
+of trash, re-evaluates the area, and performs a ”greedy pickup”
+procedure. This final stage accounts for any error in the map’s
+construction or the identified trash’s location. We found that
+ending the robotic system’s control flow with a greedy pickup
+procedure can retroactively account for processing errors of the
+system as it runs. This increases the system’s fault tolerance and
+allows for the use of cheaper equipment since pinpoint accuracy
+is not always necessary. In this paper, we present the feasibility of
+this system by testing in simulation and later using real robotic
+hardware. We show that the system is effective enough to iterate
+on its design principles to create a more sophisticated system.
+Index Terms—Autonomous trash collection, Environmental
+monitoring, Error tolerance, Multi-robot system
+I. INTRODUCTION
+Roadside trash is a massive issue currently managed by
+manual labor - a woefully inadequate solution [7]. Despite
+being a nationwide issue, the task of waste management is
+mostly under the jurisdiction of municipalities and it garners
+little to no attention or investment.
+To estimate the amount of litter along roadways, a research
+team selected a random sample of 240 roadway segments,
+stratified by type and by rural/urban areas [2]. The results
+indicate that there are 51.2 billion pieces of litter on roadways
+nationwide. Of this, the majority (91%, or 46.6 billion pieces)
+is less than four inches.
+In Monterey City, California, complaints about trash have
+increased since the start of the COVID-19 pandemic. Officials
+say this is not due to increased littering, but rather due to
+the inability to clean it up. Monterey County Public Works
+Maintenance Manager Shawn Atkins stated that his cleanup
+crew was so busy cleaning up from illegal dumpsites that they
+did not have time to walk the shoulders of their roads to pick
+up loose trash [14]. Caltrans, California’s public transportation
+department, has been faced with the same problem. Kevin
+Drabinski, public information officer for Caltrans District 5,
+said it’s important to Caltrans that they manage litter because
+of safety and environmental concerns. Caltrans spends $50
+million annually on litter cleanup [14].
+To address this issue, our multi-robot system uses a three-
+stage approach to autonomously map, identify, and pick up
+trash. The three modes are Mapping, Navigation, and Greedy
+Pickup. We would first have a lightweight drone fly over a
+specified area on the road and stream its visuals to a ground
+robot. That robot would generate a map using the drone’s
+input and identify trash pieces on that map. The ground robot
+would then navigate near each piece of identified trash and
+then switch to the greedy pickup mode where it scans the
+area for the suspected piece of trash. After a piece of trash
+is re-identified locally, the robot moves and collects it. Once
+either collected or not found, the ground robot then moves to
+the next piece on its map.
+Fig. 1. System Design Overview
+Our approach allows for accurate pickup without the need
+for massive processing power or overly expensive sensors.
+The system’s configuration used the open-source convolutional
+neural network You Only Look Once (YoLOv4) for image
+identification [24], the open-source visual SLAM solution
+ORBSLAM-2 for map-building [19], and open-source ROS
+navigation software for path planning and navigation. We
+simulated this system using Gazebo [13] and after receiving
+consistent results, tested it in a real-world environment. Our
+arXiv:2301.01704v1  [cs.RO]  4 Jan 2023
+
+Step 1: Mapping Robot identifies trash and maps
+Step 2: Mapping Robot transfers map data to UGV
+the surrounding area
+over Wi-Fi
+Step 3: UGV takes most efficient path to pick up
+Step 4: Human operator removes storage bin and
+trash and clear area.
+dumps trash into larger containerreal-world results, with a relatively low-powered system, in-
+dicate that our approach is a proof-of-concept for a scalable
+and viable solution to the growing worldwide litter problem.
+II. RELATED WORK
+There has been research into multi-robot systems used
+for environmental monitoring. The research for these sys-
+tems finds that multi-robot systems pose a more effective
+solution to surveying an environment than static monitoring
+[6]. There is also research into multi-robot systems that do
+autonomous trash collection [16]. This research concludes
+that for maximum efficiency, robots should be aware of their
+environment when trying to collect trash as opposed to making
+decisions based solely on their field of view (FOV). Therefore,
+these two systems, monitoring an environment, and using a
+collection algorithm to pick up trash in a dynamically changing
+environment could be combined to create the most effective
+version of an autonomous collection system. A version of this
+system has been created to autonomously collect and monitor
+plastics in rivers [9]. The system includes a central processor
+that takes in the necessary tasks of the environment and assigns
+those tasks to underwater autonomous vehicles that then pick
+up the plastics. This system concluded that a Multi-robot task
+allocation architecture [11] with a controlling center increased
+the efficiency of the system, but the hardware for working
+effectively in that environment would have to be improved for
+more effective use. Our research team structured our multi-
+robot system design to have a robot monitor the environment
+and wirelessly transmit its environmental depiction to a UGV
+that would collect the trash.
+For our system to be cheap and lightweight, existing soft-
+ware was needed that works in real-time on standard CPUs
+in a wide variety of environments. The Robotic Operating
+System (ROS) [17] is an open-source robotics framework that
+allowed each of our hardware and software components to
+communicate freely in real-time, and each software component
+used was compatible with this framework. ORB-SLAM2 was
+the ideal solution for mapping [18][15]. It uses differing
+angles of static environmental features to create a map and a
+keyframe-based SLAM approach that reduces the overall data
+size of the SLAM map considerably [1]. Since the system is
+designed with visual sensors, a software to visually identify
+trash was necessary. YoLOv4 is a CNN model trained from
+annotated images to place bounding boxes around specified
+objects in RBG images. Adaptive Monte Carlo Localization
+(AMCL) is the method of navigation used as well as the
+name of a compatible software stack used for navigation
+provided by ROS [8]. AMCL takes in odometry feedback
+from the robot’s wheels and scan data derived from the RGB-
+D Camera to navigate. After some static conversions from
+ORB-SLAM2’s native map format to a 2D occupancy grid,
+AMCL can autonomously navigate around an environment.
+These existing software stacks served as the framework for
+the multi-robot system to be built.
+III. SYSTEM DESIGN
+A. System Overview
+Fig. 2. Systems’ Communication Flow
+After initialization by a human operator, the mapping robot
+will scan an area with a visual sensor. This sensor data will be
+compiled using Simultaneous Location and Mapping (SLAM)
+technology to create a continuous digital map of the target area
+which will then be wirelessly transmitted to the unmanned
+ground vehicle (UGV). The UGV will identify pieces of
+trash in the environment using computer vision algorithms
+and construct a two-dimensional map populated with target
+coordinates of identified trash. The UGV will then create an
+efficient path between the target coordinates in the map. Once
+the UGV sets off on the calculated path, it will confirm the
+trash location using an onboard visual sensor and proceed to
+pick it up. Once the UGV has completed its rounds or the bin
+is detected as full, it will return home, and a human operator
+will empty the bin.
+B. Mode Controller
+Fig. 3. Mode Controller Flow
+The “Mode Controller” was created to switch between the
+three separate software components of the system: Mapping,
+Navigation, and Greedy Pickup. The Mode Controller is a
+ROS node that communicates with the Mapping, Navigation,
+and Greedy Pickup nodes, turning them on or off as needed.
+The Mode Controller starts in idle before putting the system
+into the Mapping mode. Once mapping finishes, the mode
+controller turns off Mapping mode. The map is then passed on
+
+Stage 1
+Mapping With
+ORBslam
+Realsense
+map
+Stage 2
+General
+RGB + depth camera feed
+Mode Control
+Navigation
+Stage 3
+Throughput
+Greedy Pickup
+YOLO
+Control1.Mapping &
+2. Navigation
+TrashID
+3. Greedy
+Idle
+Pickupto the Navigation mode alongside the coordinates of identified
+trash. The Mode Controller then turns on Navigation mode.
+Once a trash coordinate has been reached by the Navigation
+mode, the Mode Controller next turns Navigation off and
+Greedy Pickup on, picking up the trash. Navigation mode is
+once again activated. Navigation and Greedy Pickup modes
+will alternate until all trash is removed from the environment.
+Once all the trash is picked up and no marked coordinates
+remain on the map, the Mode Controller turns back to idle
+and awaits further instruction.
+C. Mapping
+The system navigates the surrounding area and maps its
+environment using ORB-SLAM2. This repository is designed
+to be used within ROS as a ROS node. In our default RGB-D
+configuration, the node subscribes to 2 topics (RGB and depth
+image topics) and in turn, publishes all necessary data built
+by the ORB-SLAM2 system. This includes a point cloud of
+all map key points, the current camera pose, the full camera
+path trajectory, and a morphologically transformed version of
+the projected occupancy grid [21]. In experimentation, the
+maps were initially filled with noise that led to an inability
+to navigate the space, figure 4. Morphological operations
+are commonly used tools in image processing to clean up
+an image. By “eroding” and “opening” the space, errant
+data points that were being misidentified as occupied were
+removed. By “closing” the space gaps caused by the sparse
+data, holes in our map were closed, and smooth, continuous
+maps were generated, figure 5. The product was an occupancy
+grid very close to real-world surroundings with a real-time,
+lightweight mapping solution.
+D. Trash Identification
+Simultaneously, as an area is being mapped, the system also
+detects trash. To recognize where on the map a piece of trash
+is, the mapping robot first finds the location of a piece of
+trash relative to itself. The system to locate trash was devised
+using multiple components: YOLOv4, the odometry data of
+the robot, and the depth camera feed provided by the Realsense
+RGB-D camera, figure 7.
+The first step in the trash identification pipeline is image
+identification using YOLOv4. YOLOv4 is a convolutional
+neural network that we trained with a custom dataset of
+over 1000 images, each taken of varying pieces of trash
+from the perspective of the robot. Each image was hand-
+labeled and fed into the machine learning model using an
+80-15-5 split between training, validation, and testing sets.
+The model was trained and runs in our software stack using
+a customized open-source ROS wrapper for YOLOv4 [24].
+The image identification model runs simultaneously while the
+environment is being mapped using the RGB camera feed and
+returns “bounding boxes” around identified trash pieces in the
+image, providing coordinates relative to the camera’s image
+frame, figure 6.
+These bounding boxes provide 2D pixel locations for the
+trash in the image but do not contain any information about
+Fig. 4. Raw Occupancy Grid
+Fig. 5. Morphologically Transformed Occupancy Grid
+where the trash lies in the environment. Therefore, the next
+step is to identify the angle of the closest piece of trash relative
+to the camera. This is accomplished by using the center pixel
+x-coordinate of an identified piece of trash. Using the field of
+view of the camera, an imaginary triangle can be created to
+discover the angle of the trash relative to the camera in the
+real world by using pixels as the coordinate system.
+The FOV angle is 69.4 degrees, its opposite side is 640
+pixels, and it is known to be an isosceles triangle, the re-
+maining side lengths and angles can be extrapolated as this
+is considered a trigonometrically “solved” triangle. Using this
+triangle the angle of the identified trash piece is calculated
+using the inverse tangent function, as shown in the following
+equation and in figure 9.
+θ = tan−1 trashx − 320
+462.139
+(1)
+The next step in the localization process is to determine the
+distance between the camera and the piece of trash. This is
+accomplished using the depth camera feed provided by the
+RGB-D camera. This camera outputs a grayscale image in
+which each pixel is a 16-bit value representing the distance to
+that pixel in millimeters directly from the center of the camera.
+The depth picture can be indexed as a matrix using the 2D
+coordinates given by YOLO’s bounding box to determine the
+exact distance between the camera and any piece of trash.
+Once the distance between the camera and the trash has been
+calculated, all information necessary to localize the piece of
+
+Fig. 6. Trash Bounding Boxes
+Fig. 7. Depth Feed
+trash relative to the robot has been acquired. Using a second
+triangle with coordinates in meters, both the angle of the trash
+relative to the robot as well as the distance between the trash
+and the robot can be extrapolated.
+The first unknown variable encountered is the distance
+between the trash and the center of the robot base, d. Using the
+distance between the camera and the center of the robot base
+s, as well as the distance between the trash and the camera
+taken from the depth camera feed (depth), d can be solved
+using the Law of Cosines as shown below.
+c2& = a2 + b2 + 2ab cos(c)
+(2)
+c& =
+�
+a2 + b2 + 2ab cos(c)
+(3)
+d& =
+�
+(depth2 + r2 + 2(depth)(s)(cos(180◦ − θ))
+(4)
+Once d is known, the final variable which needs solving is
+β. This can be solved using the Law of Sines.
+sin X
+x
+& = sin Y
+y
+(5)
+sin(β)
+depth & = sin(180◦ − θ)
+d
+(6)
+β& = sin−1
+�depth sin(180◦ − θ)
+d
+�
+(7)
+Once the angle to the piece of trash relative to the robot and
+the distance between these two points became known, these
+values were added to the robot’s current position to realize the
+piece of trash on the map. However, some difficulties arose
+when the computer did not process the images fast enough.
+Algorithm 1 Mapping Trash to a Map
+Input: Robot’s path r in the map m, YOLO Bounding
+Box b
+Output: Pose of piece of trash in the map p, orientation
+o of robot relative to p
+confident pieces cp ← empty
+▷ init array to hold all
+confident pieces
+for every item in b do
+if items i’s trash confidence is greater than ct then
+cp ← i
+end if
+end for
+yolotimestamp yt ← b[0].timestamp
+for pose pr in r do
+time difference td ← abs(yt − pr)
+if you don’t have a closest pose timestamp cpr to the
+yolo timestamp yt then
+cpr ← pr
+smallest time difference std ← td
+else
+if td < std then
+std ← td
+end if
+end if
+end for
+for each timestamp, image in depth camera history do
+Find the closest depthimage di taken to b
+end for
+Set robots orientation o from when the picture was
+taken
+for trashpiece tp in cp do
+get distance d of tp from di
+get tp’s angle theta from robot base
+trash x distance tdx ← o + (d ∗ cos θ)
+trash y distance tdy ← o + (d ∗ sin θ)
+p ← tdx, tdy
+end for
+YOLOv4, when run on the Intel NUC, processed images at a
+throughput of 0.5-0.8 FPS with about 4-5 seconds of latency
+from when the image was originally taken. This created a
+large gap between the time when the image was taken and the
+current position of the robot. To account for the processing
+latency, the path of the robot as it was mapping is logged
+with timestamps for every position in its path from ORB-
+SLAM2. Once the mapping robot received a successful trash
+detection, the ROS timestamp given from YOLOv4 from when
+that image was taken was passed to the path, and a Pose is
+output. It is from this Pose that distance d and angle β are
+added to localize the piece of trash relative to the map itself.
+In the figure 11, the thin blue line is the path of the robot
+as it maps the area. The red arrow is the current position of
+the robot in the map. The cyan arrow is the Pose where the
+robot was when the YOLOv4 image was taken. From this cyan
+Pose, a red trash detection is then finally placed on the map.
+
+rubbish:0.99Fig. 8. FOV Diagram
+Fig. 9. FOV Trignometric Calculations
+Fig. 10. Robot-Trash Trignometric Calculations
+Every trash detection is plotted, and a separate anti-clustering
+node averages these together, getting an approximate location
+of the piece of trash.
+E. Anti-clustering
+Initially, we found consistency issues with the trash identifi-
+cation. Either images of the same piece of trash were processed
+more than once, or the trash’s estimated position became
+inaccurate as the SLAM map updated. This problem led to
+a large amount of noise, causing up to and exceeding thirty
+detections for two pieces of trash in one single trial. In some
+limited cases, our YoLOv4 model would also erroneously
+classify a random background object as trash. To sort through
+the noisy detections, each new trash detection was run through
+Fig. 11. Robot Trajectory in the map
+a filter. Every time a piece of trash was detected, a ROS
+subscriber would listen to the detection and determine if it was
+a new piece or detection of a piece of trash already found. A
+clustered piece of trash is denoted by the green mark in figure
+11.
+To accomplish this, all the detected pieces of trash were
+stored at their initial positions. If any new trash detection
+was within a set radius of a previously detected piece of
+trash, the new trash detection became combined with the
+established piece by taking a rolling average of the detections.
+The calculations are seen in the equation 8, where p1x/y is the
+existing trash detection’s respective x and y coordinate, p2x/2y
+is the new trash detection’s x and y coordinates, and a is the
+amount of times p1 has been averaged to that point.
+p1x = p1xa + p2x
+a + 1
+p1y = p1ya + p2y
+a + 1
+(8)
+This anti-clustering algorithm decreased the total amount of
+detections to accurately reflect the number of trash pieces seen
+in the environment. To avoid erroneous trash detections, aver-
+age trash locations without enough detections were determined
+as “noisy” and filtered out. Trash points were only published to
+the navigation stack if it had three or more detections averaged
+to one point.
+F. General Navigation and Path Planning
+General navigation consists of two parts: localization and
+path planning. The Robot first receives the 2D occupancy
+grid from our mapping software, alongside the trash detection
+coordinates. Once these data are received, the robot then
+navigates to within two meters of the nearest detected trash
+point. Navigation only needs to navigate near a trash location
+since greedy pickup is routinely effective within a two meter
+distance, and the anti-clustering algorithm accounts for noise
+in our trash detections.
+To get the goal pose g you need it’s orientation, and x/y
+coordinates. The equation 9 describes how to get the angle
+for the pose where p1 is the starting robot pose and p2 the
+trash pose. In equation 10 the distance the goal pose is from
+the robot is calculated to have it within Greedy Pickup range,
+
+-ldentifiedTrashPiece
+trash_depth (m)
+180°-0
+0.6096m
+KobukiRobot--ldentifiedTrashPiece
+640Pixels
+RGBCameraFrame
+CameraFOV-
+69.4°
+RealsenseD435
+RGB-DCamera-ldentifiedTrashPiece
+trash x
+320-(640-trash_x)-
+462.139 PixelsAlgorithm 2 Anti-Clustering Algorithm
+Input: Trash Pose tp
+Output: Poses of clustered trash pieces ctp
+averaged Trash poses atp tupled with times averaged ta
+(atp, ta) ← empty
+if atp = 0 then
+atp ← tp
+ta ← 1
+else
+for every pose tuple pt in atpt do
+get x and y bottom and top around the trash’s
+location
+xb, xt, yb, yt
+if xb ≤ tpx ≤ xt & yx ≤ tpy ≤ yt then
+pose tuple x ptx = (ptx * ta + tpx) / (ta + 1.0)
+pose tuple y pty = (pty * ta + tpy) / (ta + 1.0)
+ta + 1
+end if
+end for
+if tp is not in atp then
+atp ← tp
+end if
+ctp ← all trash poses tp in atp where tpi’s tai > 2
+times
+end if
+where d3 is the goal distance, d1 is the distance between
+the trash and the robot, and d3 is Greedy Pickup’s range. In
+equation 11 the coordinates of the goal pose is found where
+gx/y is represents the coordinates respectively.
+θ = arctan(p1y − p2y
+p1x − p2x
+)
+(9)
+d3 = d1 − d2
+(10)
+gx = d3 + p1x cos(θ)gy = d3 + p2x sin(θ)
+(11)
+Fig. 12. Robot Path Planning in Rviz
+The Navigation and path planning stack was based on the
+ROS-provided open-source AMCL software stack. This soft-
+ware loads and localizes the robot in a mapped environment
+and its DWA planner creates a path between the identified
+points of trash. We created another software module that feeds
+our target coordinates from our trash detection phase into
+AMCL’s path planner to follow the most efficient path between
+the robot’s current location and the nearest possible trash point,
+figure 12.
+G. Greedy Pickup
+Once the Navigation portion of the software stack places
+the robot within 2 meters of the piece of trash, Greedy Pickup
+is activated. Greedy Pickup is an asynchronous algorithm that
+ignores all navigation and map factors and solely focuses on
+seeking out the nearby trash directly.
+When the greedy pickup is activated, it rotates in a given
+direction to look for trash using YoLOv4. Once it receives a
+trash detection, it calculates the position of the trash using the
+same algorithm explained in the Trash Identification section.
+After the robot localizes the trash, it turns back towards the
+trash at its precise angle, turns on the collection mechanism’s
+motor, and moves exactly 0.2m past the trash’s location to
+ensure proper collection. Once this occurs, the motor turns off
+and navigation resumes.
+Algorithm 3 Greedy Pickup
+Input: Detected Trash pose list tpl
+set timeout time to
+set confidence threshold ct
+for Trash pose tp in tpl do
+Determine whether tp is to the left or right of the robot
+in the Map
+Spin in direction of tp, scanning for confirmation
+if Robot gets tp ≥ ct then
+Robot Stop spinning
+Use III-D algorithm to find relevant Robot orienta-
+tion and trash pose
+Get destination angle
+Robot turns to destination angle
+Robot Turn on collection mechanism and drive over
+detected piece of trash
+else
+Robot exceeded to looking for trash
+end if
+end for
+Output: Poses of clustered trash pieces ctp
+For the collection mechanism to turn on or off, the NUC
+sends a serial packet to the connected Arduino with only three
+bytes in sequence, either [0x59, 0x59, 0x59] to start the motor
+or [0x4E, 0x4E, 0x4E] to stop the motor. Once the Arduino
+receives a start packet, it then outputs a PWM signal to two
+of its GPIO pins which control the L928N motor controller.
+The PWM signal gradually increases from a low duty cycle to
+a higher duty cycle to control the current spikes on the 12V
+line from the Robot base to the Motor. In the initial design on
+bench power, starting the motor from 0 to full power produced
+an initial current spike of approximately 1.9A before settling
+around 0.9-1.1A when in normal motion or picking up an
+
+object. The initial spike was over the 1.5A limit provided by
+the 12V port accessible on the robot base. To eliminate this
+spike, a slow ramp-up of the duty cycle of the PWM signal
+was introduced from 20% duty cycle to a maximum of 80%
+linearly over the course of 5 seconds. This removes the initial
+current spike and ensures that the motor can both properly
+power the collection mechanism and does not exceed the 1.5A
+current limit.
+H. ROS Middleware
+ROS was used to connect all the functions of this system.
+Every design block in figure 2 functions as a node, or multiple
+nodes, which subscribe and publish information to the other
+nodes. ROS would also be used to network between the
+different robots in the multi-robot system over WiFi or other
+wireless protocols.
+IV. EXPERIMENT
+A. Goal
+The purpose of this experiment is to assert the feasibility
+of this system design before iterating on the hardware used to
+make it scalable and adaptable. The robot was evaluated on
+its ability to accurately map the enclosure, identify, and mark
+the pieces of trash, choose an efficient path, and pick the trash
+up.
+The robotic system is meant to clear trash as large and heavy
+as an average 600mL Spring Valley Water bottle weighing ap-
+proximately 0.64 kg. The system’s robotic base, the TurtleBot
+2, has a load limitation of approximately 5 kilograms [20],
+which presents an upper weight limit on the total load. Since
+the robot’s additional components (external frame, storage
+container, etc.) are estimated to weigh approximately 3 kg,
+the trash load must weigh at most 2 kg.
+The UGV operates in narrow environmental parameters. The
+weather must be clear with no rain since the electronic systems
+onboard the UGV are non-weatherproofed. In addition, due to
+the wheels that come default with the robotic base (Kobuki
+Mobile Base), our prototype can only operate on relatively flat,
+smooth, evenly colored surfaces, with no extreme movement
+in the background.
+B. Testing In Simulation
+Simulated testing was done in the Gazebo Robotics simu-
+lator, 13. This simulator was included in the base Turtlebot
+SDK and includes a near true-to-life recreation of the entire
+turtlebot system. The use of ROS allows for the navigation
+stack and greedy pickup to be run against the simulator and
+behave exactly identically to reality. This simulator was instru-
+mental in the initial testing of movement and navigation as it
+allowed our team to test many different speed parameters and
+movement algorithms without risking any physical damage to
+the robot. All data associated with movement and mapping
+were recorded and played back in a simulated environment to
+recreate and reevaluate our physical testing. This allowed for
+useful visualizations and assessments of what the robot was
+processing at any given time.
+Fig. 13. Simulation Testing Still
+C. Testing the Hardware
+1) Experiment System: The system’s design is meant to
+function with a mapping robot and a ground vehicle. To
+simulate the mapping vehicle in this experiment the UGV
+plays both roles. The UGV first passed through the area to
+get a map and identify trash. Then taking that information to
+navigate and pick up the trash.
+2) The Robot:
+The hardware design uses a modified
+Turtlebot-2 as a base design, figure 15, that has four main
+components: the depth camera (480p RGB-D Intel Realsense
+D435) [10], the computer (7-year-old Intel NUC with 8GB
+of RAM) which runs a GNU-Linux OS along with ROS to
+manage sensor data collection and real-time processing, the
+mobile robotic base (Kobuki Mobile Base) [12], and a custom-
+designed collection mechanism. The camera relays RGB and
+depth images which are processed to identify and target trash.
+The Kobuki’s motor and wheels relay odometric feedback
+that helps confirm the UGV’s current location. The collection
+mechanism attaches to the front of the Kobuki Mobile base,
+plugs into power and data ports on the robot, and uses a rotary
+brush to pick up the trash.
+3) The Collection Mechanism: The collection mechanism
+is a custom-designed addition to the Turtlebot Robot. Its
+mechanical construction consists of 20-20 aluminum bars
+connected by 90-degree brackets. To allow free range of
+motion, caster wheels were affixed to the bottom of the frame.
+When the collection mechanism motor is activated, it sweeps
+trash up a ramp into a plastic storage container. A camera
+mount was printed so the Realsense could be attached to the
+front of mechanism [23]. A funnel was added to the front of
+the collection mechanism to rein in the trash that may have
+been missed slightly, figure 18.
+The design of the electronics system for the collection
+mechanism is an Arduino Mega that is connected to the NUC
+using a USB cable, figure 17. The Arduino is then connected
+to an L982N Motor driver breakout board over PWM-enabled
+GPIO pins (5V). This motor driver breakout board is supplied
+with 12V by the Kobuki base from a 12V, 1.5A Max port. The
+output of the L982N is the DC motor which drives the chain
+and the brush.
+
+Fig. 14. Hardware Overview Diagram
+Fig. 15. Turtlebot2 Base System
+The brush itself was hand-designed and fabricated since
+there is no commercial brush model that fit the design speci-
+fications for the mobile robot, figure 16.
+D. The Environment
+The environment where the robot was tested was a room
+with a random configuration of chairs and obstacles put around
+an open space. Therefore, the map would be created each
+time in a dynamic environment and the system would have
+to account for a new configuration. In that space, trash was
+put in different locations for all tests. We tested up to four
+pieces of trash in the environment at a time.
+E. Tests
+1) Testing Mapping/ Image Identification accuracy/ Ac-
+counting for Latency: To test the trash detection and how
+well the processing latency was accounted for, expected maps
+with the approximate trash locations were checked against the
+created ones.
+2) Testing the Greedy Pickup Algorithm: To test Greedy
+Pickup, the mode was enabled with varying pieces of trash
+within two meters of the robot. The piece of trash would start
+out of the UGV’s perception range. The UGV would have to
+scan for the trash, identify it and then collect it. This trial was
+Fig. 16. Custom Designed Brush
+Fig. 17. Motor Circuitry
+Fig. 18. Final Construction
+run with trash at 2 meters, 1 meter, and half a meter distance
+from the UGV.
+3) Full System Trials: Full system trials were then con-
+ducted, starting with the mapping of an environment and
+labeling trash points in that environment. Then to navigating
+to the labeled pieces of trash in the environment and collecting
+the pieces of trash using Greedy Pickup.
+V. RESULTS
+A. Accounting for Processing Latency
+The map creation and the image identification, figure 23,
+were shown to be an accurate, figure 19, fast and lightweight
+way of monitoring an area and identifying pieces of trash.
+B. Greedy Pickup Results
+Greedy pickup is shown to be accurate to an extent with
+a success rate of 77.78%, figure 20. There were multiple
+failures in these trials that were caused by the design of
+the collection mechanism. The collection mechanism was not
+equipped with odometric wheels that could relay its’ location
+so, at times it would overturn, and the caster wheels introduced
+an error the system was not aware of and could not account
+
+Intel Realsense
+Camera
+RGB + depth images
+collection
+Intel NUC
+mechanism
+motor feedback + control
+Kobuki
+Mobile Base12VDCMotor
+CURRENT SENSING A
+CURRENT SENSING
+OUTPUT
+OUTPUT 2
+OUTPUT
+GNT
+SUPPLY VOLTAGE VS
+INPUT
+ENABLE
+INPUT
+ENABLEA
+LOGIC SUPPLY VOLTAGE VSS
+INPUT
+INPUT 2
+GND
+OME
+L928N Motor Driver
+TXO
+8
+12V
+REOET
+RX3
+TX3
+MEGA
+91
+ROTAFig. 19. Accurate Map Creation with Identified Trash Points Result
+Fig. 20. Greedy Pickup Results
+Fig. 21. Full System Run-through Results
+Fig. 22. Real Time Trash Identification and Map Creation
+for. This caused the UGV to over or under turn occasionally
+and not successfully collect the trash. The Greedy Pickup
+algorithm has shown the ability to increase the error tolerance
+of the system, however, there are more improvements to the
+algorithm that could be made to make it more error-tolerant.
+For example, an added “lock on” mechanism that when a piece
+of trash is in sight it would keep the piece of trash in the center
+of its FOV as it moves forward to collect the piece of trash.
+Ensuring that the failure-causing edge cases are accounted for.
+Fig. 23. Still of UGV Picking up Water Bottle in System Test
+C. Full System Results
+These results, with a 68% success rate, figure 21, show
+that there are possible errors that can be introduced to this
+system and that they compound on themselves as the task gets
+more complex. During these trials, the collection mechanism
+introduced errors during navigation and greedy pickup. The
+collection mechanism introduced errors by being slightly out
+of position. Since the Collection mechanism did not have
+sensor feedback to tell the UGV it was not in the correct
+position the little errors compounded into failures to collect
+trash.
+VI. CONCLUSION
+Our contribution is a multi-robot system design applied
+to monitoring and managing roadside litter. We developed a
+system that can map, identify, and pick up pieces of trash. The
+system is designed to be relatively cheap and scalable which
+could be done because we introduced different algorithms that
+account for errors in a system that has less precise sensors.
+Our Greedy Pickup algorithm accounts for errors in trash iden-
+tification, and trash can be accurately identified in a system
+with low processing power. Our custom-designed collection
+mechanism, designed to pick up the main offending types of
+trash found on the side of the road, was also introduced.
+This system builds off previously known algorithms: AMCL
+and DWA for navigation, Orbslam-2 for map creation, and
+YOLO.v4 for trash identification. It also build off multi-robot
+systems to monitor a dynamic environment and combines that
+with information with a dynamic collection algorithm, Greedy
+Pickup, to create an efficient collection process.
+The next steps for this system would to be refine the
+mechanical design of the UGV, such as adding odometry
+sensors onto the collection mechanism’s wheels and improving
+the wheels to be able to drive over more difficult terrains.
+There are also improvements to be made to the Greedy Pickup
+algorithm such as adding a “Lock on” capability. Better logic
+could be applied to the navigation, for example, a future
+iteration of this system could use a graph search algorithm
+
+Mapping and Trash Identification Results
+8%
+92%
+ Successful Maps
+■ Failed MapsGreedy Pickup Success
+16
+14
+12
+10
+8
+6
+4
+2
+0
+2 meter
+1 meter
+0.5 meter
+Robot's Distance from Trash
+Total Trials
+Successful trialsFull System Tests
+Aount of Trash in the Environment
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+Successful Trials
+ Total Trials[ Interact Move Camera
+ Select
+Focus CameraMeasure2D Pose Estimate2DNavGoalPublishPoint
+Q
+RGBImage
+RGB View
+ORB-SLAM2Ima
+RViz
+SLAMView
+OOYOLOV4
+Map View
+YOLOView
+199.22
+Wall Elapsed:416.55
+ Experimentalto find the most efficient path between every piece of trash.
+An aerial robot can also be introduced as the mapping robot in
+the next iteration of this system for testing, to more accurately
+represent the issues that would arise from changes in the
+perspective of the system. More replicas of the UGV could be
+added into the system as well, so there would be multiple trash
+collectors in one environment at the same time. The system
+could also have mapping and trash collection happening in
+parallel instead of the modes being sequential.
+VII. ACKNOWLEDGMENTS
+This is a video of a full run-through our system did with
+two pieces of trash in the environment [5]. This is the central
+repository in our organization that sets up the system [4].
+This is the Greedy Pickup repository which holds all the
+functions laid out in this paper [3]. We would like to thank
+the contributions of Divya Venkatraman, Jared Raines and
+Catherine Ellingham for their work on gathering data, editing
+the paper and system networking. We’d also like to thank Dr.
+Taskin Padir and his Robotics and Intelligent Ground Vehicle
+Research Laboratory (RIVeR) [22] for allowing us to use their
+hardware to test our design.
+REFERENCES
+[1]
+Yong-bao Ai et al. “Visual SLAM in dynamic environ-
+ments based on object detection”. In: Defence Technol-
+ogy (2020). ISSN: 2214-9147. DOI: https://doi.org/10.
+1016/j.dt.2020.09.012. URL: https://www.sciencedirect.
+com/science/article/pii/S2214914720304402.
+[2]
+Keep Louisiana Beautiful. Executive Summary: Litter in
+America. URL: https://keeplouisianabeautiful.org/wp-
+content/uploads/2015/09/Litter-in-America-Executive
+Summary - FINAL.pdf. (Accessed: 10-June-2021).
+[3]
+John Chiaramonte, Jack Fenton, and Lee Milburn. cave-
+man mode. URL: https://github.com/Capstone- W3/
+caveman mode. (Accessed: 20-April-2021).
+[4]
+John Chiaramonte et al. Capstone-W3. URL: https://
+github.com/Capstone-W3. (Accessed: 11-June-2021).
+[5]
+John Chiaramonte et al. T.R.A.S.H. Demonstration.
+URL: https : / / www . youtube . com / watch ? v =
+cIwk0Kh9k7E. (Accessed: 5-July-2021).
+[6]
+Maria Valera Espina et al. “Multi-robot Teams for
+Environmental Monitoring”. In: Innovations in Defence
+Support Systems – 3: Intelligent Paradigms in Secu-
+rity. Ed. by Paolo Remagnino, Dorothy N. Monekosso,
+and Lakhmi C. Jain. Berlin, Heidelberg: Springer
+Berlin Heidelberg, 2011, pp. 183–209. ISBN: 978-3-
+642-18278-5. DOI: 10.1007/978- 3- 642- 18278- 5 8.
+URL: https://doi.org/10.1007/978-3-642-18278-5 8.
+[7]
+Conserve Energy Future. The Catastrophic Effects of
+Littering on Humans. URL: https : / / www. conserve -
+energy-future.com/littering-effects-humans-animals-
+environment.php. (Accessed: 14-June-2021).
+[8]
+Brian Gerkey. AMCL Package Summary. URL: https:
+//wiki.ros.org/amcl?distro=melodic. (Accessed: 10-
+March-2022).
+[9]
+Le Hong, Weicheng Cui, and Hao Chen. “A Novel
+Multi-Robot Task Allocation Model in Marine Plastics
+Cleaning Based on Replicator Dynamics”. In: Journal
+of Marine Science and Engineering 9.8 (2021). ISSN:
+2077-1312. DOI: 10.3390/jmse9080879. URL: https:
+//www.mdpi.com/2077-1312/9/8/879.
+[10]
+Intel. Intel® RealSense™ Depth Camera D435. URL:
+https://ark.intel.com/content/www/us/en/ark/products/
+128255 / intel - realsense - depth - camera - d435 . html.
+(Accessed: 5-July-2021).
+[11]
+Alaa Khamis, Ahmed Hussein, and Ahmed Elmogy.
+“Multi-robot Task Allocation: A Review of the State-
+of-the-Art”. In: vol. 604. May 2015, pp. 31–51. ISBN:
+978-3-319-18299-5. DOI: 10.1007/978-3-319-18299-
+5 2.
+[12]
+Iclebo Kobuki. Kobuki User Guide. URL: http://kobuki.
+yujinrobot . com / wiki / online - user- guide/. (Accessed:
+11-June-2021).
+[13]
+Nathan Koenig and Andrew Howard. “Design and Use
+Paradigms for Gazebo, An Open-Source Multi-Robot
+Simulator”. In: ().
+[14]
+Stephanie Melchor. Roadside Trash A Growing Prob-
+lem. URL: https://www.montereyherald.com/2021/01/
+23/roadside-trash-a-growing-problem/. (Accessed: 10-
+June-2021).
+[15]
+Ra´ul Mur-Artal, J. M. M. Montiel, and Juan D. Tard´os.
+“ORB-SLAM: A Versatile and Accurate Monocular
+SLAM System”. In: IEEE Transactions on Robotics
+31.5 (2015), pp. 1147–1163. DOI: 10.1109/TRO.2015.
+2463671.
+[16]
+Shinkyu Park, Yaofeng Desmond Zhong, and Naomi
+Ehrich Leonard. “Multi-Robot Task Allocation Games
+in Dynamically Changing Environments”. In: 2021
+IEEE International Conference on Robotics and Au-
+tomation (ICRA). 2021, pp. 8678–8684. DOI: 10.1109/
+ICRA48506.2021.9561809.
+[17]
+Morgan Quigley et al. “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, May 2009.
+[18]
+raulmur. ORB SLAM2. URL: https : / / github . com /
+raulmur/ORB SLAM2. (Accessed: 11-June-2021).
+[19]
+rayvburn. ORBSLAM2 ROS. URL: https://github.com/
+rayvburn / ORB - SLAM2 ROS. (Accessed: 11-June-
+2021).
+[20]
+Clearpath Robotics. TurtleBot 2 - Open Source Personal
+Research Robot. URL: https://clearpathrobotics.com/
+turtlebot - 2 - open - source - robot. (Accessed: 10-June-
+2021).
+[21]
+Adrian Rosebrock. OpenCV Morphological Operations.
+URL: https://pyimagesearch.com/2021/04/28/opencv-
+morphological-operations/. (Accessed: 11-June-2021).
+[22]
+Padir Taskin. RIVeR. URL: https : / / robot . neu . edu/.
+(Accessed: 7-July-2021).
+
+[23]
+Makerbot Thingiverse. Intel RealSense D435 Camera
+Mount for OpenBuilds 20x20mm Extrusion. URL: https:
+//www.thingiverse.com/thing:2965905.
+[24]
+Tossy0423. darknet ros. URL: https : / / github . com /
+Tossy0423 / darknet ros / tree / master/. (Accessed: 11-
+June-2021).
+

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+Geometry-based approximation of waves propagating
+through complex domains∗
+Davide Pradovera†
+Monica Nonino†
+Ilaria Perugia†
+February 1, 2023
+Abstract
+We consider wave propagation problems over 2-dimensional domains with piecewise-linear bound-
+aries, possibly including scatterers. Under the assumption that the initial conditions and forcing terms
+are radially symmetric and compactly supported (which is common in applications), we propose an ap-
+proximation of the propagating wave as the sum of some special nonlinear space-time functions: each
+term in this sum identifies a particular ray, modeling the result of a single reflection or diffraction ef-
+fect. We describe an algorithm for identifying such rays automatically, based on the domain geometry.
+To showcase our proposed method, we present several numerical examples, such as waves scattering off
+wedges and waves propagating through a room in presence of obstacles.
+Keywords: wave propagation, model reduction, scattering, geometrical optics, diffraction
+AMS subject classifications: 35L05, 35Q60, 65M25, 78A45, 78M34
+1
+Introduction
+The discretization of numerical models for the simulation of complex phenomena results in high-dimensional
+systems to be solved, usually at an extremely high cost in terms of computational time and storage memory.
+Among these models, wave propagation problems represent an extremely interesting topic: relevant applica-
+tions can be found, e.g., in the field of array imaging, where acoustic, electromagnetic, and elastic waves in
+scattering media are modeled by the reflectivity coefficient, which is often unknown. Some examples in this
+direction can be found in [5, 6, 7, 30], where inverse scattering problems are used to infer the reflectivity of
+one or more scatterers embedded either in a known and smooth medium, or in a randomly inhomogeneous
+medium. Another example of application of wave propagation problems is numerical acoustics, where the
+goal is to simulate the propagation of sound in a room, in presence of obstacles and walls with different
+absorbing and/or reflecting properties, see [28].
+Wave propagation problems in the time-harmonic setting (the Helmholtz problem, cast in the frequency
+domain) have been widely studied. See, e.g., [4, 13, 19, 24, 25, 27, 28]. However, our focus here are problems
+in the time domain, whose numerical simulation is expensive, mainly because one needs to use both a fine
+spatial mesh and a carefully chosen time step in order to satisfy the CFL condition [11, 16]. In the interest
+of making these simulations feasible, model order reduction (MOR) [3, 9, 14, 17] represents a promising
+framework, whose goal is to reduce the computational cost of solving the problem of interest.
+In this context, it is well known [12, 15] that wave propagation problems are characterized by a slowly
+decaying Kolmogorov n-width.
+Because of this, classical linear-subspace MOR methods are not able to
+reproduce the behavior of the wave propagation without relying on a very high-dimensional linear manifold.
+This makes linear surrogate models unappealing, since they do not yield significant speed-ups. In recent
+years, many approaches have been proposed to overcome the intrinsic “difficulty” of problems with slowly
+∗M. Nonino and I. Perugia have been funded by the Austrian Science Fund (FWF) through project F 65 “Taming Complexity
+in Partial Differential Systems” and project P 33477.
+†Faculty
+of
+Mathematics,
+University
+of
+Vienna,
+Oskar-Morgenstern-Platz
+1,
+1090
+Vienna,
+Austria
+(da-
+[email protected], [email protected], [email protected]).
+1
+arXiv:2301.13613v1  [math.NA]  31 Jan 2023
+
+D. Pradovera, M. Nonino, and I. Perugia
+Geometry-based approximation of waves in complex domains
+decaying Kolmogorov n-width, with the target of making MOR more efficient. To this end, such methods
+rely on nonlinear and/or hybrid space-time approaches. For more details, we refer to [8, 10, 18, 21, 26, 29, 31].
+In this work, we focus on wave propagation over 2-dimensional spatial domains, possibly including ob-
+stacles. We limit our investigation to domains with piecewise-linear boundaries and a constant wave speed.
+The initial conditions and forcing terms are assumed to be compactly supported and radially symmetric
+around a “source point”. This situation arises in many of the above-mentioned applications. Under these
+assumptions, we propose to approximate the solution of the problem of interest with the sum of some special
+nonlinear space-time functions, which we call “rays”. Each ray models a reflection or diffraction effect, and
+is composed of different parts:
+• the free-space radially symmetric solution of the wave equation, modeling the space-time propagation
+of the ray;
+• a spatial indicator function, determining the light cone of each ray;
+• a nonlinear spatial term encoding the angular modulation of the ray, which is crucial when modeling
+diffraction effects.
+The number of terms appearing in the sum is determined by the number of reflection and diffraction effects
+that are required to faithfully approximate the target wave, which ultimately depends on the geometry of
+the physical domain.
+Among the advantages of the proposed approach, we mention the fact that each ray is separable into
+time-radial and angular components (in the “polar coordinates” sense). As we will see, we can leverage this
+to reduce drastically the computational cost and the storage memory that are required by our approximation,
+with respect to competitor methods.
+The rest of the paper is structured as follows. In Section 1.1 we present the problem of interest. In
+Section 2 we introduce the main ingredients of our method, and we describe the “training phase” of the
+algorithm, i.e., the construction of the approximated wave. In Sections 3 and 4 we detail how we model
+reflection and diffraction effects, respectively. The latter section is rather extensive, since diffraction is much
+harder to model than reflection, and requires special care. In Section 5 we present some numerical results to
+showcase our method. Both simple benchmarks (wedges) and more complicated tests (2D room model with
+scatterers) are considered. Some final considerations follow in Section 6.
+1.1
+Target problem
+We are interested in the numerical approximation of the solution of the wave equation in complex domains.
+In this work, we consider 2-dimensional domains only. However, most of our discussion generalizes to 3D.
+We defer a discussion on this till Section 6.
+We denote by Ω ⊂ R2 the physical domain in which the wave equation is considered. We assume that
+Ω is either a closed polygon or a set-subtraction of polygons (to allow for multiply connected domains).
+We denote by ne and nv the number of edges and vertices of ∂Ω, respectively. We study the propagation
+of waves in Ω over a given time interval of interest [0, T]. The model problem is the wave equation with
+constant (unit) wave speed:
+�
+�
+�
+�
+�
+�
+�
+�
+�
+∂ttu(x, t) = ∆u(x, t) + f(x, t)
+for (x, t) ∈ Ω × (0, T),
+u(x, 0) = u0(x)
+for x ∈ Ω,
+∂tu(x, 0) = u1(x)
+for x ∈ Ω,
+∂νu(x, t) = 0
+for (x, t) ∈ ∂Ω × (0, T],
+(1)
+with ∆ the Laplacian operator, defined, in 2 dimensions, as ∆ = �2
+j=1 ∂xjxj. The homogeneous Neumann
+condition (i.e., the last equation above) models the whole boundary ∂Ω as sound-hard [11]. More generally,
+all or parts of ∂Ω may be modeled as sound-soft via a Dirichlet-type condition: u(x, t) = 0.
+We assume that the initial conditions u0 and u1, as well as the forcing term f, have radial symmetry
+around a given point. Without loss of generality, we will take such point to be the origin of R2:
+u0(x) = η0(∥x∥), u1(x) = η1(∥x∥), f(x, t) = η2(∥x∥ , t)
+∀(x, t) ∈ Ω × (0, T),
+(2)
+2
+
+D. Pradovera, M. Nonino, and I. Perugia
+Geometry-based approximation of waves in complex domains
+with ∥x∥2 = �2
+j=1 x2
+j. We further assume that the functions ηj have compact support, namely, that there
+exist R > 0 such that ηj(ρ) = 0 for all ρ > R and j = 0, 1, 2. Moreover, to avoid incompatibilities with the
+boundary conditions, for simplicity we will only consider the situation where the supports of the functions
+ηj are fully contained in Ω.
+2
+Approximation framework
+Before we can model boundary effects (reflection and diffraction), we need to understand how the solution
+u would behave if no boundary were present. To this aim, we consider the wave equation in free space
+�
+�
+�
+�
+�
+∂ttU(x, t) = ∆U(x, t) + f(x, t)
+for (x, t) ∈ R2 × (0, ∞),
+U(x, 0) = u0(x)
+for x ∈ R2,
+∂tU(x, 0) = u1(x)
+for x ∈ R2,
+(3)
+which we have obtained from (1) by replacing Ω with the whole plane.
+Due to radial symmetry (of the initial conditions and of the forcing term), we can recast the problem in
+polar coordinates. This allows us to define the free-space solution in the radial coordinate Ψ, as the solution
+of
+�
+�
+�
+�
+�
+�
+�
+�
+�
+∂ttΨ(ρ, t) = �∆Ψ(ρ, t) + η2(ρ, t)
+for (ρ, t) ∈ (0, ∞) × (0, ∞),
+Ψ(ρ, 0) = η0(ρ)
+for ρ ∈ [0, ∞),
+∂tΨ(ρ, 0) = η1(ρ)
+for ρ ∈ [0, ∞),
+∂ρΨ(0, t) = 0
+for t ∈ (0, ∞),
+(4)
+where �∆ is the Laplace operator in polar coordinates (under radial symmetry), i.e.
+�∆ = ∂ρρ + 1
+ρ∂ρ, and
+U(x, t) = Ψ(∥x∥ , t) for all x ∈ R2. Note that, by the compact support of the initial conditions and of
+the forcing term, and by the finite (unit) speed of propagation of the wave equation, we have Ψ(ρ, t) = 0
+whenever ρ > t + R.
+Remark 2.1. Generally, the free-space solution Ψ is not available analytically, except for very simple choices
+of initial conditions and forcing term. Accordingly, in most applications, the function Ψ will need to be
+replaced with a suitable approximation. To this effect, one could discretize (4), e.g., with a finite element
+approximation (in space) and some timestepping scheme (in time). See Section 5 for more details on how
+this can be carried out.
+Our goal is to approximate, for all (x, t) ∈ Ω × [0, T], the solution u(x, t) of the wave equation problem
+(1) with the following sum of special functions:
+u(x, t) ≈ �u(x, t) =
+N
+�
+n=1
+Ψ(∥x − ξn∥ + rn, t)1Ωn(x)ζn(x − ξn)
+�
+��
+�
+�un(x,t)
+.
+(5)
+Each term �un is what we will call a “ray”. Therein, Ψ is the above-mentioned free-space radially symmetric
+solution of (4), and 1A denotes the indicator function with support A, i.e.,
+1A(y) =
+�
+1
+if y ∈ A,
+0
+if y /∈ A.
+(6)
+Moreover, in (5), we have introduced the following quantities:
+• N is the number of rays used in the approximation.
+• ξn is the location of the new source.
+• rn ≥ 0 is a spatial delay, which will be used for the synchronization of diffraction effects.
+• Ωn ⊂ Ω is the light cone (the spatial support) of a term of the sum.
+3
+
+D. Pradovera, M. Nonino, and I. Perugia
+Geometry-based approximation of waves in complex domains
+• ζn : R2 \ {0} → R is a weight function describing the angular modulation. We require that ζn be a
+positive-homogeneous functions, i.e., ζn(y) = ζn(τy) for all τ > 0 and y ∈ R2.
+Note that, due to the finite speed of propagation of the free-space solution Ψ, we have that a generic
+term �un(x, t) is zero whenever t < ∥x − ξn∥ + rn − R, i.e., for t small enough, depending on x.
+The number of rays N in the sum (5) will be determined based on how many boundary effects (reflections
+and diffractions) need to be included in �u in order to have a good approximation of the target wave u. We
+describe a strategy for automatically identifying a good N in the next section. See, e.g., Remark 2.2.
+2.1
+Building the low-rank skeleton
+Recalling that u solves the wave equation (1) in the domain Ω, we use the first term in (5), namely, �u1, to
+approximate the outgoing component of u, ignoring any effect due to the boundary ∂Ω, except for shadows.
+Then, given such �u1, we use the other terms �u2, . . . , �uN to correct this first approximation. Each extra term
+models a single effect due to a certain portion of the boundary, specifically, an edge (reflection off that edge)
+or a vertex (diffraction about that vertex).
+Going back to the first ray �u1, let us define it, by providing its “ingredients” ξ1, r1, Ω1, and ζ1, cf. (5).
+We set ξ1 = 0, the center of the initial condition, as well as r1 = 0, since no delay is necessary for this first
+term. Then, leveraging symmetry, we set ζ1 ≡ 1, which corresponds to the (physical) assumption that the
+propagation of �u1 is purely radial. Finally, we set Ω1 (the light cone around 0) as the set of points that can
+be reached from 0 via a straight line without going outside ∂Ω, i.e.,
+Ω1 = {x ∈ Ω : τx ∈ Ω ∀0 ≤ τ ≤ 1} .
+(7)
+In summary, the first term of �u is
+�u1(x, t) = Ψ(∥x∥ , t)1Ω1(x).
+(8)
+Then we can move to the subsequent terms �un, n ≥ 2.
+Their expressions depend on our choice of
+reflection and diffraction modeling, and will be provided in the upcoming sections. Instead, in the rest of
+the present section we focus on understanding how large N should be, in order for �u to provide a faithful
+approximation of u. Equivalently, we want to count the number of times the wave gets reflected or diffracted
+at the boundary ∂Ω. This is done incrementally, starting from the initial value N = 1 (no boundary effects)
+and then updating this guess as more and more boundary effects get “discovered”.
+To help us in this endeavor, we employ what we call a timetable, which, in this work, is simply a list of
+vectors, each with size ne + nv. The timetable is built incrementally starting from an empty list, appending
+one new vector every time a new term is added in the sum (5), starting from �u1. The entries of the n-th
+timetable vector are the waiting times before �un comes in contact with an edge or a vertex of ∂Ω. If it is
+impossible for �un to “cast light” (along a straight path) onto a certain edge or vertex, then the corresponding
+entry in the timetable is set to ∞. After this, it suffices to look for the smallest not-yet-explored entry of the
+timetable to identify what the next term of the approximation �u should be. Once the entry in the timetable
+has been explored, its value is set to ∞.
+We start by describing how the first vector a1 ∈ Rne+nv of the timetable (corresponding to �u1) is
+computed, and how a1 allows us to identify the (geometric) features of �u2. The vector a1 can be partitioned
+into edges-related part (the first ne entries) and vertices-related part (the last nv entries).
+• Edge times. Given a generic edge γj ⊂ ∂Ω (j = 1, . . . , ne) belonging to the domain boundary, we
+define the corresponding entry of a1 as
+(a1)j =
+�
+r1 + inf
+�
+∥x − ξ1∥ : x ∈ γj ∩ Ω1
+�
+if the set is non-empty,
+∞
+otherwise.
+(9)
+Note that we have taken the shortest path from ��1 to γj, and that we have denoted the closure of the
+light cone Ω1 as Ω1.
+• Vertex times. Given a generic vertex yj ⊂ ∂Ω (j = 1, . . . , nv) of the domain boundary, we set
+(a1)ne+j =
+�
+r1 + ∥yj − ξ1∥
+if yj ∈ Ω1,
+∞
+otherwise.
+(10)
+4
+
+D. Pradovera, M. Nonino, and I. Perugia
+Geometry-based approximation of waves in complex domains
+Note that we have included the delay r1 (which is actually zero here) as a way to streamline Eqs. (9) and (10)
+for the upcoming section. See Fig. 1 for a diagram showcasing these formulas.
+(a1)1
+(a1)2
+(a1)5
+(a1)17
+Ω1
+Ω \ Ω1
+γ1
+γ2
+γ3
+γ4
+γ5
+y3
+y4
+y6
+ξ1
+(a1)3 = ∞
+(a1)4 = ∞
+(a1)14 = ∞
+(a1)15 = ∞
+...
+Figure 1: Computation of some timetable entries. The boundary ∂Ω has 11 sides, so that, e.g., (a1)14 is
+related to y3 and (a1)17 is related to y6. The shadowed area Ω \ Ω1 is in darker grey.
+The smallest entry of a1 is the time at which the first “boundary event” (reflection or diffraction) can
+happen1. The index of the smallest entry tells us whether the event is a reflection (index 1 ≤ j ≤ ne) or a
+diffraction (index ne + 1 ≤ j ≤ ne + nv), and also what edge/vertex causes the event. From here, we use the
+models described in Sections 3 and 4 to build �u2, by computing ξ2, r2, Ω2, and ζ2.
+Then, the second timetable vector a2 can be computed by replacing all subscripts “1” by “2” in Eqs. (9)
+and (10). This is followed by the construction of �u3, and so on. The process continues until all not-yet-
+explored entries of the timetable are larger than T +R. Indeed, starting from this time instant, the would-be
+next terms of �u do not affect the approximation anymore, since, due to the finite speed of wave propagation,
+they only act (on Ω) after the end of the time horizon, i.e., for t > T. The total number of rays N is simply
+the number of vectors in the timetable.
+We summarize the overall procedure for the construction of the rays �un in Algorithm 1. For ease of
+presentation, once an entry of the timetable has been explored, it is set to ∞ as a way for the algorithm to
+ignore it from that point forward.
+Algorithm 1 Step-by-step construction of the surrogate model
+Set N ← 1, find Ω1 as in (7), and define �u1 as in (8)
+Define a1 ∈ Rne+nv using Eqs. (9) and (10)
+Set i ← 1 and j ← arg minj=1,...,ne+nv (a1)j
+while (ai)j ≤ T + R do
+Set (ai)j ← ∞ and N ← N + 1
+if j ≤ ne then
+Find ξN, rN, ΩN, and ζN as in Section 3
+← Reflection from edge j
+else
+Find vertex index j′ ← j − ne
+Find ξN, rN, ΩN, and ζN as in Section 4
+← Diffraction from vertex j′
+end if
+Define �uN from ξN, rN, ΩN, and ζN, as in (5)
+Define aN ∈ Rne+nv using Eqs. (9) and (10), with “N” replacing “1” in subscripts
+Set (i, j) ← arg mini=1,...,N,j=1,...,ne+nv (ai)j
+end while
+1We say “can happen” since not all vertices can cause diffraction, when hit from a certain point source.
+This issue is
+discussed in Section 4, cf. Assumption 4.3.
+5
+
+D. Pradovera, M. Nonino, and I. Perugia
+Geometry-based approximation of waves in complex domains
+γ
+ξi
+ξn
+y(x)
+x
+θr
+θi
+φi(y(x))
+φn(x)
+βγ
+ξn
+ξi
+y(x)
+x
+Figure 2: Graphical representation of a reflection off edge γ. On the left, the law of reflection prescribes
+θr = θi. We show the straight line �γ supporting γ with a dotted stroke. For a given observation point x,
+y(x) denotes the point of incidence of the reflected ray. On the right, computation of the light cone Ωn
+(light-grey area) and its complementary shadow zone Ω \ Ωn (dark-grey area) for the reflected ray, in the
+presence of a rectangular obstacle. The dashed portion of edge γ denotes the shadow γ \ γ(i). The shadow
+zone consists of two connected components.
+Remark 2.2. In trapping domains, see, e.g., Section 5.2, the number of terms N might be rather large
+due to waves repeatedly “bouncing back and forth” between two or more edges/vertices. A large N, although
+necessary for a good approximation of all wavefronts, is undesirable since it increases the computational cost
+of both the construction of the surrogate �u and its evaluation.
+As a compromise, one could remove all terms �un that are smaller than a certain tolerance tol, uniformly
+over x and t. This can be done as a post-processing step (thus speeding up the evaluation of �u but not
+its construction) or while building the surrogate itself. This can be achieved with a simple modification of
+Algorithm 1, by introducing a test on the magnitude of each soon-to-be-added wave contribution �un, discarding
+terms that are too small.
+3
+Modeling reflection
+We now present the strategy for modeling reflection due to an edge γ of the domain boundary ∂Ω. We
+rely on the well-known geometrical optics model, which describes wave propagation in terms of rays, not
+accounting for any diffraction [23]. We assume that we are adding a new ray �un to the surrogate model (5),
+due to a reflection phenomenon caused by ray �ui. Specifically, we assume that a ray coming from source
+point ξi hits the edge γ ⊂ ∂Ω, i.e., that γ ∩ Ωi ̸= ∅. We need to prescribe several ingredients.
+Spatial correction rn.
+We just transfer rn over from the incoming wave: rn = ri. Indeed, as we will see
+in Section 4, we require the term rn only when modeling diffraction.
+Source point ξn.
+We use the method of images, which gives the position of ξn as the reflection of ξi with
+respect to the edge γ:
+ξn = 2 arg min
+z∈�γ
+∥z − ξi∥ − ξi,
+(11)
+where �γ ⊂ R2 is the straight line on which edge γ lies. See Fig. 2 (left).
+Weight function ζn.
+Let x − ξn be a generic point where we wish to evaluate the weight function ζn.
+We define the incidence point y(x) as the intersection (if any) between edge γ and the segment from ξn to
+x. See Fig. 2 (left). According to the method of images, the amplitude of the reflected wave is equal (up to
+sign) to the amplitude of the incoming wave:
+ζn(x − ξn) = (2σγ − 1)ζi(y(x) − ξi).
+(12)
+6
+
+D. Pradovera, M. Nonino, and I. Perugia
+Geometry-based approximation of waves in complex domains
++
+=
+Figure 3: Example of reflection off an edge in the presence of an obstacle, from Fig. 2. Neumann conditions
+are imposed on all edges. Source wave (left), reflected wave (middle), and superimposition of the two (right).
+Note how the obstacle creates a shadow zone for source and reflected waves. For simplicity, in this plot we
+are not showing any reflection or diffraction effects due to the rectangular obstacle, since they would be
+modeled at different stages of the algorithm.
+In the equation above, the quantity σγ is related to the kind of boundary conditions that are imposed on γ:
+if γ is an edge with Neumann boundary conditions, we set σγ = 1 (ζn and ζi have the same sign), whereas
+we set σγ = 0 if we have Dirichlet boundary conditions on γ (ζn and ζi have opposite signs).
+Now, recall that we are assuming all weight functions to be positive-homogeneous: ζi(x − ξi) = ζi(τ(x −
+ξi)), for all τ > 0. Accordingly, as we are in 2D, ζi(x − ξi) is only a function of the direction (with sign)
+υi(x) = (x − ξi)/ ∥x − ξi∥, or, equivalently, of the angle φi(x) between υi(x) and the positive x1-axis. See
+Fig. 2 (left) for a graphical depiction. Specifically, with an abuse of notation, let ζi(x − ξi) = ζi(φi(x)) and
+ζn(x − ξn) = ζn(φn(x)), where the “new” angle-dependent functions ζi and ζn are 2π-periodic. By (12), we
+deduce the property
+ζn(φn(x)) = (2σγ − 1)ζi(φi(y(x))) = (2σγ − 1)ζi(2βγ − φn(y(x))) = (2σγ − 1)ζi(2βγ − φn(x)),
+(13)
+where βγ is the angle between edge γ and the positive x1-axis. This uniquely identifies ζn given ζi and βγ.
+Light cone Ωn.
+We first identify what portion of γ is actually “lit” by �ui: γ(i) = γ ∩ Ωi. Note that we
+may have γ ̸= γ(i), for instance when obstacles are present between ξi and γ. See Fig. 2 (right) for an
+illustration. Then, roughly speaking, we define the new light cone Ωn as the union of all rays from ξn that
+pass through γ(i). To be more precise, given x ∈ Ω, let y(x) be the intersection (if any) between γ and the
+line segment from ξn to x. Also, if y(x) exists, we define τ0(x) = ∥y(x) − ξn∥ / ∥x − ξn∥ ∈ (0, 1), which
+satisfies y(x) = ξn + τ0(x)(x − ξn). The new light cone is defined as
+Ωn =
+�
+x ∈ Ω : y(x) ∈ γ(i) and ξn + τ(x − ξn) ∈ Ω ∀τ0(x) < τ ≤ 1
+�
+.
+(14)
+Figure 3 represents a possible output of the numerical algorithm. In this case, we simulate only the
+reflections, thus discarding, for the time being, any effect due to diffraction. It is clear that, by modeling
+reflection effects only, we may obtain a discontinuous approximation of the solution of our target problem,
+where the discontinuity happens exactly at the shadow boundaries (the boundaries of light cones). As we
+will see in the next section, introducing diffraction in our approximation will allow us to obtain a continuous
+approximation �u.
+4
+Modeling diffraction
+Here, we describe a strategy for modeling waves diffracted by a vertex of the domain boundary ∂Ω. This is
+required in building a new ray �un whenever the smallest unexplored entry of the timetable is related to a
+vertex, i.e., j > ne in Algorithm 1, cf. Section 2. We need to identify several ingredients.
+Source point ξn.
+We employ the (standard, see, e.g., [23]) assumption that diffraction emerges as a wave
+outgoing from a point source located at the diffraction point yj′ = yj−ne (we are employing the notation of
+Algorithm 1). This motivates the choice of the center ξn = yj′.
+7
+
+D. Pradovera, M. Nonino, and I. Perugia
+Geometry-based approximation of waves in complex domains
+∂Ω
+γ′
+γ
+ξ
+α
+φ
+π − θ
+θ
+3π − 2α − θ
+∂Ω
+γ′
+γ
+ξ
+α
+θ − π
+3π − 2α − θ
+θ
+Figure 4: Diagrams for the two cases of scattering for concave corners (0 < α < π): without (left plot) and
+with shadow zone (right plot). The dashed lines are reflection boundaries. The dash-dotted line is a shadow
+boundary. Shadow regions are absent if and only if π − α ≤ θ ≤ π. The angular coordinate 0 < φ < 2π − α
+is measured starting from one of the two adjacent edges of ∂Ω.
+∂Ω
+γ′
+γ
+ξ
+α
+φ
+π + θ − α
+θ
+Figure 5: Diagrams for the scattering at convex corners (π < α < 2π). The source point ξ is virtual, being
+used to model reflection off of edge γ. The dash-dotted line is the shadow boundary due to edge γ′. The
+shadow region is present if and only if α − π < θ < π. The angular coordinate 0 < φ < 2π − α is measured
+starting from one of the two adjacent edges of ∂Ω.
+Light cone Ωn.
+Since the diffracted wave propagates in all geometrically allowed directions, we define the
+support Ωn as the set of all points that are visible (along straight-line paths) from ξn, i.e.,
+Ωn = {x ∈ Ω : ξn + τ(x − ξn) ∈ Ω ∀0 < τ ≤ 1} .
+(15)
+Modeling diffraction is substantially more complicated than modeling reflection. For this reason, before
+we can describe how the remaining unknown quantities rn and ζn are defined, cf. (5), we need to introduce
+several assumptions.
+Assumption 4.1 (Separability). Diffracted waves are separable into radial-temporal and angular compo-
+nents around the diffraction point ξn.
+Otherwise stated, �un(x, t) can be expressed (at least locally) as
+ψn(∥x − ξn∥ , t)ζn(x − ξn), where ζn is positive-homogeneous, i.e., ζn(z) is independent of ∥z∥ (as long
+as z ̸= 0). Using an abuse of notation, we will express ζn as a function of φ only, with φ defined as the
+angular coordinate around ξn. See Figs. 4 and 5.
+This, together with the following assumption on the angular component ζn, will allow us to recover the
+approximation structure presented in Section 2.
+Assumption 4.2 (Piecewise-linear angular component). The angular component ζn is a piecewise-linear
+function of the angular coordinate φ, with discontinuities at all reflection and shadow boundaries. Using the
+geometrical optics approximation, we can explicitly compute the locations of such discontinuities:
+• at concave corners (see, e.g., Fig. 4), φ1 = |π − θ| = max{π−θ, θ−π} and φ2 = 2π−α−|π − α − θ| =
+min {θ + π, 3π − 2α − θ};
+• at convex corners (see, e.g., Fig. 5), φ3 = π + θ − α.
+Now we are ready to describe our full diffraction model, which satisfies Assumptions 4.1 and 4.2, as well
+as the following three standard requirements.
+8
+
+D. Pradovera, M. Nonino, and I. Perugia
+Geometry-based approximation of waves in complex domains
+Assumption 4.3 (Characterization of diffracting vertices). A vertex ξn emits a diffraction wave in “re-
+sponse” to �ui only if both following conditions are met:
+• ξn is visible from ξi, i.e., ξn ∈ Ωi;
+• one of the following is true:
+– the domain Ω is locally concave near ξn, with ξi being located on the “concave side” of ξn, i.e.,
+0 < α < π and 0 ≤ θ ≤ 2π − α in Fig. 4;
+or
+– the domain Ω is locally convex near ξn and a “shadow zone” is present, i.e., π < α < 2π and
+π − α < θ < π in Fig. 5.
+Assumption 4.4 (Continuity of the full approximation). The full wave approximation �u is continuous, in
+particular across reflection and shadow boundaries.
+Assumption 4.5 (Conservation of mass). Diffracted waves have zero “net mass”, i.e.,
+�
+R2 �un(x, t)dx = 0,
+leading to mass conservation of the full wave approximation �u. (Note that we are stating mass conservation
+in free space to ignore further reflections and diffractions of �un, which are also assumed to conserve mass.)
+Spatial correction rn.
+As in Algorithm 1, let i be the index of the term �ui that causes the diffraction
+�un. With the objective of satisfying (5) and Assumption 4.4, we define the radial component ψn as
+ψn(∥x − ξn∥ , t) = Ψ(∥x − ξn∥ + ∥ξn − ξi∥ + ri
+�
+��
+�
+=:rn
+, t).
+(16)
+By direct inspection of this definition, we can see that, by our choice of rn, we are “aligning” the wavefronts
+of the diffracted waves with the wavefronts of the reflected wave at the reflection boundaries (the shadow
+boundary of the reflected waves, if any) and with the wavefronts of the incoming wave �ui at its shadow
+boundary (if any). For instance, it is easy to see that, using (16), a point close to the diffraction point
+(x ≈ ξn) is within the support of the diffracted wave �un only for t ≥ rn − R, i.e., only when the wave �ui has
+crossed the distance from ξi to ξn.
+Weight function ζn.
+According to Assumption 4.2, we define the discontinuous piecewise-linear function
+ζn : [0, 2π − α] → R as
+ζn(φ) =
+�
+�
+�
+�
+�
+z1(φ1−φ)+z2φ
+φ1
+for 0 < φ < φ1 := |π − θ| ,
+z3(φ2−φ)+z4(φ−φ1)
+φ2−φ1
+for φ1 < φ < φ2 := 2π − α − |π − α − θ| ,
+z5(2π−α−φ)+z6(φ−φ2)
+2π−α−φ2
+for φ2 < φ < 2π − α,
+(17)
+for concave corners, and
+ζn(φ) =
+� z1(φ3−φ)+z7φ
+φ3
+for 0 < φ < φ3 := π + θ − α,
+z8(2π−α−φ)+z6(φ−φ3)
+2π−α−φ3
+for φ3 < φ < 2π − α,
+(18)
+for convex corners. The scalars z1, . . . , z8 are nodal values of ζn: ζn(0) = z1, ζn(φ+
+1 ) = z3 for concave corners,
+ζn(φ−
+3 ) = z7 for convex corners, etc. These values are chosen so as to satisfy:
+• The boundary conditions at the edges ending at ξn, i.e., γ and γ′.
+• Assumption 4.4 at the discontinuity angles φ1, φ2, and φ3. To this aim, we prescribe values for the
+jumps (z3 − z2), (z5 − z4), and (z8 − z7).
+• Assumption 4.5. Given the radial-angular decomposition of �un from Assumption 4.1, this is equivalent
+to the condition
+� 2π−α
+0
+ζn(φ)dφ = 0.
+9
+
+D. Pradovera, M. Nonino, and I. Perugia
+Geometry-based approximation of waves in complex domains
++
+=
++
+=
+Figure 6: Examples of diffraction at the concave (top) and convex (bottom) corners from Fig. 4 (right) and
+Fig. 5. Neumann conditions are imposed on all edges. In each row of plots, we have: discontinuous wave
+without diffraction (left), diffraction wave (middle), and continuous wave with diffraction (right). Note that,
+in the convex case, we are not showing the wave �ui that causes the reflection off edge γ nor the reflection
+and scattering of �ui off edge γ′.
+In the case of a convex corner, this set of condition uniquely identifies the four degrees of freedom. See
+Section 4.1 for the formulas and for their derivation.
+However, in the case concave case, an additional
+condition is required. In this work, we set this last condition as described in Section 4.2. We show in Fig. 6
+the results obtained with our diffraction modeling in two simple illustrative cases.
+Before proceeding further, we deem it important to make the following remark.
+Remark 4.6. Our proposed strategy is able to deliver only a fairly crude approximation of diffraction effects.
+(We refer to Section 5.1 for a validation of our model.) However, it has the great advantage of being extremely
+simple to build and to evaluate. Thanks to the modularity of our approach, it would be surely possible to
+replace our diffraction model with more sophisticated ones (e.g., removing Assumptions 4.1 and 4.2), in the
+interest of achieving a better approximation of the exact solution. To this aim, we mention that a wide body
+of works has been dedicated to the modeling of diffraction in the time-harmonic (Helmholtz) setting: among
+others, we name the geometrical [20] and uniform [23] theories of diffraction. However, the authors have
+not been able to find any satisfactory all-purpose time-domain diffraction modeling in the literature.
+4.1
+Convex diffraction coefficients
+Consider the situation depicted in Fig. 5 and the notation introduced therein. Also, we rely on the quantities
+ξn, rn, Ωn, and i introduced in Section 4. For diffraction to happen, cf. Assumption 4.3, �ui must be a wave
+reflected off either edge γ or γ′. Indeed, a convex vertex ξn cannot be “hit” from outside the domain Ω by
+the source wave �u1, nor by any wave reflected off a different edge, nor by any diffracted wave centered at
+some vertex of ∂Ω.
+For this reason, the shadow boundary {φ = φ3} must belong to ∂Ωi (the boundary of the light cone Ωi),
+at least locally around ξn. Without loss of generality, we assume that �ui is a wave reflected off edge γ, so
+that Ωi consists (locally) of point whose angular coordinate is 0 < φ < φ3. This means that φ3 < φ < 2π −α
+is a shadow zone. The alternative case (of reflection off edge γ′) can be obtained by symmetry.
+Let σγ = 0 if γ is a Dirichlet edge and σγ = 1 if it is a Neumann edge. Define σγ′ similarly for edge γ′.
+To satisfy the conditions described in Section 4, the quantities z1, z7, z8, z6 appearing in the angular weight
+ζn, cf. (18), must satisfy the conditions:
+• Boundary condition at γ: z1 = σγz7.
+• Boundary condition at γ′: z6 = σγ′z8.
+• For �u to be continuous at φ = φ3, there must be a jump to account for the fact that �ui is nonzero for
+φ → φ−
+3 but zero for φ → φ+
+3 : given the angular component of �ui at the shadow boundary, namely,
+10
+
+D. Pradovera, M. Nonino, and I. Perugia
+Geometry-based approximation of waves in complex domains
+hi := ζi(ξn − ξi), we impose z8 − z7 = hi.
+• Conservation of mass: for all t > 0,
+0 =
+�
+R2 �un(x, t)dx =
+� ∞
+0
+� 2π−α
+0
+ψn(ρ, t)ζn(φ)ρdφdρ
+=
+�� 2π−α
+0
+ζn(φ)dφ
+� �� ∞
+0
+ψn(ρ, t)ρdρ
+�
+=
+�z1 + z7
+2
+φ3 + z8 + z6
+2
+(2π − α − φ3)
+� �� ∞
+0
+ψn(ρ, t)ρdρ
+�
+,
+which leads to the condition z1+z7
+2
+φ3 + z8+z6
+2
+(2π − α − φ3) = 0.
+With simple algebra, we now obtain
+z7 =
+(σγ′ + 1)hi(φ3 + α − 2π)
+(σγ′ + 1)(2π − α) + (σγ − σγ′)φ3
+,
+z8 =
+(σγ + 1)hiφ3
+(σγ′ + 1)(2π − α) + (σγ − σγ′)φ3
+,
+(19)
+as well as z1 = σγz7 and z6 = σγ′z8. See Fig. 6 (bottom) for an example of the resulting diffraction wave.
+4.2
+Concave diffraction coefficients
+Consider the setup depicted in Fig. 4 and the notation introduced therein. Also, we rely on the quantities
+ξn, rn, Ωn, and i introduced in Section 4. Let σγ = 0 if γ is a Dirichlet edge and σγ = 1 if it is a Neumann
+edge. Define σγ′ similarly for edge γ′. Without loss of generality, we assume that 0 < θ ≤ π − α
+2 , since the
+other case can be easily obtained by symmetry. In this setting, γ is in the light cone of �ui (at least locally
+around ξn). Accordingly, let �ui′ be the wave component obtained by reflection of �ui off edge γ.
+If 0 < θ ≤ π − α, γ′ is not in the light cone of �ui, so that:
+• for 0 < φ < φ1 = π − θ, both �ui and �ui′ are present;
+• for φ1 < φ < φ2 = π + θ, only �ui is present, since the light cone Ωi′ ends at {φ = φ1};
+• for φ2 < φ < 2π − α, neither �ui nor �ui′ is present, i.e., we have a shadow zone, since the light cone Ωi
+ends at {φ = φ2}.
+Otherwise, assume that π − α < θ ≤ π − α
+2 . In this case, γ′ is also in the light cone of �ui (at least locally
+around ξn). We denote the wave component obtained by reflection of �ui off edge γ′ by �ui′′. Then:
+• for 0 < φ < φ1 = π − θ, both �ui and �ui′ are present;
+• for φ1 < φ < φ2 = 3π − 2α − θ, only �ui is present, since the light cones Ωi′ and Ωi′′ end at {φ = φ1}
+and at {φ = φ2}, respectively;
+• for φ2 < φ < 2π − α, both �ui and �ui′′ are present.
+To satisfy the conditions described in Section 4, the quantities z1, . . . , z6 appearing in the angular weight
+ζn, cf. (17), must satisfy the conditions:
+• Boundary condition at γ: z1 = σγz2.
+• Boundary condition at γ′: z6 = σγ′z5.
+• For �u to be continuous at φ = φ1, there must be a jump whose height is the angular component of �ui′
+at φ = φ1, i.e., hi′ := ζi′(ξn −ξi′); we impose z3 −z2 = hi′. Note that, by the law of reflection, cf. (11),
+hi′ = τhi, with τ = 2σγ − 1 and hi := ζi(ξn − ξi).
+• For �u to be continuous at φ = φ2, there must be a jump whose height depends on whether θ ≤ π − α
+or not:
+11
+
+D. Pradovera, M. Nonino, and I. Perugia
+Geometry-based approximation of waves in complex domains
+– If 0 < θ ≤ π − α, the jump equals the angular component of �ui at φ = φ2, i.e., hi := ζi(ξn − ξi);
+we impose z5 − z4 = hi.
+– If π − α < θ ≤ π − α
+2 , the jump equals minus the angular component of �ui′′ at φ = φ2, i.e.,
+hi′′ := ζi′′(ξn − ξi′′); we impose z5 − z4 = −hi′′. Note that, by the law of reflection, cf. (11),
+hi′′ = (2σγ′ − 1)hi, with hi := ζi(ξn − ξi).
+In summary, z5 − z4 = τ ′hi, with τ ′ = 1 if 0 < θ ≤ π − α and τ ′ = 1 − 2σγ′ if π − α < θ ≤ π − α
+2 .
+• Conservation of mass: for all t > 0,
+0 =
+�
+R2 �un(x, t)dx =
+� ∞
+0
+� 2π−α
+0
+ψn(ρ, t)ζn(φ)ρdφdρ
+=
+�� 2π−α
+0
+ζn(φ)dφ
+� �� ∞
+0
+ψn(ρ, t)ρdρ
+�
+�
+��
+�
+=:C(t)
+=
+�z1 + z2
+2
+φ1 + z3 + z4
+2
+(φ2 − φ1) + z5 + z6
+2
+(2π − α − φ2)
+�
+C(t),
+which leads to the condition z1+z2
+2
+φ1 + z3+z4
+2
+(φ2 − φ1) + z5+z6
+2
+(2π − α − φ2) = 0.
+One constraint is missing for the values z1, . . . , z6 to be uniquely determined. Specifically, some simple
+algebra shows that, for any δ ∈ R, the following set of values satisfies all the above conditions:
+z2 =
+τhi + δ
+(σγφ1 + φ2)/(φ1 − φ2),
+z5 =
+τ ′hi + δ
+(φ1 + σγ′φ2 − (σγ′ + 1)(2π − α))/(φ1 − φ2),
+(20)
+together with z1 = σγz2, z3 = z2 + τhi, z4 = z5 − τ ′hi, and z6 = σγ′z5. Note that the jump heights τhi and
+τ ′hi appear in the numerators of z2 and z5, respectively.
+Our diffraction model is simply the one given by δ = 0. Intuitively, this corresponds to a “balanced”
+partitioning of the mass of the diffracted wave into the components related to the two (reflection and/or
+shadow) boundaries φ1 and φ2. See Fig. 6 (top) for an example of the resulting diffraction wave.
+To further highlight the (mostly beneficial) effects of the choice δ = 0, we note that:
+• in the symmetric case θ = π − α
+2 , δ = 0 leads to a symmetric ζn: (1 − 2σγ′)z2 = (1 − 2σγ)z5;
+• in the case θ = π − α, the second transition happens at γ′, i.e., φ2 = 2π − α, and the choice δ = 0
+yields z4 = 0, so that the diffraction wave vanishes at γ′ (which is physically sound);
+• in the “grazing incidence” case θ = 0, the two transition coalesce into one, i.e., φ1 = φ2, and the choice
+δ = 0 leads to ζn(φ) = 0 for all φ, which is unphysical; see the following remark for a possible solution.
+Remark 4.7. In the “grazing incidence” case θ = 0 (which corresponds to φ1 = φ2 = π), the diffraction
+wave �un does not cure the discontinuity of �u at the boundary {φ = π}. This is because, in some sense, the
+two discontinuities of �un at φ1 and φ2 cancel each other out. For a similar reason, a small θ ≈ 0 will result
+in a continuous total wave, but a sharp gradient will be present for φ1 < φ < φ2.
+By tweaking the value of δ, we can obtain an alternative diffraction model, which guarantees Assump-
+tion 4.4 even in the case of grazing incidence. To this aim, we can set
+δ =
+�
+(σγ + 1) σγφ1 + φ2 − (σγ′ + 1)(2π − α)
+(σγ − σγ′)φ1 + (σγ′ + 1)(2π − α)φ1τ+
++(σγ′ + 1)
+φ1 + σγφ2
+(σγ − σγ′)φ2 + (σγ′ + 1)(2π − α)(φ2 − 2π + α)τ ′
+�
+hi
+φ1 − φ2
+.
+(21)
+Roughly speaking, this corresponds to a different “balancing” of the mass of the diffracted wave into the
+components related to the two boundaries φ1 and φ2.
+This being said, in our numerical tests, such alternative model, albeit recovering a continuous total wave,
+resulted in a reduced accuracy of approximation. Specifically, using the value of δ above, we have observed
+an exaggerated magnitude of the diffraction wave, especially in the shadow zone.
+12
+
+D. Pradovera, M. Nonino, and I. Perugia
+Geometry-based approximation of waves in complex domains
+Example
+exterior
+incidence
+∥�u(·, T)∥L2(Ω)
+∥�u(·, T) − uFEM(·, T)∥L2(Ω)
+index
+angle α
+angle ω
+#1
+4.712
+0.984
+2.50 · 10−1
+3.65 · 10−4
+#2
+5.093
+0.603
+2.02 · 10−3
+#3
+2.761
+1.363
+1.05 · 10−2
+#4
+2.761
+3.277
+2.00 · 10−2
+Table 1: Setup for the four wedge examples. The angle ω is as in Fig. 7.
+ω
+ω
+ω
+ω
+−1
+0
+1
+Figure 7: Initial conditions for the wedge examples, indexed #1 through #4 from left to right. The (dashed)
+distance between the center of the Gaussian and the boundary vertex is 4 units in all cases.
+5
+Numerical results
+In our experiments, we require a “reference” solution of (1) to validate our results. To this effect, we use the
+solution uFEM obtained by discretizing (1) with:
+• the P1-finite element method (P1-FEM) with mass-lumping, over a regular triangulation (mesh) of the
+physical domain Ω;
+• explicit leapfrog timestepping with a uniform time step that satisfies the CFL condition on the chosen
+mesh.
+See [11, 16] for more details on this discretization strategy.
+If the domain Ω is unbounded, we first need to truncate it in such a way that reflections from the non-
+physical truncation boundary do not affect the solution in the region of interest for t < T. Recalling that
+the problem data are supported in a ball of radius R and center 0, this can be done, e.g., by truncating Ω
+at the sphere with radius R + T and center 0. In our tests, we rely on FEniCS [1] to carry out the P1-FEM
+discretization on 2-dimensional domains Ω.
+Instead, note that with our proposed approach, modeling unbounded domains is straightforward. Indeed,
+we can simply ignore any reflection or diffraction from its “infinitely far” vertices/edges.
+All our tests are performed in Python 3.8 on a machine with an 8-core 3.60 GHz Intel® processor
+and 64 GB of RAM. For reproducibility, our code is made available at https://github.com/pradovera/
+ray-wave-2d.
+5.1
+Some simple wedges
+As a way to assess our proposed method in simple settings, we consider four different “wedge” domains.
+Similarly to the diagrams in Figs. 4 and 5, we define Ω to be one of the portions of the plane R2 delimited by
+straight lines intersecting at a point. Locally around such point, Ω “looks” like either Fig. 4 or Fig. 5, with
+α being the outer angle. The specific choices of wedge angles α are reported in Table 1 for the four cases.
+We set up a problem of the form (1), with u0 an isotropic Gaussian with standard deviation 0.2. The
+center of u0 is at a point located at a 4-unit distance from the wedge vertex, in the direction determined by
+the “incidence angle” ω. See Fig. 7 for a representation of the initial conditions in the four cases. We set
+u1 = f = 0, we enforce Neumann boundary conditions on the whole ∂Ω, and we seek the solution at the
+final time T = 5, i.e., 1 time unit after the wave crest has reached the wedge vertex.
+13
+
+D. Pradovera, M. Nonino, and I. Perugia
+Geometry-based approximation of waves in complex domains
+0
+1
+2
+3
+4
+5
+0
+2
+4
+6
+t
+ρ
+−1
+−0.5
+0
+0.5
+1
+Figure 8: Free-space solution Ψ. The dashed line denotes the upper bound of the “causality cone” of Ψ, i.e.,
+ρ = t + R, with R = 1.
+�u
+uFEM
+�u − uFEM
+Example #1
+Example #2
+Example #3
+Example #4
+−0.1
+0
+0.1
+−2 · 10−4
+0
+2 · 10−4
+−0.1
+0
+0.1
+−5 · 10−3
+0
+5 · 10−3
+−0.1
+0
+0.1
+−1 · 10−2
+0
+1 · 10−2
+−0.1
+0
+0.1
+−5 · 10−2
+0
+5 · 10−2
+Figure 9: Results for the four wedge examples. Each row pertains to a different example. In each row, from
+left to right: surrogate solution, FE solution, and error. The color scales for the first two columns are the
+same. All results are shown at the final time t = T.
+14
+
+AD. Pradovera, M. Nonino, and I. Perugia
+Geometry-based approximation of waves in complex domains
+To this aim, we employ our proposed approach, see Section 2. First, we compute an approximation of
+the free-space solution Ψ, which solves (4), by employing P1-FEM with explicit leapfrog timestepping. Note
+that, since (4) is cast in polar coordinates, we only need to discretize a 1D interval with P1-FEM. Since the
+initial condition u0 is supported within the unit disk, we have R = 1, and it suffices to approximate Ψ(ρ, t)
+for (ρ, t) ∈ [0, T + R] × [0, T]. Since this space-time domain is only 2-dimensional, we can afford even a very
+fine discretization. In our experiments, we employ a 1001 × 2001 uniform Cartesian space-time grid, i.e.,
+the mesh size is δx = T +R
+1000 and the time step is δt =
+T
+2000. This satisfies the CFL condition. We show the
+resulting Ψ (which, in fact, we should denote by ΨFEM) in Fig. 8.
+After this preliminary step, we use the timetable-based strategy from Section 2 to identify reflection
+and scattering effects, which are then added up to give the final approximation �u. We show the resulting
+�u(·, T) in Fig. 9. In this figure, we also display a reference solution uFEM(·, T), which we obtain by direct
+discretization of (1) by P1-FEM and leapfrog timestepping, as described at the beginning of Section 5.
+In all four examples, we see that �u and the reference uFEM seem qualitatively close. Notably, we can
+observe a good representation of the most prominent wavefronts, which are due to propagation of either the
+main “free-space” wave or to its reflections. Indeed, those wave contributions are reconstructed exactly: the
+only errors are the ones due to FE approximation and timestepping, which affect both uFEM and �u (the
+latter through the approximation of Ψ). Instead, some differences are present when comparing diffraction
+effects, which arise as circular waves about the wedge vertex. We can quantitatively observe this in the last
+column of both Table 1 and Fig. 9.
+In example #1, we observe a very small error, which, in fact, is simply the (FEM and timestepping)
+discretization error. This is related to the fact that the wedge has exterior angle α =
+3
+2π, which makes
+diffraction unnecessary in approximating the wave u: reflections are enough2.
+In the other examples, diffraction effects are necessary to correctly identify u. While a good qualitative
+behavior can be observed in Fig. 9, we can see in Table 1 that a modest error is present. Specifically, we
+report the L2(Ω)-norm of �u and of the error �u − uFEM at the final time t = T, defined as
+∥v∥L2(Ω) =
+��
+Ω
+v(x)2dx
+�1/2
+.
+(22)
+We see the largest error in example #4, where the relative L2(Ω)-approximation error amounts to 8%.
+This was to be expected, since this last example is rather close to the setting of grazing incidence (α+ω ≈ 2π),
+which, as discussed in Section 4.2, is approximated rather poorly by our diffraction model. Qualitatively,
+the bad approximation quality is apparent in the form of a rather sharp gradient in the corresponding plot
+of �u in Fig. 9 (bottom left).
+5.1.1
+Building a cavity out of wedges
+As a slightly more complicated example, we now combine the four wedges from the previous section to obtain
+the open cavity represented in Fig. 10. In this case, more reflection and diffraction effects will arise, due
+to the trapping nature of the domain. Our initial conditions and forcing term are the same as before, but
+now all edges are sound-soft. Accordingly, we model them using Dirichlet boundary conditions. The time
+horizon is T = 9.
+Using our strategy from Section 2, we build the approximation �u, which contains 47 wave terms (1 source
+wave, 32 reflected waves, and 14 diffraction waves). We compare the approximation �u with the reference
+solution uFEM, obtained as described at the beginning of Section 5.
+We show the results of the comparison in Fig. 10, at 4 time instants t ∈ {0, 3, 6, 9}. Once more, we see
+a good qualitative agreement between �u and uFEM, with the most important features of u being identified
+well.
+Up to t = 3, only reflections have happened, so that the error �u − uFEM consists only of FEM
+and timestepping errors. On the other hand, for larger times, we see “error waves” of small-to-moderate
+amplitude propagating from the 3 vertices of Ω that generate diffraction effects. These correspond to errors
+in diffraction modeling.
+2To intuitively understand why, let γ1 and γ2 be the two sides forming ∂Ω. The domain Ω is partitioned exactly into (i)
+the light cone of the reflection off γ1 and then off γ2 and (ii) the light cone of the reflection off γ2 and then off γ1. For this
+reason, the diffraction effects due to these two rays cancel out. Incidentally, the same phenomenon can be expected whenever
+the interior angle 2π ��� α is of the form π
+n , with n ∈ {2, 3, . . .}.
+15
+
+D. Pradovera, M. Nonino, and I. Perugia
+Geometry-based approximation of waves in complex domains
+�u
+uFEM
+�u − uFEM
+t = 0
+t = 3
+t = 6
+t = 9
+−1
+−0.5
+0
+0.5
+1
+−0.1
+0
+0.1
+−5
+0
+5
+·10−4
+−0.1
+0
+0.1
+−5
+0
+5
+·10−2
+−0.1
+0
+0.1
+−5
+0
+5
+·10−2
+Figure 10: Results for the cavity domain. Each row corresponds to a different time instant t ∈ {0, 3, 6, 9},
+from top to bottom. In each row, from left to right: surrogate solution �u(·, t), FE solution uFEM(·, t), and
+error �u(·, t) − uFEM(·, t). The color scales for the first two columns are the same.
+16
+
+D. Pradovera, M. Nonino, and I. Perugia
+Geometry-based approximation of waves in complex domains
+5.2
+A tall room
+We consider a simplified sound propagation problem in a room. For simplicity, we consider a 2-dimensional
+problem, thus assuming an infinitely tall room, and modeling line sources (in the z-direction) as point sources.
+The complicated domain Ω ⊂ R2 is depicted in Fig. 11. It is composed of two communicating “rooms”
+with sound-hard walls, as well as of a third large room (above), which is modeled as infinitely large. In the
+main room, three sound-soft triangular obstacles are also present.
+Setting once more u1 = f = 0, we are interested in modeling the propagation of an initial condition
+u0 modeled as a Ricker wavelet centered at 0, see Fig. 11 (top left), over the time horizon t ∈ [0, T], with
+T = 20. To this aim, we employ our proposed method from Section 2.
+As in the previous example, we start by computing an approximation of the free-space solution Ψ = Ψ(ρ, t)
+for (ρ, t) ∈ [0, T + R] × [0, T], see (4), with R being the radius of the support of the initial condition u0.
+Again, we use P1-FEM with leapfrog timestepping for this.
+Since many reflective surfaces face each other, the domain Ω is trapping. Accordingly, we expect the
+number N of waves in the approximation �u to be rather large. In the interest of reducing the number of
+such terms, we can employ the on-the-fly parsimonious strategy described in Remark 2.2, removing all wave
+terms �un whose magnitude is smaller than tol = 10−2. After this, N ≈ 1.4 · 103 terms are left. Although
+this value of N may seem large, the evaluation of the corresponding surrogate �u is rather quick, due to the
+explicit nature of each wave contribution (and to the fact that their supports are smaller than the whole Ω).
+We show the resulting u(·, t) for the four times t ∈ {0, 7.5, 15, 20} in Fig. 11. There, we can see why so
+many terms are necessary for the approximation of u: we must model many reflection and diffraction effects.
+Since energy escapes the system only through the top “door”, the wave will persist for quite a long time.
+Accordingly, a larger T will make a larger N necessary.
+In order to better inspect this effect, we show the trace of the solution at the arbitrarily chosen point
+xtrace = (−1, −2) in Fig. 12. We notice that oscillations persist for t > 10. We use this last plot also to
+validate our results. To this aim, we compare three results:
+• The surrogate �u obtained as described above, with tol = 10−2.
+• The surrogate �u obtained with our strategy, but with tol = 10−3. This leads to an increased number
+of rays N ≈ 7.3 · 103.
+• The reference solution uFEM obtained by the P1-FEM with leapfrog timestepping, as described at the
+beginning of Section 5. The mesh size must be chosen small enough to resolve both the initial condition
+and the domain Ω well. In our case, we have a mesh with approximately 1.4 · 106 elements. To satisfy
+the CFL condition on this mesh, we choose a time step ∆t ≈ 7 · 10−3.
+We can observe that the two surrogates obtained with our approach give very similar results. Indeed,
+the cutoff tolerance tol affects the results only for large t > 15, due to the accumulation of “small” waves
+that are excluded from the coarser surrogate but included in the finer one.
+Moreover, taking the FE solution as reference, we see that most of the peaks of the surrogates are
+aligned with the FE ones (i.e., the “phase” of the wave is well approximated), but there are some noticeable
+discrepancies in their amplitudes. This is due to the fact that, in our approach, reflection is modeled exactly,
+whereas the magnitudes of the diffraction waves are only roughly approximated. For this reason, we should
+not expect the amplitude error to get smaller if we reduce tol.
+The only “real” way of improving the
+approximation is using a better diffraction modeling.
+As a final result, we also report:
+• The so-called “offline” time, i.e., the time required to compute the numerical solution. For �u, this
+means executing Algorithm 1. For uFEM, this means building the mesh, assembling the FE stiffness
+and (lumped) mass matrices, and carrying out the timestepping.
+• The so-called “online” time, i.e., the time required to evaluate the numerical solution (�u or uFEM) at
+a single (x, t)-point.
+They can be found in Table 2.
+17
+
+D. Pradovera, M. Nonino, and I. Perugia
+Geometry-based approximation of waves in complex domains
+−1
+0
+1
+t = 0
+−0.1
+0
+0.1
+t = 7.5
+−0.1
+0
+0.1
+t = 15
+−0.1
+0
+0.1
+t = 20
+Figure 11: 2-dimensional domain Ω modeling a room. Top left plot: initial condition �u(·, 0) = u(·, 0) = u0,
+a Ricker wavelet; we also show the point xtrace as a cross. Top right plot: intermediate solution �u(·, 7.5).
+Bottom left plot: intermediate solution �u(·, 15). Bottom right plot: final solution �u(·, 20).
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+−0.1
+0
+0.1
+t
+u(xtrace, t)
+�u (tol = 10−2)
+�u (tol = 10−3)
+uFEM
+Figure 12: Value of solution at point xtrace = (−1, −2).
+Method
+�u (tol = 10−2)
+�u (tol = 10−3)
+uFEM
+Offline
+46.66 [s]
+252.9 [s]
+188.9 [s]
+Online
+2.04 [ms]
+8.48 [ms]
+27.67 [µs]
+Table 2: Timings for the room test case. To obtain more statistically significant results, each displayed time
+is the average over 3 (resp. 103) runs of the offline (resp. online) phase with identical parameters.
+18
+
+D. Pradovera, M. Nonino, and I. Perugia
+Geometry-based approximation of waves in complex domains
+We can observe the increased training and evaluation time that results from decreasing tol. Moreover, we
+see that, in this example, our proposed approach is more competitive offline, but less so online. Somewhat
+surprisingly, we have observed that the most expensive step in evaluating �u (taking about half of the online
+time) is determining whether an evaluation point is in the light cones Ωn. The reason for this is that they
+can have rather complicated shapes, cf. Section 3.
+Evaluating the FE solution at a space-time point is an extremely cheap operation (essentially corre-
+sponding to a vector dot product) while evaluating �u is more expensive, requiring the evaluation of O(N)
+nonlinear functions. However, the FE solution comes with the serious drawback of memory usage. Indeed, in
+our example, storing uFEM as a (∼ 1.5 · 106) × (∼ 2.9 · 103) array of double-precision floating-point numbers
+requires approximately 34 GB.
+Concerning the timing results, we also wish to mention that the online times in Table 2 should be
+interpreted carefully. Indeed, the online time for the FE solution is artificially deflated by the fact that
+xtrace is a vertex of the FE mesh: each evaluation of uFEM corresponds to extracting a vector entry. If xtrace
+had not been a vertex of the mesh, the online time could have been larger by at least one order of magnitude,
+if not more, depending on the FE implementation. Moreover, we note that accessing point-evaluations of
+uFEM at arbitrary times a posteriori, namely, after the timestepping has been carried out, is feasible only if
+enough memory is available to store the whole “timestepping history”. Considering the numbers mentioned
+in the previous paragraph, this might not be possible in practice, especially for more complex and/or larger
+domains.
+5.2.1
+A time-harmonic source
+One of the advantages of our approach is that it allows changing the source terms of the problem in a
+seamless way. Notably, under minor technical constraints (e.g., the support of the new source term should
+not be larger than the old one), this kind of change does not require training a new surrogate.
+To showcase this, we approximate the wave propagating from a time-harmonic point source at x = 0
+with angular frequency ω > 0. In our tests, we pick ω ∈ {2π, 10π}. To this aim, we define u as the solution
+of the following (ω-dependent) problem:
+�
+�
+�
+�
+�
+�
+�
+�
+�
+∂ttu(x, t) = ∆u(x, t) − ω2 sin(ωt)δ0(x)
+for (x, t) ∈ Ω × (0, T),
+u(x, 0) = 0
+for x ∈ Ω,
+∂tu(x, 0) = 0
+for x ∈ Ω,
+∂νu(x, t) = 0
+for (x, t) ∈ ∂Ω × (0, T],
+(23)
+where δ0 denotes the usual 2-dimensional “delta function” centered at x = 0.
+As usual, we define Ψ = Ψ(ρ, t) as the (ω-dependent) solution of the free-space version of (23) in radial-
+temporal coordinates. Note that, in free space, i.e., without boundary effects3, the forcing term in (23) is
+equivalent to the following non-homogeneous Dirichlet-like condition at ρ = 0:
+U(0, t) = Ψ(0, t) =
+� t
+0
+� t′
+0
+−ω2 sin(ωt′′)dt′′dt′ = sin(ωt)
+∀t > 0.
+(24)
+Accordingly, the free-space solution Ψ has space-time support {(ρ, t) ∈ [0, ∞)2, ρ ≤ t}, which is a subset of
+the free-space solution Ψ from the previous section, namely, {(ρ, t) ∈ [0, ∞)2, ρ ≤ t + R}. As such, to obtain
+an approximation for the wave u generated by the time-harmonic source for an arbitrary ω, it suffices to
+plug the corresponding Ψ in each term of the surrogate �u from the previous section! We show the results of
+our approximation in Figs. 13 and 14.
+We note that, if we had chosen to apply the FEM to approximate the wave u generated by the time-
+harmonic source, we would have been forced to carry out a new simulation from scratch for every frequency
+to be studied. To this end, we would have needed to choose a mesh with ω-dependent resolution: the mesh
+size should be small enough for the well-known pollution effect (see, e.g., [22, 2]) to be absent.
+3If Ω is bounded, reflected or diffracted waves will generally bounce back to x = 0. As such, the value of u(0, t), u being
+the solution of (23), will be different from the source signal sin(ωt). For this reason, we cannot turn the forcing term in (23)
+into a condition like (24), except in free space.
+19
+
+D. Pradovera, M. Nonino, and I. Perugia
+Geometry-based approximation of waves in complex domains
+−0.4
+−0.2
+0
+0.2
+0.4
+−0.2
+0
+0.2
+Figure 13: Surrogate solution found with the proposed approach. Left plot: ω = 2π. Right plot: ω = 10π.
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+−0.2
+−0.1
+0
+0.1
+0.2
+t
+u(xtrace, t)
+ω = 2π
+ω = 10π
+Figure 14: Value of solution at point xtrace = (−1, −2) for different excitation frequencies.
+In our proposed approach, we also have a constraint on the mesh resolution. However, it only applies
+to the problem defining the free-space solution Ψ, which is 1-dimensional in space. Hence, having to refine
+the mesh represents a much smaller obstacle to efficiency. In particular, for a fixed time horizon T, the
+computation of �u becomes more and more efficient, when compared to the computation of uFEM, as the
+frequency ω increases.
+6
+Conclusions
+We have presented a method for approximating waves propagating through complex 2-dimensional domains
+with polygonal boundaries. Our method relies on the automatic identification of reflection and diffraction
+effects caused by the domain geometry. Each effect is modeled through a relatively simple nonlinear expres-
+sion. In our numerical tests, we have observed rather a good approximation quality, with the main features
+of the target wave being well identified. As a way to improve the approximation accuracy, we recall that any
+diffraction model could replace the current one.
+In terms of complexity, our method requires the solution of a simplified 1D-in-space problem, much
+simpler than the original 2D-in-space one. We expect such improved accuracy to increase even further if
+one were to consider 3D instead of 2D problems. However, in order to generalize our method to 3 space
+dimensions, a suitable diffraction model would be necessary. This is one of our ongoing research directions.
+Another favorable aspect of our algorithm is its potential to be run on parallel architectures, since
+the computation of different rays can be carried out independently.
+This is not the case for standard
+timestepping-based discretizations, due to their intrinsically sequential nature.
+Other envisioned extensions of our technique involve the cases of domains with curvilinear boundaries
+and of propagation media with non-uniform properties (e.g., density and refraction index). Specifically, this
+latter case would effectively result in a non-uniform wave speed, with reflections and diffractions happening
+20
+
+D. Pradovera, M. Nonino, and I. Perugia
+Geometry-based approximation of waves in complex domains
+also within the domain Ω.
+Finally, we recall that, in many applications, the ultimate target is understanding how the wave u solving
+(1) depends on underlying parameters µ, e.g., the forcing term f, the shape of the domain Ω, etc. In this
+setting, MOR methods try to construct a surrogate model of the form �u = �u(x, t; µ), providing a good
+approximation of u over a whole range of parameter values. Even though our technique was presented here
+in the non-parametric setting, we believe that it potentially allows incorporating the parameter dependence
+in a natural and efficient way. In our opinion, this might be achievable by leveraging the simple and in-
+terpretable structure of the rays (free-space solution, light cone, and angular modulation). As a simple
+preliminary example, we showcased this in Section 5.2.1 for a parametric source term, with the parame-
+ter being the frequency. We are currently investigating how to extend our method to more complicated
+parametric problems.
+References
+[1] M. S. Alnæs, J. Blechta, J. Hake, and Others. The FEniCS Project version 1.5. Archive of Numerical
+Software, 3(100), 2015.
+[2] I. M. Babuˇska and S. A. Sauter. Is the pollution effect of the fem avoidable for the helmholtz equation
+considering high wave numbers? SIAM Journal on Numerical Analysis, 34(6):2392–2423, 1997.
+[3] P. Benner, M. Ohlberger, A. Cohen, and K. Willcox. Model reduction and approximation: theory and
+algorithms. SIAM, 2017.
+[4] F. Bonizzoni, F. Nobile, I. Perugia, and D. Pradovera. Least-Squares Pad´e approximation of parametric
+and stochastic Helmholtz maps. Advances in Computational Mathematics, 46(3):46, 2020.
+[5] L. Borcea, V. Druskin, A. V. Mamonov, and M. Zaslavsky.
+Robust nonlinear processing of active
+array data in inverse scattering via truncated reduced order models. Journal of Computational Physics,
+381:1–26, 2019.
+[6] L. Borcea, V. Druskin, A. V. Mamonov, M. Zaslavsky, and J. Zimmerling.
+Reduced order model
+approach to inverse scattering. SIAM Journal on Imaging Sciences, 13(2):685–723, 2020.
+[7] L. Borcea, G. Papanicolaou, C. Tsogka, and J. Berryman. Imaging and time reversal in random media.
+Inverse Problems, 18(5), 2002.
+[8] P. Buchfink, S. Glas, and B. Haasdonk.
+Symplectic Model Reduction of Hamiltonian Systems on
+Nonlinear Manifolds. arXiv preprint arXiv:2112.10815, 2021.
+[9] P. Buchfink, B. Haasdonk, and S. Rave. PSD-Greedy Basis Generation for Structure-Preserving Model
+Order Reduction of Hamiltonian Systems. In Proceedings of ALGORITMY, pages 151–160, 2020.
+[10] N. Cagniart, Y. Maday, and B. Stamm. Model Order Reduction for Problems with Large Convection
+Effects, pages 131–150. Springer International Publishing, Cham, 2019.
+[11] G. Cohen, A. Hauck, M. Kaltenbacher, and T. Otsuru. Different Types of Finite Elements. In S. Marburg
+and B. Nolte, editors, Computational Acoustics of Noise Propagation in Fluids - Finite and Boundary
+Element Methods, pages 57–88. Springer Berlin Heidelberg, Berlin, Heidelberg, 2008.
+[12] R. A. DeVore. The theoretical foundation of reduced basis methods. Model reduction and approximation:
+theory and algorithms, 15:137, 2017.
+[13] S. Esterhazy and J. M. Melenk. On stability of discretizations of the Helmholtz equation. In Numerical
+analysis of multiscale problems, pages 285–324. Springer, 2012.
+[14] S. Glas, A. T. Patera, and K. Urban. A reduced basis method for the wave equation. International
+Journal of Computational Fluid Dynamics, 34(2):139–146, 2020.
+21
+
+D. Pradovera, M. Nonino, and I. Perugia
+Geometry-based approximation of waves in complex domains
+[15] C. Greif and K. Urban. Decay of the Kolmogorov N-width for wave problems. Applied Mathematics
+Letters, 96:216–222, 2019.
+[16] E. Hairer, G. Wanner, and C. Lubich. Geometric Numerical Integration, volume 31 of Springer Series
+in Computational Mathematics. Springer Berlin Heidelberg, Berlin, Heidelberg, 2002.
+[17] J. S. Hesthaven and C. Pagliantini. Structure-preserving reduced basis methods for Poisson systems.
+Mathematics of Computation, 90(330):1701–1740, 2021.
+[18] J. S. Hesthaven, C. Pagliantini, and N. Ripamonti. Rank-adaptive structure-preserving reduced basis
+methods for Hamiltonian systems. arXiv preprint arXiv:2007.13153, 2020.
+[19] R. Hiptmair, A. Moiola, and I. Perugia. Trefftz discontinuous Galerkin methods for acoustic scattering
+on locally refined meshes. Applied numerical mathematics, 79:79–91, 2014.
+[20] J. B. Keller. Geometrical theory of diffraction. J. Opt. Soc. Am., 52(2):116–130, 1962.
+[21] K. Lee and K. T. Carlberg. Deep conservation: A latent-dynamics model for exact satisfaction of physical
+conservation laws. Proceedings of the AAAI Conference on Artificial Intelligence, 35(1):277–285, 2021.
+[22] S. Marburg.
+A Unified Approach to Finite and Boundary Element Discretization in Linear Time–
+Harmonic Acoustics. In S. Marburg and B. Nolte, editors, Computational Acoustics of Noise Propagation
+in Fluids - Finite and Boundary Element Methods, pages 1–34. Springer Berlin Heidelberg, Berlin,
+Heidelberg, 2008.
+[23] D. A. McNamara, C. W. I. Pistorius, and J. A. G. Malherbe. Introduction to the uniform geometrical
+theory of diffraction. Artech House Norwood, MA, 1990.
+[24] J. M. Melenk and S. Sauter. Wavenumber explicit convergence analysis for Galerkin discretizations of
+the Helmholtz equation. SIAM Journal on Numerical Analysis, 49(3):1210–1243, 2011.
+[25] A. Moiola, R. Hiptmair, and I. Perugia. Plane wave approximation of homogeneous Helmholtz solutions.
+Zeitschrift f¨ur angewandte Mathematik und Physik, 62(5):809–837, 2011.
+[26] C. Pagliantini. Dynamical reduced basis methods for Hamiltonian systems. Numerische Mathematik,
+148(2):409–448, 2021.
+[27] S. F. Potter and M. K. Cameron. Jet marching methods for solving the eikonal equation. SIAM Journal
+on Scientific Computing, 43(6):A4121–A4146, 2021.
+[28] S. F. Potter, M. K. Cameron, and R. Duraiswami. Numerical geometric acoustics: an eikonal-based
+approach for modeling sound propagation in 3D environments. arXiv preprint arXiv:2208.13002, 2022.
+[29] J. Reiss, P. Schulze, J. Sesterhenn, and V. Mehrmann. The shifted proper orthogonal decomposition:
+A mode decomposition for multiple transport phenomena.
+SIAM Journal on Scientific Computing,
+40(3):A1322–A1344, 2018.
+[30] P. H. Tournier, I. Aliferis, M. Bonazzoli, M. de Buhan, M. Darbas, V. Dolean, F. Hecht, P. Jolivet,
+I. El Kanfoud, C. Migliaccio, F. Nataf, C. Pichot, and S. Semenov. Microwave tomographic imaging of
+cerebrovascular accidents by using high-performance computing. Parallel Computing, 85:88–97, 2019.
+[31] G. Welper. Interpolation of functions with parameter dependent jumps by transformed snapshots. SIAM
+Journal on Scientific Computing, 39(4):A1225–A1250, 2017.
+22
+

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+A stabilized local integral method using RBFs
+for the Helmholtz equation with applications
+to wave chaos and dielectric microresonators
+L. Ponzellini Marinelli[1,2]
+[email protected]
+L. Raviola[1]
+[email protected]
+[1] Faculty of Exact Sciences, Engineering and Surveying,
+National University of Rosario, Rosario, Argentina.
+[2] Rosario Physics Institute, UNR-CONICET,
+Rosario, Argentina.
+January 3, 2023
+Abstract
+Most problems in electrodynamics do not have an analytical so-
+lution so much effort has been put in the development of numerical
+schemes, such as the finite-difference method, volume element meth-
+ods, boundary element methods, and related methods based on bound-
+ary integral equations.
+In this paper we introduce a local integral
+boundary domain method with a stable calculation based on Radial
+Basis Functions (RBF) approximations, in the context of wave chaos
+in acoustics and dielectric microresonator problems. RBFs have been
+gaining popularity recently for solving partial differential equations
+numerically, becoming an extremely effective tool for interpolation on
+scattered node sets in several dimensions with high-order accuracy
+and flexibility for nontrivial geometries. One key issue with infinitely
+1
+arXiv:2301.00069v1  [math.NA]  30 Dec 2022
+
+smooth RBFs is the choice of a suitable value for the shape param-
+eter which controls the flatness of the function. It is observed that
+best accuracy is often achieved when the shape parameter tends to
+zero. However, the system of discrete equations obtained from the
+interpolation matrices becomes ill-conditioned, which imposes severe
+limits to the attainable accuracy. A few numerical algorithms have
+been presented that are able to stably compute an interpolant, even
+in the increasingly flat basis function limit. We present the recently
+developed Stabilized Local Boundary Domain Integral Method in the
+context of boundary integral methods that improves the solution of
+the Helmholtz equation with RBFs. Numerical results for small shape
+parameters that stabilize the error are shown. Accuracy and compar-
+ison with other methods are also discussed for various case studies.
+Applications in wave chaos, acoustics and dielectric microresonators
+are discussed to showcase the virtues of the method, which is com-
+putationally efficient and well suited to the kind of geometries with
+arbitrary shape domains.
+1
+Introduction and motivation
+Dielectric microresonators, also known as dielectric microcavities, have at-
+tracted interest in the last decades due to technological applications like
+microlasers and and as systems with intrinsic theoretical interest for its con-
+nections with quantum billiards and wave chaos [2, 9, 20].
+A quantum billiard is a system in which a free particle is confined within
+a 2D domain and whose dynamics is governed by the Schr¨odinger equation
+iψt(x, t) = −∆ψ(x, t),
+x ∈ Ω ⊂ R2, t > 0.
+(1)
+where ψ(x, t) = 0 for x ∈ Γ being Γ the boundary of the domain Ω.
+When searching the time harmonic solutions of this system in the form
+ψ(x, t) = ˜ψ(x)eikt, the spatial dependence, ˜ψ(x), satisfies the well-known
+Helmholtz Equation (HE)
+�
+∆ + k2� ˜ψ(x) = 0,
+x ∈ Ω ⊂ R2, t > 0.
+(2)
+In this case, the eigenvalues to equation (2) are related to the energy of the
+particle.
+On the other hand, a similar situation arises when trying to solve the
+problem of light waves propagating inside a dielectric medium satisfying the
+2
+
+Maxwell equations. Also in this case, the search for time harmonic solutions
+leads to the Helmholtz equation for the spatial dependence of the electro-
+magnetic field [2].
+For generic domains, the equation (2) cannot be solved analytically to find
+stationary states. So we must resort to finding efficient and reliable numerical
+methods to solve this equation. There are many numerical techniques to
+address this equation such as the finite element method (FEM), the finite
+volume method (FVM), the Boundary Element Method (BEM) or spectral
+methods (PS) [19]. However, several of these require the construction of a
+specific mesh or refinement to efficiently address certain numerical problems
+on non-trivial geometries.
+The BEM transforms the formulated Partial Differential Equations (PDE)
+into integral equations, that is, into an integral form over the boundary
+[1, 13]. In BEM the PDE that describes the physical problem is transformed
+into a Boundary Integral Equation (BIE), which is achieved by using Green’s
+identities to then apply this integral formulation over points distributed in
+the domain. Many local integral methods are based on an integral formula-
+tion on small, strongly overlapping stencils with local interpolations.
+In recent decades, methods involving the Radial Basis Functions (RBF)
+have become an extremely effective tool in non-trivial geometries for inter-
+polation in sets of scattered nodes and for numerically approximating PDE.
+There are many modern books dealing with theory, implementations and ap-
+plications [3, 4, 6]. One advantage is that when the distribution nodes are
+created, it is possible to achieve local refinement in critical areas depending
+on the specific problem [5]. Particularly, this is interesting to resolve local-
+ized structures like the scarred states observed in quantum chaos phenomena
+[18].
+Using infinitely differential RBFs like Gaussians, exponential convergence
+can be shown. A practical obstacle is the ill-conditioning of the interpolation
+matrix when the shape parameter ε that defines the Gaussian RBF tends
+to zero. It is known that when this parameter is reduced, the interpolation
+accuracy of the method improves considerably but the numerical conditioning
+of the problem worsens if it is solved with a direct type numerical method.
+That is, there is a conflict between accuracy and the constraint known as the
+uncertainty principle [17].
+In this paper we present the Stabilized Localized Boundary-Domain Inte-
+gral Method (SLBDIM) [16] in the context of Helmholtz type equations. This
+is a new stable integral local numerical method for approximating elliptic-
+3
+
+type PDE solutions to solve Boundary Value Problems (BVP) in 2D that
+uses local interpolations with RBF for low values ε > 0. This technique is
+a combination of meshless methods, local integral formulations and bound-
+ary elements in multi-domains independent of a structured mesh and that
+only requires an unstructured distribution of nodes of the domain Ω and its
+boundary Γ = ∂Ω that allows to deal with complex geometries. For local
+interpolations, the Gaussian RBFs ϕ(r) = e−(εr)2 are used when ε → 0 in
+local interpolations in stable form.
+Numerical results are shown for a small shape parameter that stabilizes
+the error. Comparisons with other methods in several cases are also dis-
+cussed. It is shown that the method is computationally efficient and suit-
+able for geometries that come from applications of wave chaos and dielectric
+microresonators. In particular, we solve differential problems with Dirichlet-
+type boundary conditions over square domains with quasi-uniform point dis-
+tributions.
+2
+The Stabilized Localized Boundary Domain
+Integral Method for Helmholtz equations
+2.1
+Problem description and local integral method
+We consider the following Boundary Value Problem (BVP) on an open,
+bounded and simply connected domain Ω ⊂ R2
+(BV P)
+� L [u] (x) = f(x),
+x ∈ Ω,
+(3a)
+B [u] (x) = g(x),
+x ∈ Γ = ∂Ω,
+(3b)
+where L[ . ] = ∆ + λ is an elliptic differential Helmholtz-type operator,
+∆ =
+∂
+∂x2 +
+∂
+∂y2 is tha Laplacian, λ ∈ R (when λ = k2 > 0, k is the wave-
+number) and f(x) is the smooth source term. B[ . ] is the boundary operator
+with the boundary conditions (BC).
+The BC are Dirichlet, Neumann or mixed over Γ = Γ1∪Γ2 and Γ1∩Γ2 = ∅
+�
+�
+�
+u(x) = g1(x),
+x ∈ Γ1,
+(4a)
+∂u(x)
+∂n
+= g2(x),
+x ∈ Γ2,
+(4b)
+with g1 and g2 known data, and ∂u(x)
+∂n
+the outward normal derivative of the
+unknown field u.
+4
+
+We propose that PDE (3a) can be written as
+∆u (x) = f(x) − λu (x) = b (x, u (x)) ,
+(5)
+where u (x) is the potential in the point x ∈ Ω.
+We consider x ∈ Ω ⊂ R2
+∆u∗ = δ(x − ξ),
+(6)
+where δ(x − ξ) is Delta’s delta centered at ξ ∈ Ω with fundamental solution
+u∗(x, ξ) = 1
+2πln(r),
+r = ∥x − ξ∥.
+(7)
+From equation (5)
+∆u (x) = b ⇔
+�
+Ω
+u∗ (x, ξ) ∆u (x) dΩx =
+�
+Ω
+u∗(x, ξ)b dΩx.
+(8)
+Applying Green’s second identity for u that satisfies (5) and u∗ that
+satisfies (6)
+�
+Ω
+(u∗∆u − u∆u∗) dΩx =
+�
+Γ
+�
+u∗ ∂u
+∂n − u∂u∗
+∂n
+�
+dΓx,
+(9)
+we obtain
+u(ξ) =
+�
+Ω
+u∗ (x, ξ) b dΩx −
+�
+Γ
+�
+u∗ (x, ξ) ∂u(x)
+∂n
+− u(x)u∗(x, ξ)
+∂n
+�
+dΓx.
+(10)
+From equation (10) we have a formula for the integral representation of
+the PDE over a subregion Ωi with boundary Γi. The interior collocation
+point xi is obtained as before from the fundamental solution and Green’s
+second identity
+u(ξ) =
+�
+Γi
+q∗ (x, ξ) u (x) dΓx −
+�
+Γi
+u∗ (x, ξ) q (x) dΓx +
+�
+Ωi
+b u∗ (x, ξ) dΩx,
+(11)
+where q = ∂u
+∂n is the normal derivative of the unknown field, u∗ is the fun-
+damental Laplacian solution and q∗ =
+∂u∗
+∂n is the normal derivative of the
+fundamental solution.
+5
+
+Using the well-known Green-Dirichlet function (FGD), G (x, ξ), and its
+normal derivative Q (x, ξ) [8] in (11) we obtain a new integral formulation of
+the form
+u(ξ) =
+�
+Γi
+Q (x, ξ) u (x) dΓx +
+�
+Ωi
+b G (x, ξ) dΩx.
+(12)
+since the integral over Γi involving G in (11) vanishes since its value is zero.
+In addition, if the non-homogeneous term b of the PDE can be split
+b (x, u (x)) = f (x) − λu (x) ,
+(13)
+where the funcion source f is data.
+The integral representation (12) in each subregion of integration Ωi is
+u(ξ) =
+�
+Γi
+Q(x, ξ)u(x) dΓx+
+�
+Ωi
+G(x, ξ)f(x) dΩx+
+�
+Ωi
+���λu (x) G(x, ξ) dΩx,
+(14)
+where ξ is the interior source point. The collocation technique is done only
+at interior points of the domain.
+2.2
+Local interpolations with RBF
+A function ϕ : Rd → R is an RBF if there exists φ : [0, ∞) → R such that
+ϕ (x) = φ(r),
+r = ∥x − xj∥,
+(15)
+where ∥.∥ is the Euclidean norm on Rd and depends on the distance to a
+center xj ∈ Rd. If it depends on the shape parameter ε > 0, then ϕε
+j (x) =
+φ(r, ε) is often noted.
+In the LBDIM the field u is locally interpolated with RBF {ϕj}n
+j=1 with
+centers of the stencil Θx = {xj}n
+j=1
+u (x) ≈
+n
+�
+j=1
+αjϕj(x),
+(16)
+where the interpolation matrix Ai is such that
+(Ai)jk = ϕk(xj) = φ(∥xj − xk∥),
+j, k = 1, . . . , n
+(17)
+6
+
+The term b of (13) is interpolated with RBF {χj}m
+j=1 with centers of the
+stencil Θy = {yj}m
+j=1
+�b (u (x) , ∇u (x)) ≈
+m
+�
+j=1
+βjχj (x) ,
+(18)
+where the interpolation matrix �Ai is such that
+(�Ai)jk = χk(yj) = χ(∥yj − yk∥), j, k = 1, . . . , m
+(19)
+The RBFs are eventually of the same type and with the same centers. If
+we take the same RBF bases with the same centers, the result is {ϕj}n
+j=1
+and {χj}m
+j=1 for m = n although they could be different depending on the
+application problem or numerical experience.
+The local integral formulation of (14) is of the form
+u(ξ)
+≈
+n
+�
+j=1
+αj
+��
+Γi
+Q(x, ξ)ϕj(x) dΓx
+�
++
+m
+�
+j=1
+βj
+��
+Ωi
+G(x, ξ)χj (x) dΩx
+�
++
+�
+Ωi
+G(x, ξ)f(x) dΩx.
+(20)
+If Θ = {x1, . . . , xN} is the discretization of domain Ω and ξ = xi ∈ Θ is
+the collocation point, the discretized formulae of the unknown field is
+ui = u (xi) =
+n
+�
+j=1
+αj�hij +
+m
+�
+j=1
+βj�gij + �fi,
+(21)
+where αj and βj come from equations (16) and (18). The coefficients �hij, �gij
+and �fi are of the form
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�hij =
+�
+Γi
+Q (x, xi) ϕj (x) dΓx,
+(22a)
+�gij =
+�
+Ωi
+G (x, xi) χj (x) dΩx,
+(22b)
+�fi =
+�
+Ωi
+G (x, xi) f (x) dΩx,
+(22c)
+which are calculated by Gauss-Legendre quadratures.
+7
+
+Defining the vectors α = [α1, . . . , αn]T and β = [β1, . . . , βm]T as interpo-
+lation coefficients, the discretized form (21) of u can be expressed as
+ui = �hT
+i α + �gT
+i β + �fi,
+(23)
+where �hi = [�hi1, . . . ,�hin]T and �gi = [�gi1, . . . , �gim]T are the influence coeffi-
+cients, and �fi ∈ R is data.
+The vector α arises from the local system by interpolating with the RBF
+basis {ϕj}n
+j=1
+Aiα = di ⇔ α = A−1
+i di
+(24)
+and the vector β arises from the local system by interpolating with the RBF
+basis {χj}m
+j=1
+�Aiβ = �bi ⇔ β = �A−1
+i �bi = �A−1
+i
+�
+A�biα
+�
+= �A−1
+i
+�
+A�biA−1
+i di
+�
+,
+(25)
+where A�bi is the calculation matrix of the vector �bi with known coefficients
+(A�bi)jk = �b (ϕk (yj) , ∇ϕk (yj)) ,
+j = 1, . . . , m, k = 1, . . . , n.
+(26)
+Substituting (24) and (25) in the discretized form (23), we obtain the
+discretized matrix form for ui in terms of di
+ui =
+�
+�hT
+i A−1
+i
++ �gT
+i �A−1
+i A�biA−1
+i
+�
+di + ˜fi.
+(27)
+Rewriting (27) we obtain an algorithmic procedure to avoid the compu-
+tation of inverses A−1
+i
+and �A−1
+i
+(see [14])
+ui = zTdi + �fi
+donde zT = �hT
+i A−1
+i
++ �gT
+i �A−1
+i A�biA−1
+i
+(28)
+which are assembled into a global sparse-like system and numerically resolved
+with Generalized Minimal Residual (GMRES).
+2.3
+Stability with Gaussian RBFs
+Convergence in global interpolations with ε-dependent RBFs can be studied
+in a stationary way (n = cte. and ε → 0) or non-stationary (ε = cte. and
+increasesn). In the case of Gaussian RBFs, they produce convergence of
+order O(e
+− const
+(εh)2 ) (superspectral).
+8
+
+The RBF interpolation matrix is
+A(ε) =
+�
+����
+φ(∥x1 − x1∥, ε)
+φ(∥x1 − x2∥, ε)
+. . .
+φ(∥x1 − xn∥, ε)
+φ(∥x2 − x1∥, ε)
+φ(∥x2 − x2∥, ε)
+. . .
+φ(∥x2 − xn∥, ε)
+...
+...
+...
+...
+φ(∥xn − x1∥, ε)
+φ(∥xn − x2∥, ε)
+. . .
+φ(∥xn − xn∥, ε)
+�
+���� .
+When ε is small, the RBFs become almost linearly dependent (’flat’)
+forming a bad basis of functions and generating ill-conditioned interpolation
+matrices A(ε) in a good interpolation space. To avoid this problem in [7, 10]
+numerical techniques were developed that stabilize the solutions of linear
+systems where the RBFs that form the matrix of the system take arbitrarily
+small shape parameters. The RBF-QR method developed for global inter-
+polations of scattered nodes using Gaussian RBFs is numerically stable for
+nearly zero parameters. The idea of the RBF-QR algorithm is to change
+the base {φj} to a new base {ψj} using combinations of polynomial powers,
+Chebyshev polynomials and trigonometric functions.
+3
+Implementation of the SLBDIM
+The new matrix form for u of (27) at each node is
+ui =
+�
+lT
+i Bi
+−1 + �l
+T
+i �
+Bi
+−1B˜biBi
+−1�
+di + ˜fi,
+(29)
+where li = [. . . , lik, . . .]T and �li = [. . . ,�lik, . . .]T are the column vectors.
+For internal stencils, the local interpolation matrix is
+Bi
+ψ = V
+�
+In
+�
+R
+T
+�
+,
+(30)
+where (Bi
+ψ)jk = ψk(xj) and Vjk = Vk(xj) for j, k = 1, . . . , n ([7] for details).
+For boundary stencils, the local matrix interpolation matrix is Bi has
+two blocks,
+Bi =
+� Bi
+ψ
+Bi
+Bψ
+�
+,
+(31)
+where the first matrix block is
+(Bi
+ψ)jk = ψk(xj),
+(32)
+9
+
+for j = 1, . . . , nint (interior nodes) and k = 1, . . . , n (boundary nodes), and
+the second matrix block is
+(Bi
+Bψ)jk = Bψk(xj)
+(33)
+for j = nint + 1, . . . , n and k = 1, . . . , n.
+To avoid calculating Bi
+−1 and �
+Bi
+−1 when ε → 0 we follow an algorithmic
+procedure. The inclusion of this technique in the local integral method allows
+to stabilize the numerical error of the approximation of the Helmholtz-type
+equations.
+This Stabilized Domain and Boundary Local Integral Method
+(SLBDIM) was presented at [16] for Poisson problems, convection-diffusion
+equations and elliptic PDEs. Another strategy of stability technique for local
+integral methods that uses RBF interpolation functions was presented in [15].
+4
+Numerical examples on several billiars
+In this section we report two numerical experiments to show the accuracy and
+efficiency of the proposed numerical scheme to solve Helmholtz-type equa-
+tions in two dimensions. Implementations and numerical experiments were
+performed using MATLAB version R2017a numerical calculation software on
+a PC with 7.5 GB of RAM and an Intel Core i7-7500U 7th Generation CPU.
+running at 2.70GHz.
+The reported errors are the standard error L2 (L2-Error)
+L2-Error
+=
+�
+�N
+i=1(uexac
+i
+−uapprox
+i
+)
+2
+�N
+i=1(uexac
+i
+)
+2
+(34)
+and the root mean square error (RMS):
+RMS
+=
+�
+�N
+i=1(uexac
+i
+−uapprox
+i
+)
+2
+N
+.
+(35)
+4.1
+Polygonal billiars: case 1
+This Helmholtz-type PDE is given over the rectangular domain Ω = [−1, 1]×
+[−1, 1]
+� ∆u(x) − k2u(x)
+=
+f(x),
+x = (x, y) ∈ Ω,
+u(x)
+=
+g(x),
+(x, y) ∈ Γ = ∂Ω,
+(36)
+10
+
+where f(x, y) = 2 cos(x2 + y) − (4x2 + 1 + k2)sin(x2 + y) and the parameter
+k = 9. The BCs of this BVP are of the Dirichlet type, the analytical solution
+being u(x, y) = sin(x2+y). In our case, we will use the local integral method
+presented in its original form with Gaussian RBF kernels φ(r) = e−(εr)2 (we
+will call it LBDIM) and in its stabilized form (SLBDIM).
+There are several ways to discretize the Ω domain with distributions of
+nodes. In our case we will use the algorithm for generating quasi-uniform
+distributions developed in [5] for 2D. These distributions were created with
+a fast-forward method that generates a set of nodes from a density function
+starting from the Γ boundary towards the interior of the domain.
+-1
+-0.5
+0
+0.5
+1
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.8
+1
+-1
+1
+-0.5
+0.5
+1
+0
+0.5
+0.5
+0
+1
+0
+-0.5
+-0.5
+-1
+-1
+Figure 1: Quasi-uniform 2D node distribution for Nint = 916 internal col-
+location points and Ncol = 124 boundary points with Dirichlet BC (left).
+Analytical solution of BVP (right).
+We compare the L2-Error of the formulation of the LBDIM and the SLB-
+DIM using the Gaussian RBFs in the local interpolations varying the pa-
+rameter in the form ε ∈ [1, 10].
+Figure 2 shows that as ε decreases, the
+accuracy increases but the LBDIM is destabilized and the convergence is
+interrupted all for cases N = 400, 916, 1610, 3604 quasi-uniform nodes. In
+turn, we observe that as we increase the number of nodes on the domain
+and the boundary, the errors decrease. This plot shows that for local in-
+terpolation with Gaussian RBF lead to a loss in accuracy for small shape
+parameters.
+However, the best performance is obtained by the stabilized
+local integral method to address this Helmholtz-type equation with known
+analytical solutions. The error for N = 916, 1610, 3604 is of order 1 × 10−8.
+The application of the RBF-QR kernel makes the system well-posed to solve
+11
+
+them with a direct method in the LBDIM. In this numerical experiment the
+size of the stencil is n = 50.
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+10 -8
+10 -6
+10 -4
+10 -2
+10 0
+LBDIM (N=400)
+SLBDIM (N=400)
+LBDIM (N=916)
+SLBDIM (N=916)
+LBDIM (N=1610)
+SLBDIM (N=1610)
+LBDIM (N=3604)
+SLBDIM (N=3604)
+Figure 2: Comparison of the L2-Error between LBDIM and SLBDIM versus
+the shape parameter ε.
+In Figure 3 we show the isolines of the error log10(L2-Error) for the range
+of the shape parameter [1, 10] and for different sizes of stencils n=10:10:100.
+As n increases, the linear systems increase, worsening the conditioning of
+the interpolation matrices. To understand the importance of local stability
+technique, both graphs of this figure must be observed simultaneously. The
+yellow region at the top left shows the region of error instability due to poor
+numerical conditioning while in the isolines of the graphs on the right, the
+region dark blue shows how 1 × 10−8 could be kept in order. As N increases
+from 916 to 3604 this numerical behaviour is similar reading the figure row-
+wise.
+12
+
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+30
+40
+50
+60
+70
+80
+90
+100
+-7
+-6
+-5
+-4
+-3
+-2
+-1
+Figure 3: Accuracy isolines (log10(L2-Error)) with Nint = 916, 1610, 3604
+interior points varying the shape parameter ε and the stencil size n.
+In [12] this same Helmholtz type PDE is worked with mixed type BC.
+In said work it can be seen that for N = 900 nodes the L2-Error 1 × 10−5
+is reached using the Radial Basis Function - Finite Difference (RBF-FD)
+technique using a kernel hybrid of the Gaussian of type φ(r) = αe−(εr)2 +βr3.
+13
+
+4
+6
+6
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+30
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+80
+90
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+14
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+20
+4
+4
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+8
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+10
+10
+1
+2
+3
+4
+5
+6
+7
+8
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+10
+10
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+30
+40
+50
+60
+70
+80
+90
+3
+4
+5
+6
+7
+8
+9
+10
+Figure 4: Condition number isolines (log10(κ(Ai)) with Nint=916,1610,3604
+interior points varying the shape parameter ε and the stencil size n.
+In Figure 4 the isolines condition number log10(κ(Ai) is shown.
+The
+range of the shape parameter is [1, 10] and the for different sizes of stencils
+are n=10:10:100. As n increases, the conditioning of the local interpolation
+matrices increases. The yellow region at the top left shows the region of
+the condition number up to 1 × 1020. In the isolines of the graphs on the
+right column, the region dark blue shows better conditioning up to 1 × 1010.
+This ten order of magnitude are significant when when using linear solvers
+14
+
+numerically. Also we can observe thar as N increases from 916 to 3604 the
+conditioning behaviour is similar reading the figure row-wise.
+In Figure 3 it was observed that the error plots suggest smaller values
+of ε0 for better accuracy, whereas in this figure the condition isolines plots
+suggest the larger values of ε for better stability. This numerical results are
+interpreted as the well-known uncertainty principle in RBF local interpola-
+tions [17]. The idea behind this principle is that one cannot simultaneously
+achieve good conditioning and high accuracy using RBF basis. The relation
+between numerical stability and accuracy may be reviewed from different
+perspectives as in our case to obtain a stable formulation our option was to
+find a better basis in the same space of approximation using RBF-QR [7] in
+the local boundary domain integral method.
+4.2
+Polygonal billiars: case 2
+Consider the following two-dimensional Helmholtz equation
+� ∆u(x, y) + k2u(x, y)
+=
+f(x, y),
+Ω = [0, 1] × [0, 1],
+u(x, y)
+=
+g(x, y),
+Γ = ∂Ω,
+(37)
+where k2 = 2, f(x, y) = 2x − 4y and the exact solution is given by u(x, y) =
+sin(
+√
+3x)sinh(y) + cos(
+√
+2y) + x − 2y, and g(x, y) is chosen to match the
+exact one, thus giving BC of type Dirichlet. We use quasi-uniform nodes
+within the domain and stencils of size n = 25 counting the collocation center
+as shown in Figure 5.
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Figure 5: Quasi-uniform node distribution with N = 900 interior nodes (left).
+Stencil node sets with n = 25 (right).
+15
+
+In Table 1 we show the accuracy of the SLBDIM for the shape parameter
+ε = 1 and for a range of low values, ε ∈ {1 × 100, 1 × 10−1, 1 × 10−2, 1 ×
+10−3, 1 × 10−4, 1 × 10−5}. The number of quasi-uniform interior points of
+the domain, N, varies from 121 to 900. It can be seen that for fixed ε = 1,
+the ´orders of magnitude decrease from 1 × 10−6 to 1 × 10−8 starting at 441
+nodes. In turn, the convergence of the method is observed for low values of
+the shape parameter, reaching RMS of the order 1 × 10−8 from 225 nodes.
+The ε shown is where the best error is reached in that range.
+N
+SLBDIM
+SLBDIM
+ϵ
+RMS
+low ϵ
+RMS
+121
+1.0
+1.2028E-06
+0.1
+2.1405E-07
+225
+1.0
+5.8570E-07
+0.1
+5.0834E-08
+361
+1.0
+3.9338E-07
+0.01
+3.3821E-08
+441
+1.0
+7.8581e-08
+0.1
+3.3866E-08
+530
+1.0
+5.2907E-08
+0.00001
+3.5984E-08
+628
+1.0
+4.3843E-08
+0.00001
+3.6887E-08
+Table 1: RMS for low shape parameters ε ∈ {1 × 10−1, . . . , 1 × 10−5}.
+In [11] this differential problem with mixed BC over the same domain
+is investigated using Multiquadric RBF kernels ϕ(r, ε) =
+�
+1 + (εr)2 and a
+new RBF with N ∈ [50, 350] placement points. The results obtained in said
+reference reach errors of the order of 1 × 10−5 for ε ∈ [0.4].
+5
+Summary
+In this work we have introduced a new local integral method to compute reso-
+nances in dielectric cavities with different shapes. We have discussed numer-
+ical solutions, the node quasi-uniform node distributions over the domains
+and cavities with corners. Numerical results for Helmholtz-type equations
+were obtained using a stabilized local integral method that uses interpola-
+tions with RBF Gaussians. This method does not depend on a mesh, so it
+can be easily adapted to problems with complex geometries from . The good
+performance of the method has been shown with good results as shown in
+numerical tests 1 and 2 comparing with other results in the literature. Test 1
+shows the advantage of using the SLBDIM to find regions of convergence of
+the L2-Error of the order 1×10−8 when the shape parameter approaches zero.
+16
+
+In test 2, a low shape parameter range is studied reaching the same order of
+the RMS. Having investigated the computational efficiency of the method,
+the future work consists of approaching some applications in wave chaos and
+dielectric microresonators, which is adequate to deal with geometries that
+come from arbitrary domains without analytical solutions.
+References
+[1] C. Brebbia and D. Dominguez. Boundary Elements. An Introductory
+Course. 2nd Ed. WIT Press, Computational Mechanics Publications,
+Southampton and Boston, 1998.
+[2] H. Cao and J. Wiersig. Dielectric microcavities: Model systems for wave
+chaos and non-hermitian physics. Reviews of Modern Physics, 87:61–111,
+2015.
+[3] G. Fasshauer. Meshfree Approximation Methods with MATLAB. World
+Scientific Publishing Co., Hackensack, NJ, USA, 2007.
+[4] G. Fasshauer and M. McCourt. Kernel-based Approximation Methods
+using MATLAB. World Scientific Publishing Co., Hackensack, NJ, USA,
+2015.
+[5] B. Fornberg and N. Flyer. Fast generation of 2-D node distributions
+for mesh-free PDE discretizations. Computers and Mathematics with
+Applications, 69:531–544, 2015.
+[6] B. Fornberg and N. Flyer. A Primer on Radial Basis Functions with Ap-
+plications to the Geosciences. Society for Industrial and Applied Math-
+ematics, Philadelphia, PA, USA, 2015.
+[7] B. Fornberg, E. Larsson, and N. Flyer. Stable Computations with Gaus-
+sian Radial Basis Functions. SIAM Journal of Scientific Computing,
+33:869–892, 2011.
+[8] M. Greenberg. Applications of Green’s Functions in Science and Engi-
+neering. Dover Publications, Mineola, New York, 2015.
+17
+
+[9] D. Kaufman, I. Kosztin, and K. Schulten. Expansion method for station-
+ary states of quantum billiards. American Journal of Physics, 67:133–
+141, 1999.
+[10] E. Larsson, E. Lehto, A. Heryudono, and B. Fornberg. Stable Compu-
+tation of Differentiation Matrices and Scattered Node Stencils on Gaus-
+sian Radial Basis Functions. SIAM Journal of Scientific Computating,
+35:A2096–A2119, 2013.
+[11] J. Lin, W. Chen, and K. Sze. A new radial basis function for helmholtz
+problems. Engineering Analysis with Boundary Elements, 36(12):1923–
+1930, 2012.
+[12] P. Mishra, G. Fasshauer, M. Sen, and L. Ling. A stabilized radial basis-
+finite difference (RBF-FD) method with hybrid kernels. Computers &
+Mathematics with Applications, 77(9):2354–2368, 2019.
+[13] P. Partridge and C. B. andL.C. Wrobel. The Dual Reciprocity Boundary
+Element Method. Computational Mechanics Publications co-published
+with Elsevier Applied Science, Southampton Boston, 1992.
+[14] L. Ponzellini Marinelli. Estabilidad num´erica de un m´etodo local inte-
+gral basado en funciones de base radial para problemas de valores de
+contorno. Universidad Nacional de Rosario, 2021:164 p´aginas, 2021.
+[15] L. Ponzellini Marinelli. Stabilizing radial basis functions techniques for
+a local boundary integral method.
+Revista de la Uni´on Matem´atica
+Argentina, 64:in press, 2021.
+[16] L. Ponzellini Marinelli, N. Caruso, and M. Portapila. A stable com-
+putation on local boundary-domain integral method for elliptic PDE.
+Mathematics and Computers in Simulation, 180:379–400, 2021.
+[17] R. Schaback. Error estimates and condition numbers for Radial Basis
+Function interpolants. Advances in Computational Mathematics, 3:251–
+264, 1995.
+[18] H.-J. St¨ockmann. Quantum Chaos: An Introduction. Cambridge Uni-
+versity Press, Cambridge, UK, 1999.
+18
+
+[19] L. Trefethen. Spectral Methods in Matlab. Society for Industrial and
+Applied Mathematics, Philadelphia, PA, USA, 2000.
+[20] J. Wiersig. Boundary element method for resonances in dielectric mi-
+crocavities.
+Journal of Optics A: Pure and Applied Optics, 5:53–60,
+2003.
+19
+

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+Automated Misconfiguration Repair of Configurable
+Cyber-Physical Systems with Search: an Industrial
+Case Study on Elevator Dispatching Algorithms
+Pablo Valle
+Mondragon University
+Mondragon, Spain
+[email protected]
+Aitor Arrieta
+Mondragon University
+Mondragon, Spain
+[email protected]
+Maite Arratibel
+Orona
+Hernani, Spain
+[email protected]
+Abstract—Real-world Cyber-Physical Systems (CPSs) are usu-
+ally configurable. Through parameters, it is possible to configure,
+select or unselect different system functionalities. While this
+provides high flexibility, it also becomes a source for failures
+due to misconfigurations. The large number of parameters these
+systems have and the long test execution time in this context
+due to the use of simulation-based testing make the manual
+repair process a cumbersome activity. Subsequently, in this
+context, automated repairing methods are paramount. In this
+paper, we propose an approach to automatically repair CPSs’
+misconfigurations. Our approach is evaluated with an industrial
+CPS case study from the elevation domain. Experiments with a
+real building and data obtained from operation suggests that our
+approach outperforms a baseline algorithm as well as the state of
+the practice (i.e., manual repair carried out by domain experts).
+Index Terms—Cyber-Physical Systems, Repair, Debugging,
+Configurable Systems.
+I. INTRODUCTION
+Cyber-Physical Systems combine digital cyber computa-
+tions with parallel physical processes [1]–[3]. In such sys-
+tems, digital technologies, such as computational units, low
+and high-level software and communication protocols interact
+among them to control a physical process through sensors and
+actuators [1]. In practice, most CPSs deal with parameters.
+For instance, a heavy duty lifting system involved more than
+2,000 configuration parameters [4]. The behavior of CPSs can
+significantly change depending on these parameters. This often
+causes misconfigurations, even when selecting parameters that
+are within the ranges provided by the manufacturer [5]. A
+recent study showed that 19.6% of UAV-specific bugs were
+caused by parameters [6]. Garcia et al. [7] found that 27.25%
+of autonomous vehicle bugs were caused by incorrect con-
+figurations. In our industrial case study, which involves the
+traffic dispatching algorithm of a system of elevators, around
+55% of the issues assigned to the traffic team are solved
+through configuration changes. Therefore, it is paramount to
+leverage automated and scalable techniques to automatically
+repair CPS misconfigurations. However, this involves four core
+challenges:
+1) Challenge 1 – Expensive execution of the tests: It
+is well-known that executing CPS tests is highly time-
+consuming [8]–[15]. This is because, as the execution
+of tests is carried out at system level, CPSs involve
+compute-intensive models to simulate the physical part of
+the system (e.g., models of electrical engines, dynamics
+of a system). This makes the computation of the fitness
+to assess how close the algorithm is from repairing the
+misconfiguration expensive. For instance, in our industrial
+case study, executing a test case takes around 5 minutes.
+2) Challenge 2 – Large configuration space: Since con-
+figurable CPSs involve many parameters, the amount of
+possible configurations that a CPS can have is huge.
+Subsequently, testing all of these configurations is compu-
+tationally unfeasible [16]–[21]. Furthermore, it is usually
+unknown which the reason (i.e., the parameters) that
+causes the misconfiguration is.
+3) Challenge 3 – Multiple requirements: Multiple fail-
+ing requirements may exist. Some of them might be
+independent from one-another [15], while others may be
+conflicting (e.g., in our case study, better energy con-
+sumption could lead to passengers needing to wait more).
+Therefore, the repair algorithm shall be approached as a
+many-objective optimization problem.
+4) Challenge 4 – Prioritize severe failures: The repair
+technique needs to resolve failures in their order of sever-
+ity [15]. For instance a test case that shows a passengers’
+average waiting time (AWT) of 55 seconds is more
+critical than one showing 35 seconds. Therefore, similar
+to other CPS repairing techniques [15], our algorithm
+shall give priority to more critical test cases over the less
+critical ones.
+On the one hand, there are approaches that target the prob-
+lem of repairing misconfigurations [22], [23] of configurable
+software. However, such approaches only cover the second
+aforementioned challenge. On the other hand, Swarmbug [24]
+focuses on repairing misconfigurations of swarm robots, which
+can be considered CPSs. However, Swarmbug [24] solely
+focuses on one specific objective (e.g., not crashing), therefore,
+not tackling the third and fourth challenges that our industrial
+case study requires.
+arXiv:2301.01487v1  [cs.SE]  4 Jan 2023
+
+In this paper we propose an automated repairing approach
+specifically targeting CPSs’ misconfigurations. Specifically,
+we tackle this by recasting the misconfiguration repair problem
+to that of a many-objective search problem. To deal with
+the aforementioned first challenge, we propose an algorithm
+that follows a single population-based approach. Multiple
+population-based algorithms, such as genetic algorithms, are
+not appropriate for this context because the repair process
+requires interaction with the simulator for executing test cases.
+Such algorithms require a large population, and the large test
+execution time would lead the algorithm to require too much
+time to converge. This could eventually lead to scalability is-
+sues in the context of CPSs. To deal with the second challenge,
+our repairing approach implements a strategy that permits
+measuring the suspiciousness of each parameter. This permits,
+as the search process evolves, increasing the probability of
+selecting suspicious parameters to provide a new patch. As a
+result, in the beginning of the search, our approach focuses
+on exploring which the critical parameters can be. As the
+search evolves, the algorithm starts to focus on the exploitation
+by targeting suspicious parameters. To deal with the third
+challenge, our approach includes a Pareto-optimal archive-
+based strategy to select and evolve potential misconfiguration
+patches. This permits focusing on more than one requirement
+at the same time when repairing the misconfiguration. To deal
+with the last challenge, search objectives are prioritized based
+on their severity level.
+Our main contributions can be highlighted as follows:
+1) We propose a scalable and automated approach to repair
+misconfigurations in CPSs.
+2) We integrate the approach with an industrial case study
+from Orona, one of the largest elevator companies in
+Europe. The case study involves the traffic dispatching
+algorithm, a highly configurable software system.
+3) We empirically evaluate our approach by using a real
+scenario in which Orona’s engineers had to manually
+intervene in the misconfiguration repair process. Our
+repairing technique not only outperforms a baseline al-
+gorithm, but also the manually derived repairing patches
+by Orona’s domain experts.
+4) We extract key lessons learned from the application of
+our approach in an industrial case study, and provide ap-
+plicability guidelines in order our approach to be adopted
+by other CPS developers.
+The rest of the paper is structured as follows: Section II
+explains our industrial case study, how the testing is carried out
+and why misconfigurations occur. In Section III we present our
+approach to repair misconfigurations in our industrial context.
+Section IV presents how we evaluated our approach. We ex-
+tract key lessons learned and we explain the required changes
+in our approach to be applied in other CPSs in Section V.
+We position our work with relevant studies in Section VI. We
+conclude and present future work in Section VII.
+II. INDUSTRIAL CASE STUDY
+Our repair algorithm is applied in an industrial case study
+from the elevation domain. This section explains the different
+details of the case study.
+The Cyber-Physical System: Figure 1 shows an overview
+of the CPS. A system of elevators is a complex CPS, whose
+goal is to transport passengers from one floor to another safely
+while trying to provide the highest comfort as possible. In this
+system, a passenger registers a call in a floor by pushing a call
+button. This information is transferred to the traffic master
+through a Controller Area Network (CAN) bus. The traffic
+master, after collecting other CPS information (e.g., position of
+each elevator, elevator occupancy), assigns one of the available
+elevators to each active call. This assignation can be carried
+out through different objectives (e.g., reducing the passengers’
+waiting times, reducing energy consumption). When the call
+is assigned, the elevator attends the passenger.
+Elevator 1
+Elevator 2
+Elevator 3
+Floor 1
+Floor 2
+Floor N
+Controller Area Network
+Controller 1
+Controller 2
+Controller 3
+Traffic Master (SUT)
+Fig. 1: Overview of our industrial case study
+The System Under Test (SUT): Our SUT is the traffic
+dispatching algorithm (i.e., dispatcher), which is an important
+module inside the traffic master. To deal with different func-
+tionalities and priorities, the dispatcher is highly configurable
+through parameters. Different traffic dispatching algorithms
+exist in Orona, and each of them encompasses one config-
+uration file. The number of potential configurations of each
+dispatcher is over trillions.
+Test Executions: Three different phases are undertaken
+when testing the dispatching algorithm [25], [26]: the
+Software-in-the-Loop (SiL), the Hardware-in-the-Loop (HiL)
+and Operation. Our algorithm is designed for the first phase,
+i.e., the SiL test level. At this stage, a domain-specific
+simulator, i.e., Elevate1, takes as input (1) the dispatching
+algorithm’s executable, (2) the building installation, (3) the
+configuration ��le and (4) the passenger file. The passenger
+file is considered the test input, and it involves a set of
+1https://peters-research.com/index.php/elevate/
+
+passengers traveling through different floors in a building.
+Each passenger has different attributes, such as, its arrival
+time (i.e., time at which the passenger arrives to the floor
+and pushes the button), arrival floor (i.e., floor at which the
+passenger arrives), destination floor (i.e., floor at which the
+passenger is traveling to), passenger weight, etc. When a test
+is executed, Elevate returns a file with the results of the
+simulation (e.g., waiting time required by each passenger, their
+traveling time, energy consumption, distance traveled by each
+elevator). This information is parsed and the necessary test
+oracles are employed to assess the quality of the execution of
+the test.
+Functional performance requirements: When executing
+test cases, besides considering certain functional require-
+ments, we focus on “functional performance requirements”.
+Functional performance is defined as “the properties derived
+indirectly from the output of the system, rather than the
+system’s efficient usage of the computational resources” [26].
+These properties are directly employed for evaluating the
+functional performance requirements of Orona’s dispatching
+algorithms. The properties involve metrics from the elevator
+traffic domain, such as the Average Waiting Time (AWT) of
+passengers, the Average Transit Time (ATT) of passengers,
+Longest Waiting Time (LWT), Longest Transit Time (LTT),
+number of engine starts, traveled distance by each elevator
+or consumed energy. Note that configuration changes affect
+functional performance requirements, whereas functional re-
+quirements (e.g., ensuring that reverse journeys do not take
+place) are, in principle, not affected by such changes.
+Why misconfigurations occur and how they are handled:
+The dispatcher has different parameters to accommodate dif-
+ferent functionalities that have a direct impact on the CPS per-
+formance. However, it is noteworthy that a configuration may
+perform well in one installation of elevators, while not well
+in another one, causing a misconfiguration. This is because
+the performance of a system of elevators largely depends on
+(1) the type of building and its composition and (2) how its
+traffic flow is. Regarding the former, the performance can vary
+depending on aspects like number of elevators in a building,
+the number of floors the building has, whether all elevators
+attend all floors or not, etc. For some types of buildings, some
+configurations are more appropriate than others. As for the
+latter, the traffic is also different depending on the type of
+buildings. For instance, the traffic flow is completely different
+in a hospital and in a residential building. While in a hospital
+inter-floor travels are common, in a residential building most
+of the calls are from the base floor to the floor where the
+apartment is and vice-versa. When a system of elevators shows
+a poor performance, its traffic flow is reproduced at the SiL
+test level to debug and try to improve its performance through
+changing parameters. If a new set of parameters improves
+the system performance, then, the original configuration is
+considered a misconfiguration. It is important to note that
+in our industrial case study, a misconfiguration might not be
+detected nor foreseen before the system is in operation due to
+the CPS exposition to uncertainty [27], [28].
+III. CPS MISCONFIGURATION REPAIR METHOD
+Algorithm 1 shows an overview of our repairing algo-
+rithm. The algorithm takes as input (1) a faulty configu-
+ration file C, composed of N number of parameters, i.e.,
+C = {p1, p2, ..., pN}; and (2) a test suite, composed of M
+failing test cases, i.e., TS = {tc1, tc2, ..., tcM}. The first step
+of the algorithm consists on assessing the failing configuration
+file, where all the parameter values are parsed (Line 1) and
+all test cases are executed (Line 2). When the failing test
+suite is executed, the values returned by the oracle are used to
+initialize the Archive (Line 3) and the suspiciousness scores
+of parameters initialized (Line 4). After that, the algorithm
+enters into a while loop (Lines 5-11) that ends when the
+termination criteria are met. These criteria involve (1) fixing
+the misconfiguration or (2) exceeding the running time.
+Algorithm 1: Overview of our search-based repairing
+algorithm
+Input: C //Faulty Configuration file
+TS //Test Suite
+Output: Archive //Archive containing improved
+configurations
+1 Patch0 ← getValues(C);
+2 InitialScore← executeTestSuite(Patch0, TS);
+3 Archive ← saveToArchive(Patch0, InitialScore);
+4 Susp ← initSusp();
+5 while terminationCriteriaNotMet do
+6
+Parent ← selectAParentArchive(Archive);
+7
+Patch1 ← generatePatch(Parent,Susp);
+8
+Score ← executeTestSuite(Patch1, TS);
+9
+Susp ← updateSusp(Patch1, Parent, Score,
+ScoreParent);
+10
+Archive ← saveToArchive(Patch1, Score);
+11 end
+12 return Archive;
+Inside this while loop, the first step consists in selecting a
+solution from the Archive (Line 6), which will be the parent.
+The solution is selected pure randomly. With the selected
+solution, a potential patch is proposed (Line 7), which consists
+of changing one or more parameters from the parent solution
+(Section III-A). This patch is assessed by executing the failing
+test suite (Line 8), and the test execution results are obtained
+and stored as Scores (Section III-B). In a fourth step, the
+suspiciousness score of each parameter is recalculated (Line
+9, Section III-C). Lastly, the Archive is updated (Line 10,
+Section III-D).
+A. Patch generation
+A patch in our context refers to a mutation of at least one
+parameter. Algorithm 2 shows our algorithm for proposing a
+potential patch. As input, it receives (1) a parent configuration,
+which corresponds to one configuration in the archive of the
+algorithm and (2) the suspiciousness ranking of all parameters.
+First, a parameter to be mutated is selected (Line 4) based on
+
+the suspiciousness of each parameter (see Section
+III-C for
+more details on how to compute the suspiciousness score).
+The higher the suspiciousness, the higher the probability of
+being selected. The parameter to be mutated is obtained by
+employing Algorithm 3. The selected parameter is mutated
+(Line 5) by giving a random value within its ranges. After
+this, it is decided whether a new parameter is mutated (Line
+8). The probability of mutating a new parameter decreases as
+the number of mutated parameters in the new patch increases.
+We ensure that one parameter is not mutated more than once.
+Algorithm 2: Patch generation algorithm
+Input: Parent //Faulty Configuration
+SuspRanking //Suspiciousness Ranking
+Output: Patch //Mutated Configuration
+1 numOfMutParams ← 0;
+2 Patch ← Parent;
+3 do
+4
+varToMutate ← selectParam(SuspRanking);
+5
+Patch ← mutate(Patch,varToMutate);
+6
+numOfMutParams ← numOfMutParams +1;
+7
+p ← rand(); //returns random value 0 to 1
+8 while p < 0.5numOfMutatedP arams;
+9 return Patch;
+Algorithm 3: Suspiciousness-based parameter selec-
+tion algorithm
+Input: SuspScore = {ss1, ss2, ..., ssN}
+Output: selected //Index of the selected parameter
+1 total ← �N
+i=1(ssi);
+2 iterativeSum←0;
+3 prob ← [];
+4 for i ← 1 to nPop do
+5
+prob[i] ← iterativeSum + SuspScore[i]/total;
+6
+iterativeSum←prob[i];
+7 end
+8 prob←orderAscending(prob);
+9 r←rand();//Returns random number 0 to 1
+10 j←0;
+11 selected=N;
+12 while j<N and selected==N do
+13
+if r<prob[j] then
+14
+selected←j;
+15
+j←j+1;
+16 end
+17 return parameter(selected); //translates index of
+selected to parameter ID
+B. Test suite execution
+After the patch is generated, this needs to be assessed. We
+assess each patch by re-executing all test cases in the test
+suite that have failed. We do not execute the passing test cases
+because executing such test cases would significantly increase
+the computational time of our approach. Furthermore, for the
+sake of increasing the efficiency of our repair algorithm, the
+process of executing test cases is parallelized. When executing
+the test suite, test oracles assess the performance of the system.
+In our context, similar to other approaches [8]–[11], [14], [15],
+[29], test oracles not only provide a boolean verdict (i.e., Pass
+or Fail), but also a confidence value. The lower the value, the
+lower the performance of the CPS in terms of the assessed
+property by such test oracle.
+These oracles’ confidence values are used as search objec-
+tives to guide the repair algorithm towards finding effective
+patches. For repairing a CPS, a total of k test oracles may exist.
+Each of these k oracles acts as an individual objective function
+in the repair algorithm. For each test case (tc) in the failing
+test suite (TS), each of these k oracles returns its confidence
+value, i.e., Conf(tc, oi) ∈ [−1, 0], where oi is the i-th
+oracle. -1 means that the severity of the failure is the highest
+contemplated one, whereas 0 means that the oracle has passed.
+The repair algorithm aims at maximizing that confidence
+value. Therefore, after executing all test cases in TS, similar
+to Abdessalemm et al., [15], we obtain the minimum value for
+each of the test oracles (i.e., the most severe value), converting
+the repair problem in a many-objective optimization problem
+that gives priority to the most severe failures, such that:
+�
+�
+�
+�
+�
+max Oracle1(Patch) = min
+tc∈T S{Conf(tc, o1)}
+...
+max Oraclek(Patch) = min
+tc∈T S{Conf(tc, ok)}
+(1)
+As previously explained, executing a test in the context
+of CPSs is time consuming. Previous studies using compute-
+intensive CPSs have leveraged surrogate models to accelerate
+the generation of test cases [8], [10], [13], [14]. That is,
+after a set of test executions, a model is trained with test
+results, and this model is employed as a substitute of the
+simulation-based test execution. This permits accelerating the
+generation of test cases. While we considered to use surrogate
+models to accelerate the repair process, we noticed that too
+many simulations were required to obtain a reliable surrogate
+model. Unlike previous approaches [8], [10], [13], [14], which
+only use the dimension of the test input, configurable CPSs
+also need to consider the dimension of parameters, which
+makes it harder to train a surrogate model. After carrying out
+a preliminary evaluation with our industrial case study, we
+noticed that the required time to obtain data for building a
+reliable surrogate model was similar or even higher than the
+time required by our repair algorithm to converge. Therefore,
+the option of using a surrogate model to accelerate the repair
+process was discarded.
+C. Measuring parameter suspiciousness
+Based on analyzing the behavior of our industrial case
+study, and by interviewing domain experts, we noticed that
+some parameters have a higher influence than others on the
+system performance. Therefore, we implement a mechanism
+to measure the suspiciousness of each parameter in C. The
+
+suspiciousness provides a score between 0 and 1, where the
+higher the suspiciousness, the higher the likelihood of the
+parameter having an influence in the system performance. The
+ultimate goal of this strategy is to give a higher probability of
+being mutated to those parameters having an influence in the
+system performance.
+All configurable parameters start with the same suspicious-
+ness score, which is 0.5. This suspiciousness remains un-
+changed until the parameter is mutated by the Patch generation
+algorithm for Nsusp times (we employed Nsusp = 5 in our
+experiments). This permits the algorithm to focus on the
+exploration phase at the beginning of the search process, while
+focusing on the exploitation as the search process evolves.
+Every time a parameter is mutated by the Patch generation
+algorithm, after assessing the patch, we extract whether the
+parameter had (1) a positive impact on the performance of the
+system, (2) a negative impact on the performance of the system
+or (3) no impact at all. A positive impact of a parameter pi
+is considered when the patch is non-dominated by any other
+patch in the system based on the test results. A negative impact
+of a parameter pi is considered when the patch is dominated
+by the solutions in the archive (i.e., including its original
+parent). The patch does not have any impact for a parameter
+pi when the result of the test shows the same performance
+as its original parent. After a parameter pi is selected Nsusp
+times, its suspiciousness starts to be computed as follows:
+susp(pi) =
+Ppi + Npi
+Ppi + Npi + Spi
+(2)
+where Ppi is the number of times that parameter pi had a
+positive impact, Npi is the number of times that the parameter
+pi had a negative impact and Spi is the number of times that
+the parameter pi had no impact at all.
+Notice that either the positive or the negative impact in-
+crease the suspiciousness of a particular parameter. This is
+because the patch is proposed by mutating the value of a pa-
+rameter by another random value within its ranges. Therefore,
+another value in a parameter that previously had a negative
+impact may have a positive impact on the CPS performance.
+Based on our analysis, the suspiciousness of the parameters
+in the context of CPSs is, in principle, unknown, even with
+domain expertise. This is, to a large extent, because CPSs
+highly depend on the context at which they operate. For
+instance, in the case of our industrial case study, a parameter
+can have a large impact on the performance of the CPS
+depending on the type of building (e.g., parameters may
+behave differently in a residential building with 2 elevators
+or in a hospital building with 4 elevators). For this reason, we
+assume there is no prior knowledge of the impact a parameter
+may have in the context of a CPS. However, our approach
+for measuring the suspiciousness of parameters can easily be
+extended to other strategies (e.g., providing the algorithm with
+an initial suspiciousness score for each of the parameters in
+the configuration).
+D. Updating the Archive
+Our algorithm uses an archive encompassing non-dominated
+solutions that are generated by including patches. The first
+configuration file being updated in the archive is the misconfig-
+uration provoking the failure. After assessing a patch (Patch1)
+by executing the failing test suite, the archive needs to be
+updated. Such patch is compared with the rest of solutions
+in the archive. The comparison is based on the notion of
+dominance, and similar to other studies [15], the archive is
+updated as follows:
+1) If Patch1 dominates at least one solution in the archive,
+Patch1 is included in the archive, and the dominated
+solutions are removed.
+2) If no element in the archive dominates Patch1, but
+Patch1 is neither dominated by any solution in the
+archive, Patch1 is included in the archive.
+3) The archive remains unchanged if Patch1 is dominated
+by at least one solution in the archive.
+By following this strategy, there is some risk that the archive
+increases in size. This would lead the algorithm to need much
+more time to converge. To overcome this problem, if the
+archive exceeds certain size, we remove solutions from it.
+Same as Abdessalem et al., [15], the maximum size of our
+archive is limited to 2 × k, k being the number of oracles.
+However, unlike [15], instead of randomly removing the
+solution from the archive, we removed the solution which had
+the longest Average Waiting Time (AWT). This decision was
+taken because in the elevation domain, this is the main metric
+used to assess the performance of a dispatching algorithm [30].
+If two or more solutions encompassed the same highest AWT,
+the choice is random among those two solutions.
+E. Stopping criteria
+The repair process stops given two criteria: (1) all test cases
+in TS pass, i.e., all oracles in all test cases return the 0
+value; or (2) the search budget is exceeded (i.e., repairing
+time was exceeded). If the latter happens, it might be the case
+where the test cases are too demanding. Therefore, the repair
+process would be converted into a parameter optimization
+problem. For instance, by analyzing our industrial case study
+with the elevator dispatching algorithm, we noticed that some
+test inputs may encompass too many passenger calls in a short
+time window. In such cases, the CPS may enter in a saturation
+state, where the only solution would be to include additional
+elevators to better attend calls, something that is out of the
+scope of the dispatching algorithm’s competence.
+F. Decision maker
+When the repairing algorithm stops due to the search budget
+being exceeded, there might be a high probability that more
+than one solution exists in the Archive. In such a case,
+a decision maker (DM) with certain rules would need to
+select one of the solutions and propose it as a patch. This
+decision maker is, in our case, domain-specific. The DM
+was a rule-based algorithm that was designed by involving
+domain experts in the process. The specified thresholds are
+
+configurable because some thresholds may be valid in certain
+buildings but not in others. The algorithm follows the next
+procedure to decide which patch to propose:
+1) It first selects all patches where the AWT is less than
+25 seconds. This is the threshold that an international
+standard considers as a good performance of a system
+of elevators [31]. Since the AWT is the most widely
+employed metric to assess the quality of a system of
+elevators [30], we gave first priority to this metric. If
+there is no solution meeting that requirement, we select
+the patch that exhibits the lowest AWT.
+2) If more than one patch remains, the DM prioritizes
+patches whose test execution showed a lower number
+of passengers waiting above 55 seconds. That threshold
+is specified to be below 10%, which was considered
+an affordable number. Domain experts considered that
+waiting nearly a minute is an anti-pattern, therefore, they
+decided to give priority to those solutions that exhibited a
+low number of passengers waiting more than 55 seconds.
+3) In a third stage, if more than one patch exists in the set of
+candidate solutions, priority is given to the ATT metric.
+The DM selects those solutions that have a lower ATT
+than 45 seconds. If there are no solution meeting that
+requirement, we select the patch that exhibits the lowest
+ATT.
+4) If multiple patch candidates keep existing, the DM selects
+those solution whose test execution showed a lower
+number of passengers having a transit time above 70
+seconds. That threshold was specified to be below 10%,
+as it was considered an affordable number.
+5) After that, in the event that more than one candidate
+patch existed, the DM selected the patch with lowest
+LWT, which was considered of higher importance than
+the LTT. If more than a patch existed, the patch with
+lowest LTT was chosen. Although the possibilities are
+remote, it is still possible to have more than one solution.
+In such a case, the similarity of the configuration files of
+the candidate patches is compared with the original con-
+figuration file through the well-known hamming distance
+metric. The one which has more similarity is chosen. The
+reasons are two-fold. On the one hand, engineers are
+not usually eager to change too many parameters from
+the original configuration file. This is because, what it is
+good for certain passenger flows, it may not be good for
+others. On the other hand, we conjecture that the higher
+the number of parameters that have been changed, the
+higher the probability that the solution is overfitted to the
+failing test suite. Therefore, by means of this mechanism,
+we aim at reducing the probability for our plausible patch
+to be overfitted.
+G. Patch confirmation
+Since we only use a failing test suite to repair the mis-
+configuration, the patch needs to be retested. This way, we
+ensure that the patch is not overfitted to the failing test suite,
+which is a core problem of automated program repair [32]–
+[35]. This can be carried out following any kind of state-of-the-
+art technique. In our case, we use a regression test oracle [26]
+and execute synthetic test inputs (i.e., test inputs based on
+templates for full-day theoretical passenger profiles [36] and
+up and down-peak profiles suggested by international elevator
+standards [31]). We ensure that the new patch does not perform
+worse than the original patch. Besides, we test its functionality
+by employing metamorphic testing with shorter test cases, as
+proposed by Ayerdi et al. [37], [38].
+IV. EVALUATION
+In our evaluation, we aimed at answering the following two
+research questions (RQs):
+• RQ1 – Sanity check: How does our approach compare
+to the baseline? To assess whether the problem to solve
+is trivial, the first RQ is a sanity check. To do so,
+we implemented an unguided version of our repairing
+algorithm.
+• RQ2 – Comparison with state of the practice: How
+does our approach compare to manual repair carried
+out by domain experts? The current practice at Orona
+is to manually repair the misconfigurations. This RQ
+aims at comparing whether our algorithm is competent
+when compared to a manual repair process carried out
+by domain experts in the company.
+A. Experimental Setup
+1) System Under Test and Building: We used Orona’s
+Conventional Group Control (CGC) traffic dispatching algo-
+rithm [30], which has also been used in other studies [27],
+[28], [37]–[39]. Furthermore, we used a real installation to
+assess our approach. The installation involved a total of three
+elevators and 12 floors. We used this installation because it was
+a real case where Orona had to manually intervene to resolve
+the misconfiguration. Furthermore, the manual misconfigura-
+tion process taken by the engineers was well documented. In
+addition, we also had access to the operational data obtained
+from the conflicting installation to be used as failing test
+inputs. In total, we used three failure-inducing test inputs,
+involving 16 hours of passenger flow each, and between 3,105
+and 3,769 passengers in total.
+The version of the algorithm we used involved a total of
+43 parameters. The total number of potential configurations
+ascends to over 9.3 × 1092, which makes the search space too
+large to employ brute force.
+2) Test oracles: By carefully analyzing the internal docu-
+ment Orona used to give solution to the conflicting installa-
+tion, we defined six oracles based on the metrics they were
+aiming to optimize. Below we explain the selected functional
+performance metrics:
+• Average Waiting Time (AWT): It measures the average
+waiting time of all passengers. The waiting time refers to
+the time since a passenger registers a call until an elevator
+arrives to attend her.
+
+• Longest Waiting Time (LWT): It measures the longest
+waiting time experienced by the passengers.
+• % of passengers with Waiting Time (WT) above 55
+seconds: It measures the percentage of passengers who
+had to wait more than 55 seconds.
+• Average Transit Time (ATT): It measures the average
+transit time of all passengers. The transit time refers to
+the time since a passenger enters a lift until it arrives to
+its destination.
+• Longest Transit Time (LTT): It measures the longest
+transit time of all passengers.
+• % of passengers with Transit Time (TT) above 70 sec-
+onds: It measures the percentage of passengers who had
+a transit time above 70 seconds.
+When repairing this misconfiguration, the domain experts
+aimed at improving as much as possible the functional per-
+formance metrics listed above. Therefore, in the context of
+this study, we opted for being aggressive with the thresholds.
+Therefore, all thresholds were set to 0. We acknowledge that
+these values are unfeasible to obtain. However, this way the
+comparison with the manual approach is fairer. Furthermore,
+we also wanted to assess the patch that the DM selected.
+3) Execution platform: Elevate version 8.19 was used as
+simulator for executing the tests. The experiments were carried
+out using a PC with a Windows 10 operating system, with a
+CPU Intel Core i5 7th generation, and a 16 Gb RAM.
+4) Baseline algorithm and state of the practice comparison:
+As baseline algorithm, we developed an unguided version of
+our repairing algorithm. Two core differences exists between
+the unguided version and the repair algorithm proposed in this
+paper: (1) the unguided version saves all configurations in the
+archive and (2) the parameters to be mutated are considered all
+to have the same suspiciousness score (i.e., the suspiciousness
+is not measured in this version). It is noteworthy that this
+baseline is stronger than a pure Random Search (RS), which
+is the usual baseline algorithm used to assess search-based
+software engineering problems [17], [19], [40]–[43]. This is
+because, RS would take the initial failing configuration and
+propose some patches based on our patch generation approach
+(Algorithm 2). However, with RS, these generated patches
+would not evolve anymore. Conversely, with our unguided
+approach, we give the option of evolving patches in the
+archive, leading to higher probabilities of finding a patch.
+As for the comparison with the state of the practice, for
+the building installation used, we had data from engineers
+from Orona. Specifically, when the issue was raised, engineers
+from Orona proposed different potential patches (i.e., different
+configurations of the dispatcher). We compared the results
+obtained by our algorithm with the patches proposed by
+the domain experts. Six different patches were provided by
+Orona’s engineers.
+5) Evaluation
+Metrics:
+As
+our
+algorithm
+is
+Pareto-
+compliant, we had to assess all the solutions in the archive as a
+whole. Because of this, and based on related guidelines [44],
+[45], we used the Hypervolume (HV) quality indicator. The
+HV is one of the most widely employed metrics to assess
+Pareto-based search algorithms [44]–[46]. The HV measures
+the volume in the objective space of a search algorithm, and
+has many advantages [46], such as, (1) being Pareto compliant,
+(2) being able to evaluate convergence and the diversity of
+a solution set simultaneously and (3) only requiring one
+reference point.
+Besides the HV quality indicator, as we designed a DM,
+we also compared each of the six objective functions used
+as performance metrics in the test oracles for the solutions
+proposed by the DM after the search budget was exceeded.
+6) Statistical tests: Since the employed algorithms are non-
+deterministic, we run each algorithm 10 times. We could not
+afford more runs given that the search budget was selected to
+be 12 hours. Therefore, in total we employed 10 (runs) × 12
+(hours) × 2 (baseline and repair algorithms) = 240 hours for
+executing the experiments.
+To assess the statistical significance, we employed the
+Wilcoxon rank sum test. We considered that there was statis-
+tical significance between both algorithms when the p-value
+was lower than 0.05. In addition, we employed the Vargha
+and Delaney ˆA12 value, which measures the probability of a
+technique being better than the other one.
+7) Algorithm configuration: We gave 12 hours of time
+budget to both, our algorithm and the baseline algorithm.
+Similar to [15], the maximum number of solutions in the
+archive was set to 12 (i.e., 6 objective function × 2). We
+also set the parameter Nsusp = 5, which means that the
+suspiciousness of a parameter is neutral (i.e., suspiciousness
+score of 0.5) until it is mutated 5 times.
+B. Analysis and Discussion of the Results
+1) RQ1 – Sanity check: Figure 2 shows the average HV
+score of the 10 runs for both, the repair algorithm proposed in
+this paper and the baseline algorithm, which is the unguided
+version of the repair algorithm. As it can be appreciated,
+the repair approach showed a higher average HV than the
+baseline after the second execution hour. By the time the
+search budget was expired, the repair algorithm showed a 29%
+average improvement over the baseline in terms of the HV
+quality indicator. It is noteworthy that the HV values are quite
+low. The reasons for this is that the HV favors knee points of
+a solution set in a Pareto-frontier [45]. As explained before,
+in our case, the specified threshold values were 0 (i.e., the
+repair algorithm aims at optimizing as much as possible all the
+functional performance metrics). Achieving such value was not
+realistic, and therefore we did not have knee values. Besides,
+6 different oracles (i.e., fitness functions) were employed to
+guide the search towards providing patches. Nevertheless, a
+low HV value makes not unfair the comparison between both
+techniques, which is the goal of the first RQ.
+These results were further corroborated by means of statisti-
+cal tests. Table I shows the ˆA12 as well as p-values (computed
+by employing the Wilcoxon rank sum test) for each of the 12
+hours when comparing the repair algorithm with the baseline.
+The ˆA12 shows the probability of the repair algorithm being
+better than its unguided version. As suggested by Romano et
+
+Fig. 2: Average value of the 10 runs for the hypervolume
+quality indicator when comparing the repair and unguided
+algorithms
+al. [47], we categorized the difference existing between the
+repair algorithm and its baseline as negligible if d < 0.147,
+as small if d < 0.33, as medium if d < 0.474 and as large
+if d >= 0.474, where d = 2| ˆA12 − 0.5|. According to this
+categorization, the difference was negligible during the first
+execution hour, small between the second and third execution
+hours and medium during the fourth execution hour. In these
+first four execution hours, there was no statistical significance
+between the repair algorithm and the baseline. Conversely,
+after the fifth hour, there was statistical significance (i.e., p-
+value < 0.05) with large effect sizes based on the related
+categorization [47], all of them in favor of our approach.
+TABLE I: RQ1 – Summary of the statistical tests when
+comparing the repair algorithm with its unguided version for
+the HV quality indicator over the execution of 12 hours. An
+ˆA12 value higher than 0.5 means that the results are in favor
+of the repair algorithm. Statistical significance is set as p-
+val<0.05
+Hour
+ˆA12
+p-val
+1
+0.51
+0.9698
+2
+0.61
+0.4273
+3
+0.65
+0.2730
+4
+0.71
+0.1212
+5
+0.80
+0.0256
+6
+0.86
+0.0081
+7
+0.89
+0.0040
+8
+0.82
+0.0172
+9
+0.85
+0.0090
+10
+0.85
+0.0090
+11
+0.90
+0.0028
+12
+0.92
+0.0017
+Besides the HV, we also analyzed the individual patches
+provided by the decision maker (DM). In this case, the aim
+of the algorithm was to reduce such metrics. Therefore, an
+ˆA12 lower than 0.5 means that the repair algorithm performed
+better. Table II summarizes the statistical tests for the ten runs
+and each individual objective function. There was statistical
+significance in half of the objective functions (i.e., LWT, ATT
+and LTT). For such cases, the effect sizes were large (i.e.,
+ˆA12 between 0.18 to 0.2). For the remaining objectives, where
+there was no statistical significance, in the case of the AWT
+and %WT>55, the effect sizes showed a negligible difference,
+whereas for the case of %TT>70, the difference was small.
+TABLE II: Summary of the statistical test results when com-
+paring the patches provided by the DM when employing repair
+algorithm against the baseline and manual repair approaches
+vs. Baseline
+vs. Manual
+ˆA12
+p-val
+ˆA12
+p-val
+AWT
+0.52
+0.9097
+0.10
+0.0014
+LWT
+0.18
+0.0165
+0.20
+0.0161
+%WT>55s
+0.47
+0.8788
+0.00
+<0.0001
+ATT
+0.20
+0.0312
+0.40
+0.4429
+LTT
+0.20
+0.010
+0.00
+<0.0001
+%TT>70s
+0.37
+0.3438
+0.00
+<0.0001
+Table III show the average value of each of the functional
+performance metrics used by the oracles for the 10 runs and
+the patches provided by the DMs. These results were somehow
+consistent with those from Table II. As it can be appreciated,
+the most striking difference relates to the LWT and the LTT
+functional performance metrics. On the contrary, for the AWT,
+%WT>55, ATT and %TT>70, the differences were not that
+large. This could be due to the nature of the DM. Note
+that for those metrics, the DM accepts values that are below
+certain thresholds (e.g., AWT < 25 seconds), whereas for LWT
+and LTT, the DM selects those patches with lowest values.
+However, in all metrics except the AWT, our algorithm showed
+lower average values.
+TABLE III: Comparison between the misconfigured configu-
+ration file, the patch provided by the DM with the manual
+repair, the average values of the patches returned by the DM
+for the baseline algorithm and the average values of the patches
+returned by the DM for the repair algorithm
+Misconf
+Manual
+Baseline DM
+Repair DM
+AWT
+25.99
+23.10
+22.66
+22.77
+LWT
+435.70
+223.00
+241.55
+213.72
+%WT >55s
+12.78
+11.99
+9.93
+9.92
+ATT
+42.01
+41.60
+41.77
+41.58
+LTT
+209.80
+220.60
+206.24
+195.56
+%TT>70s
+10.24
+10.02
+9.64
+9.45
+In conclusion, the first RQ can be answered as follows:
+Answer to the first RQ: The repair algorithm outper-
+formed the baseline algorithm. The average improvement
+extent of the repair algorithm with respect to the baseline
+was around 29% when considering the HV quality indi-
+cator. Furthermore, there was statistical significance with
+large effect sizes when comparing individual patches pro-
+posed by the DM for half of the objective functions, all
+of them in favor of the repair algorithm. All this suggests
+that the problem of repairing CPSs misconfigurations is
+non-trivial, and therefore, automated and scalable repair
+techniques are necessary.
+2) RQ2 – Comparison with manual repair: With the second
+RQ, we aimed at comparing the proposed repairing algorithm
+
+Hypervolume
+0.014
+0.012
+0.01
+C
+0.008
+ o- Repair
+0.006
+ Unguided
+-- Manual
+0.004
+-0
+G中
+0.002
+0d
+2
+4
+6
+8
+10
+0
+12
+Execution hourswith the manual process of repairing the misconfiguration by
+domain experts. Specifically, these domain experts provided a
+total of 6 patches. With those patches, and by applying the six
+oracles in our algorithm, we derived the HV metric. As can
+be seen in Figure 2, the HV was quite low. This was because
+only four patches were non-dominated, whereas our archive is
+capable of handling up to twelve patches. Therefore, those four
+patches were not able to cover a large volume in the objective
+space. Furthermore, it is important to note that the time was not
+considered here, because we do not have such information. In
+terms of the HV, the average improvement extent of our repair
+algorithm over the manually derived patches was up to 77.5%.
+For this case, we also employed the DM to select one of
+the non-dominated patches. Table II shows the statistical tests
+carried out when comparing the patches provided by the DM
+after executing the repair algorithm with the patch proposed
+by the DM after processing the four non-dominated solutions.
+As it can be appreciated, in five out of six metrics there
+was statistical significance, where the effect size showed a
+large difference according to the categorization proposed by
+Romano et al. [47]. All these effect sizes were in favor of the
+repair algorithm. On the other hand, for the case where there
+was no statistical significance, i.e., for the case of the ATT
+metric, the difference was small in terms of the ˆA12 value,
+but in favor of the repair algorithm.
+The improvement extent for each functional performance
+metric obtained by the patches provided by the DM (over 10
+runs) with respect to the manual approach can be appreciated
+in Table III. These results are consistent with the statistical
+tests, where it can be appreciated a similar average value in the
+case of the ATT. In this case, the improvement extent is higher
+in the cases of the AWT, % WT > 55, LTT and %WT>70
+when compared to the baseline algorithm. However, in relation
+to the LWT, the improvement was only of 10 seconds on
+average, unlike with the baseline, where the improvement was
+of nearly 29 seconds on average.
+In summary, the second RQ can be answered as follows:
+Answer to the second RQ: The repair algorithm
+outperformed the manual repair process. The average
+improvement extent of the repair algorithm with re-
+spect to the patches provided by the domain experts
+was around 77.5% when considering the HV quality
+indicator. Furthermore, there was statistical significance
+with large effect sizes when comparing individual patches
+proposed by the DM in five out of six objective functions.
+In addition, our approach provides a fully automated
+approach, which can therefore increase the productivity
+of engineers from Orona when dealing with misconfigu-
+rations of the traffic dispatching algorithm.
+C. Threats to Validity
+We now summarize the threats to validity of our study and
+the measures taken to mitigate them.
+An internal validity threat in our evaluation could be related
+to the parameters used in the algorithms, which were not
+changed. Three main parameters need to be configured (1)
+the time budget, which was set to 12 hours; (2) the number of
+time a parameter needs to be selected to start computing its
+suspiciousness score (i.e., Nsusp), which is set to 5; and (3) the
+number of solutions in the archive. The first two parameters
+were selected based on preliminary evaluations. Coversely, the
+maximum number of solutions in the archive was the same as
+other repair approaches targeting CPSs [15].
+As in any search-based software engineering problem, a
+conclusion validity threat involves the stochastic nature of the
+algorithms used. To mitigate such issue, we run each algorithm
+10 times. It is important to note that our technique needs a long
+time to converge because the simulations employed to assess
+potential patches are exhaustive, therefore, we could not afford
+a large number of runs. Furthermore, we applied statistical
+tests to analyze the results, as recommended by Arcuri and
+Briand [48].
+As in any study involving humans, our evaluation is also
+subject to external validity threats. One such threats refers
+to the patches proposed by engineers from Orona. It is note-
+worthy, however, that these engineers have broad experience
+and domain expertise, and that the patches they proposed
+were the ones that were later deployed in the real CPS. The
+generalizability of the results is also another external validity
+threat of our study; note, however, that we used an industrial
+case study with a real installation and data obtained from
+operation. We plan to mitigate such threat in the future by
+(1) using other case studies from a different domain and (2)
+using other real installations where misconfigurations occured.
+Lastly, construct validity threats arise when the measures
+used are not comparable across algorithms. This was mitigated
+by giving the same search budget to both algorithms (i.e., the
+repair and the unguided algorithm).
+V. LESSONS LEARNED AND APPLICABILITY
+In this section, we describe the lessons we have learned
+thorough the whole process of developing and evaluating the
+repairing algorithm. In addition, we explain the main changes
+our method would require when applying it to other CPS
+domains.
+A. Lessons Learned
+Lesson 1 – Reduction of personnel cost: The current
+state of the practice when repairing misconfigurations is purely
+manual. This requires significant personnel cost since domain
+experts are required in the process. Our fully automated
+repairing approach not only outperforms the state of the
+practice in terms of providing a better patch to repair the
+misconfiguration, but also reduces significantly the personnel
+costs that are required behind a manual repair process.
+Lesson 2 – Scalable technique: Scalability is one of the
+main concerns when testing and debugging CPSs, mainly
+due to the need of considering properties involving physical
+devices with continuous dynamics and complex concurrent
+interactions between the system and its environment (e.g.,
+people) [49]. We saw that our search-based repair algorithm
+
+converges after around 10 hours, which is affordable for
+our industrial partner as engineers can launch the automated
+misconfiguration repair tool nightly.
+Lesson 3 – Surrogate models are, in principle, not
+appropriate: Despite we did not carefully assess this, while
+we developed the algorithm, we intended to integrate surrogate
+models to accelerate the repair process. However, we saw
+that this technique required too much time to build reliable
+surrogate models. This time was similar to the time budget
+that our repair algorithm required to converge. Although we
+assessed different types of surrogate models, we still need to
+more carefully analyze this, which remains a future work.
+Lesson 4 – Challenging conflicting installation: After
+applying our experiments and showing the results to Orona’s
+engineers, we noted that the conflicting installation we selected
+was challenging. Indeed, the traffic was abnormal, with many
+unforeseen situations (e.g., having too many calls in a short
+time window) and therefore, repairing the misconfiguration
+in such installation was, according to domain experts, more
+difficult than other installations.
+B. Applicability
+The context at which we have applied our repairing ap-
+proach is the elevator dispatching algorithm of Orona. How-
+ever, we believe that the three key challenges that we tackle
+(i.e., expensive execution of tests, large configuration space
+and multiple functional performance requirements) are com-
+mon in all types of configurable CPSs. As we involved domain
+experts when developing the repair approach, several domain-
+specific design choices were considered, which would require
+adaptions when applying our approach in another domain. Be-
+low we explain different alternatives and the changes required
+for the adoption of our method in another domain.
+Test execution process: One of the first changes our method
+would require is the test execution. As explained in Section
+III-B, we use a domain-specific simulator to execute test cases
+and measure how close the algorithm is from repairing the
+misconfiguration. This process would need to be substituted
+by the simulator being used to execute the tests within
+other CPSs. In addition, we employ a parallel test execution,
+which was possible in our context. However, other simulators
+(e.g., autonomous vehicles) could require more computing
+resources. For instance, testing autonomous vehicles often
+requires rendering driving scenes in virtual scenarios using
+high-fidelity simulators [13], which may require the execution
+of test cases to be sequential. Lastly, test oracles would need
+to be defined. When using Simulink models to execute the
+tests, which is a predominant CPS testing tool [50], an option
+could be to use SOCRaTEs [11], a DSL-based test oracle
+specification and generation tool for Simulink. Specifically,
+SOCRaTEs [11] provides a quantitative measure of the degree
+of violation of a requirement, similar to what we need in our
+algorithm to guide the misconfiguration repair process.
+Removing solutions from the archive: As explained in
+Section III-D, the archive may increase in size, which may
+have a direct implication in the convergence of the repairing
+algorithm. Therefore, when the archive exceeds a predefined
+number of solutions, one of the solutions needs to be removed.
+Our algorithm removes the solution with longest AWT, given
+that this is the most widely employed metric when testing dis-
+patching algorithms [30]. In another domain, two alternatives
+can be considered. The first one, employing one of the most
+important metrics. If all metrics have a similar importance,
+the second alternative could be to randomly remove one of
+the solutions from the archive or use a crowding distance to
+remove solutions that are too close from each other.
+Decision maker: The decision maker is another component
+that we developed ad-hoc for the traffic dispatching algorithm
+by following the advise of domain experts. We recommend
+to analyze priorities of the specific CPS to make a decision.
+In case there are no clear priorities, a solution could be to
+employ a weighted approach giving the same importance to
+all objectives.
+Patch confirmation: We only employed a failing test suite
+to guide the repair process. The core reason was the high test
+execution time. Eventually, it could happen that a proposed
+patch makes a test case from the passing test suite fail. Because
+of this, we implemented a patch confirmation process by
+following a traditionally employed regression test method [26]
+combined with a newly incorporated metamorphic testing
+approach by Orona [37], [38]. The patch confirmation module
+should follow the internally standardized testing approach,
+which can vary from a company to another.
+VI. RELATED WORK
+The related work in automated program repair is huge.
+Monperrus mantains a living review on such techniques [51].
+Table IV shows a summarized classification of the related work
+analyzing four key characteristics covered by our approach.
+The first characteristic (C1) analyzes the possibility of repair-
+ing computationally expensive systems. The second one (C2),
+whether the approach is intended to repair misconfigurations.
+The third one (C3), analyzes if the approach is able to deal
+with many requirements (i.e., more than 3). And the last one
+(C4), whether the approach prioritizes critical faults over the
+less critical ones.
+TABLE IV: Related work comparison with different charac-
+teristics required by our repairing technique
+C1
+C2
+C3
+C4
+[15]
++
+-
++
++
+[22], [23]
+-
++
+-
+-
+[24]
++
++
+-
+-
+[52]–[56]
+-
+-
+-
+-
+[57]–[64]
++
+-
+-
+-
+We found that, in the field of CPSs, repairing approaches
+are still in their infancy. Indeed, to the best of our knowledge,
+only two approaches tackle the problem of repairing CPSs.
+On the one hand, Swarmbug [24] focuses on repairing mis-
+configurations of swarm robotics. Specifically, they make use
+of a mechanism called the “degree of causal contribution”
+to abstract impacts of configurations to the swarm drones
+
+via behavior causal analysis. The evaluation is carried out
+in four swarm algorithms, and the repair objectives are in-
+dividual for each of them. These involve aspects like avoiding
+obstacles or unsafe zones in order the drones not to crash.
+The approach, however, does not cover C3 and C4. On the
+other hand, Ariel [15] focuses on repairing feature interaction
+failures in automated driving systems. Similar to our approach,
+ARIEL [15] uses a many-objective and a single population-
+based approach, and also employs an archive to keep track of
+partially repaired solutions. However, unlike this paper, which
+focuses on repairing misconfigurations, ARIEL [15] repairs
+feature interaction bugs by applying modify and swift mutation
+operators to integration rules that resolve conflicts between
+automated driving system features. Therefore, ARIEL does not
+cover C2.
+CADET [22] does cover C2 as it is intended to debug
+and fix misconfigurations that cause non-functional faults.
+Xiong et al. [23] focus on repairing misconfigurations in
+software product lines by generating a list of range fixes
+to help satisfy a constraint. However, both approaches do
+not consider systems that take high computation resources to
+execute the tests. In addition, CADET [22] only covers two
+non-functional properties (i.e., latency and energy), whereas
+Xiong et al. [23] focus on satisfying individual constraints.
+Lastly, the approaches do not prioritize fixing more critical
+faults over the less critical ones. Subsequently, both techniques
+do not cover C1, C3 and C4.
+Besides these three studies, which are the most closely
+related to our approach, other studies exist in the field
+of automated program repair [52]–[64]. Similar to this ap-
+proach, some consider search techniques, such as genetic
+programming [55], [56]. GenProg [55] is one of the first
+approaches that proposed the use of meta-heuristic search to
+repair software programs. Specifically they leveraged genetic
+programming to repair C programs. However, all these ap-
+proaches focus on repairing bugs in the code. Conversely, our
+approach focuses on repairing misconfigurations in the field
+of configurable CPSs.
+Another line of research related to our approach is that
+of unified debugging [65], [66]. Such technique uses patch
+execution results to improve localizing the fault [65], [66].
+Therefore, even if the repair process is unable to repair the bug,
+unified debugging helps improving the fault localization for
+latter manual repair. Our approach follows a similar strategy,
+where we aim at localizing suspicious parameters that will
+eventually help repair the misconfiguration. However, besides
+the fact that unified debugging [65], [66] is not aimed at
+debugging misconfigurations, but bugs at the code level, it
+assumes that there is an initial suspiciousness score (i.e., at
+statement level). Conversely, our approach begins with all
+parameters having the same suspiciousness because there is
+no information about which parameters have influence in the
+system performance.
+VII. CONCLUSION AND FUTURE WORK
+Real-world CPSs, such as elevators, involve many param-
+eters. The performance of CPSs is tightly linked to such
+parameters, and therefore, misconfigurations may occur. On
+the one hand, manually dealing with such misconfigurations
+might not always be feasible. On the other hand, automated
+solutions require dealing with certain challenges, such as,
+expensive simulations to execute test cases. In this paper we
+propose an automated and scalable solution based on meta-
+heuristic search to repair misconfigurations in CPSs. Our
+approach was integrated with an industrial case study provided
+by Orona, one of the largest elevator manufacturers in Europe.
+The evaluation was carried out with a real installation in which
+domain experts from Orona had to manually intervene in
+repairing a misconfiguration. The results suggest that, besides
+automating a process that before was purely manual, our algo-
+rithm provides better patches than those provided by domain
+experts. Specifically, in five out of the six quality indicators
+employed by domain experts to assess the quality of a patch,
+our algorithm outperformed with statistical significance the
+patch provided by domain experts.
+In the future, we would like to extend our approach
+from different perspectives. In terms of the applicability, we
+would like to integrate our algorithm with other CPSs in
+which configurations have been found to be problematic (e.g.,
+unmanned aerial vehicles [5]). Furthermore, we would like
+to explore solutions to prevent potential overfitting issues
+before proposing a plausible patch. This has been one of the
+core challenges identified in automated program repair [32]–
+[35], and therefore, we should be aware of it. In terms of
+internal applicability within Orona, we would like to evaluate
+our approach in other installations where misconfigurations
+occurred. Furthermore, we would also like to transfer the
+repair algorithm beyond the traffic team and within other
+departments. Lastly, we would like to further study whether
+other strategies exist to better train and integrate surrogate
+models in the repair process.
+ACKNOWLEDGMENT
+Project supported by a 2021 Leonardo Grant for Researchers
+and Cultural Creators, BBVA Foundation. The BBVA Founda-
+tion is not responsible for the opinions, comments and contents
+included in the project and/or the results derived from it,
+which are the total and absolute responsibility of their authors.
+Aitor Arrieta is part of the Software and Systems Engineer-
+ing research group of Mondragon Unibertsitatea (IT1519-22),
+supported by the Department of Education, Universities and
+Research of the Basque Country.
+REFERENCES
+[1] P. Derler, E. A. Lee, and A. S. Vincentelli, “Modeling cyber–physical
+systems,” Proceedings of the IEEE, vol. 100, no. 1, pp. 13–28, 2011.
+[2] R. Baheti and H. Gill, “Cyber-physical systems,” The impact of control
+technology, vol. 12, no. 1, pp. 161–166, 2011.
+[3] R. Alur, Principles of cyber-physical systems.
+MIT press, 2015.
+
+[4] S. Fischer, R. Ramler, C. Klammer, and R. Rabiser, “Testing of highly
+configurable cyber-physical systems–a multiple case study,” in 15th
+International Working Conference on Variability Modelling of Software-
+Intensive Systems, 2021, pp. 1–10.
+[5] R. Han, C. Yang, S. Ma, J. Ma, C. Sun, J. Li, and E. Bertino, “Control
+parameters considered harmful: Detecting range specification bugs in
+drone configuration modules via learning-guided search,” in Proceedings
+of the 44th International Conference on Software Engineering, 2022, pp.
+462–473.
+[6] D. Wang, S. Li, G. Xiao, Y. Liu, and Y. Sui, “An exploratory study of
+autopilot software bugs in unmanned aerial vehicles,” in Proceedings
+of the 29th ACM Joint Meeting on European Software Engineering
+Conference and Symposium on the Foundations of Software Engineering,
+2021, pp. 20–31.
+[7] Garcia, Joshua and Feng, Yang and Shen, Junjie and Almanee, Sumaya
+and Xia, Yuan and Chen, and Qi Alfred, “A comprehensive study
+of autonomous vehicle bugs,” in Proceedings of the ACM/IEEE 42nd
+International Conference on Software Engineering, 2020, pp. 385–396.
+[8] R. B. Abdessalem, A. Panichella, S. Nejati, L. C. Briand, and T. Stifter,
+“Testing autonomous cars for feature interaction failures using many-
+objective search,” in 2018 33rd IEEE/ACM International Conference on
+Automated Software Engineering (ASE).
+IEEE, 2018, pp. 143–154.
+[9] R. B. Abdessalem, S. Nejati, L. C. Briand, and T. Stifter, “Testing vision-
+based control systems using learnable evolutionary algorithms,” in 2018
+IEEE/ACM 40th International Conference on Software Engineering
+(ICSE).
+IEEE, 2018, pp. 1016–1026.
+[10] C. Menghi, S. Nejati, L. Briand, and Y. I. Parache, “Approximation-
+refinement testing of compute-intensive cyber-physical models: An
+approach based on system identification,” in 2020 IEEE/ACM 42nd
+International Conference on Software Engineering (ICSE). IEEE, 2020,
+pp. 372–384.
+[11] C. Menghi, S. Nejati, K. Gaaloul, and L. C. Briand, “Generating
+automated and online test oracles for simulink models with continuous
+and uncertain behaviors,” in Proceedings of the 2019 27th acm joint
+meeting on european software engineering conference and symposium
+on the foundations of software engineering, 2019, pp. 27–38.
+[12] S. Nejati, K. Gaaloul, C. Menghi, L. C. Briand, S. Foster, and D. Wolfe,
+“Evaluating model testing and model checking for finding requirements
+violations in simulink models,” in Proceedings of the 2019 27th acm
+joint meeting on european software engineering conference and sympo-
+sium on the foundations of software engineering, 2019, pp. 1015–1025.
+[13] F. U. Haq, D. Shin, and L. Briand, “Efficient online testing for dnn-
+enabled systems using surrogate-assisted and many-objective optimiza-
+tion,” in Proceedings of the 44th International Conference on Software
+Engineering, 2022, pp. 811–822.
+[14] D. Humeniuk, F. Khomh, and G. Antoniol, “A search-based framework
+for automatic generation of testing environments for cyber-physical
+systems,” Information and Software Technology, p. 106936, 2022.
+[15] R. B. Abdessalem, A. Panichella, S. Nejati, L. C. Briand, and T. Stifter,
+“Automated repair of feature interaction failures in automated driving
+systems,” in Proceedings of the 29th ACM SIGSOFT International
+Symposium on Software Testing and Analysis, 2020, pp. 88–100.
+[16] G. Perrouin, S. Sen, J. Klein, B. Baudry, and Y. Le Traon, “Automated
+and scalable t-wise test case generation strategies for software product
+lines,” in 2010 Third international conference on software testing,
+verification and validation.
+IEEE, 2010, pp. 459–468.
+[17] A. Arrieta, S. Wang, G. Sagardui, and L. Etxeberria, “Search-based test
+case prioritization for simulation-based testing of cyber-physical system
+product lines,” Journal of Systems and Software, vol. 149, pp. 1–34,
+2019.
+[18] C. Henard, M. Papadakis, G. Perrouin, J. Klein, P. Heymans, and
+Y. Le Traon, “Bypassing the combinatorial explosion: Using similarity
+to generate and prioritize t-wise test configurations for software product
+lines,” IEEE Transactions on Software Engineering, vol. 40, no. 7, pp.
+650–670, 2014.
+[19] S. Wang, S. Ali, and A. Gotlieb, “Cost-effective test suite minimization
+in product lines using search techniques,” Journal of Systems and
+Software, vol. 103, pp. 370–391, 2015.
+[20] D. Marijan, A. Gotlieb, S. Sen, and A. Hervieu, “Practical pairwise test-
+ing for software product lines,” in Proceedings of the 17th international
+software product line conference, 2013, pp. 227–235.
+[21] A. Hervieu, D. Marijan, A. Gotlieb, and B. Baudry, “Practical mini-
+mization of pairwise-covering test configurations using constraint pro-
+gramming,” Information and Software Technology, vol. 71, pp. 129–146,
+2016.
+[22] R. Krishna, M. S. Iqbal, M. A. Javidian, B. Ray, and P. Jamshidi, “Cadet:
+Debugging and fixing misconfigurations using counterfactual reasoning,”
+arXiv preprint arXiv:2010.06061, 2020.
+[23] Y. Xiong, H. Zhang, A. Hubaux, S. She, J. Wang, and K. Czarnecki,
+“Range fixes: Interactive error resolution for software configuration,”
+Ieee transactions on software engineering, vol. 41, no. 6, pp. 603–619,
+2014.
+[24] C. Jung, A. Ahad, J. Jung, S. Elbaum, and Y. Kwon, “Swarmbug:
+debugging configuration bugs in swarm robotics,” in Proceedings of the
+29th ACM Joint Meeting on European Software Engineering Conference
+and Symposium on the Foundations of Software Engineering, 2021, pp.
+868–880.
+[25] J. Ayerdi, A. Garciandia, A. Arrieta, W. Afzal, E. Enoiu, A. Agirre,
+G. Sagardui, M. Arratibel, and O. Sellin, “Towards a taxonomy for
+eliciting design-operation continuum requirements of cyber-physical
+systems,” in 2020 IEEE 28th International Requirements Engineering
+Conference (RE).
+IEEE, 2020, pp. 280–290.
+[26] A. Gartziandia, A. Arrieta, J. Ayerdi, M. Illarramendi, A. Agirre,
+G. Sagardui, and M. Arratibel, “Machine learning-based test oracles for
+performance testing of cyber-physical systems: An industrial case study
+on elevators dispatching algorithms,” Journal of Software: Evolution and
+Process, p. e2465, 2022.
+[27] L. Han, S. Ali, T. Yue, A. Arrieta, and M. Arratibel, “Uncertainty-aware
+robustness assessment of industrial elevator systems,” ACM Transactions
+on Software Engineering and Methodology, 2022.
+[28] L. Han, T. Yue, S. Ali, A. Arrieta, and M. Arratibel, “Are elevator
+software robust against uncertainties? results and experiences from an
+industrial case study,” in Proceedings of the 30th ACM Joint European
+Software Engineering Conference and Symposium on the Foundations
+of Software Engineering, 2022, pp. 1331–1342.
+[29] A. Arrieta, M. Otaegi, L. Han, G. Sagardui, S. Ali, and M. Arratibel,
+“Automating test oracle generation in devops for industrial elevators,”
+in 2022 IEEE International Conference on Software Analysis, Evolution
+and Reengineering (SANER).
+IEEE, 2022, pp. 284–288.
+[30] G. Barney and L. Al-Sharif, Elevator traffic handbook: theory and
+practice.
+Routledge, 2015.
+[31] G. Barney, Transportation systems in buildings : CIBSE Guide
+D:
+2010.
+London:
+Chartered
+Institution
+of
+Building
+Services
+Engineers,
+2010.
+[Online].
+Available:
+https://www.worldcat.org/
+title/transportation-systems-in-buildings-cibse-guide-d-2010/oclc/
+880899711
+[32] C. L. Goues, M. Pradel, and A. Roychoudhury, “Automated program
+repair,” Communications of the ACM, vol. 62, no. 12, pp. 56–65, 2019.
+[33] M. Martinez, T. Durieux, R. Sommerard, J. Xuan, and M. Monperrus,
+“Automatic repair of real bugs in java: A large-scale experiment on the
+defects4j dataset,” Empirical Software Engineering, vol. 22, no. 4, pp.
+1936–1964, 2017.
+[34] A. Nilizadeh, G. T. Leavens, X.-B. D. Le, C. S. P˘as˘areanu, and D. R.
+Cok, “Exploring true test overfitting in dynamic automated program
+repair using formal methods,” in 2021 14th IEEE Conference on
+Software Testing, Verification and Validation (ICST).
+IEEE, 2021, pp.
+229–240.
+[35] E. K. Smith, E. T. Barr, C. Le Goues, and Y. Brun, “Is the cure
+worse than the disease? overfitting in automated program repair,” in
+Proceedings of the 2015 10th Joint Meeting on Foundations of Software
+Engineering, 2015, pp. 532–543.
+[36] M.-L. Siikonen, T. Susi, and H. Hakonen, “Passenger traffic flow
+simulation in tall buildings,” Elevator world, vol. 49, no. 8, pp. 117–123,
+2001.
+[37] J. Ayerdi, S. Segura, A. Arrieta, G. Sagardui, and M. Arratibel, “Qos-
+aware metamorphic testing: An elevation case study,” in 2020 IEEE 31st
+International Symposium on Software Reliability Engineering (ISSRE).
+IEEE, 2020, pp. 104–114.
+[38] J. Ayerdi, P. Valle, S. Segura, A. Arrieta, G. Sagardui, and M. Arratibel,
+“Performance-driven metamorphic testing of cyber-physical systems,”
+IEEE Transactions on Reliability, 2022.
+[39] J. Ayerdi, V. Terragni, A. Arrieta, P. Tonella, G. Sagardui, and M. Arrat-
+ibel, “Generating metamorphic relations for cyber-physical systems with
+genetic programming: an industrial case study,” in Proceedings of the
+29th ACM Joint Meeting on European Software Engineering Conference
+and Symposium on the Foundations of Software Engineering, 2021, pp.
+1264–1274.
+
+[40] A. Arrieta, S. Wang, U. Markiegi, A. Arruabarrena, L. Etxeberria, and
+G. Sagardui, “Pareto efficient multi-objective black-box test case selec-
+tion for simulation-based testing,” Information and Software Technology,
+vol. 114, pp. 137–154, 2019.
+[41] A. Arrieta, S. Wang, U. Markiegi, G. Sagardui, and L. Etxeberria, “Em-
+ploying multi-objective search to enhance reactive test case generation
+and prioritization for testing industrial cyber-physical systems,” IEEE
+Transactions on Industrial Informatics, vol. 14, no. 3, pp. 1055–1066,
+2017.
+[42] P. McMinn, “Search-based software testing: Past, present and future,”
+in 2011 IEEE Fourth International Conference on Software Testing,
+Verification and Validation Workshops.
+IEEE, 2011, pp. 153–163.
+[43] M. Di Penta, G. Canfora, G. Esposito, V. Mazza, and M. Bruno, “Search-
+based testing of service level agreements,” in Proceedings of the 9th
+annual conference on Genetic and evolutionary computation, 2007, pp.
+1090–1097.
+[44] S. Wang, S. Ali, T. Yue, Y. Li, and M. Liaaen, “A practical guide to select
+quality indicators for assessing pareto-based search algorithms in search-
+based software engineering,” in Proceedings of the 38th International
+Conference on Software Engineering, 2016, pp. 631–642.
+[45] M. Li, T. Chen, and X. Yao, “How to evaluate solutions in pareto-based
+search-based software engineering: A critical review and methodological
+guidance,” IEEE Transactions on Software Engineering, vol. 48, no. 05,
+pp. 1771–1799, 2022.
+[46] K. Shang, H. Ishibuchi, L. He, and L. M. Pang, “A survey on the
+hypervolume indicator in evolutionary multiobjective optimization,”
+IEEE Transactions on Evolutionary Computation, vol. 25, no. 1, pp.
+1–20, 2020.
+[47] J. Romano, J. D. Kromrey, J. Coraggio, J. Skowronek, and L. Devine,
+“Exploring methods for evaluating group differences on the nsse and
+other surveys: Are the t-test and cohen’sd indices the most appropriate
+choices,” in annual meeting of the Southern Association for Institutional
+Research.
+Citeseer, 2006, pp. 1–51.
+[48] A. Arcuri and L. Briand, “A practical guide for using statistical tests
+to assess randomized algorithms in software engineering,” in 2011 33rd
+International Conference on Software Engineering (ICSE). IEEE, 2011,
+pp. 1–10.
+[49] L. Briand, S. Nejati, M. Sabetzadeh, and D. Bianculli, “Testing the
+untestable: model testing of complex software-intensive systems,” in
+Proceedings of the 38th international conference on software engineer-
+ing companion, 2016, pp. 789–792.
+[50] R. Matinnejad, S. Nejati, L. C. Briand, and T. Bruckmann, “Test
+generation and test prioritization for simulink models with dynamic
+behavior,” IEEE Transactions on Software Engineering, vol. 45, no. 9,
+pp. 919–944, 2018.
+[51] M. Monperrus, “The living review on automated program repair,” Ph.D.
+dissertation, HAL Archives Ouvertes, 2018.
+[59] F. DeMarco, J. Xuan, D. Le Berre, and M. Monperrus, “Automatic
+repair of buggy if conditions and missing preconditions with smt,” in
+Proceedings of the 6th international workshop on constraints in software
+testing, verification, and analysis, 2014, pp. 30–39.
+[52] T. Ackling, B. Alexander, and I. Grunert, “Evolving patches for software
+repair,” in Proceedings of the 13th annual conference on Genetic and
+evolutionary computation, 2011, pp. 1427–1434.
+[53] D. Kim, J. Nam, J. Song, and S. Kim, “Automatic patch generation
+learned from human-written patches,” in 2013 35th International Con-
+ference on Software Engineering (ICSE).
+IEEE, 2013, pp. 802–811.
+[54] J. D. Knowles, R. A. Watson, and D. W. Corne, “Reducing local optima
+in single-objective problems by multi-objectivization,” in International
+conference on evolutionary multi-criterion optimization. Springer, 2001,
+pp. 269–283.
+[55] C. Le Goues, T. Nguyen, S. Forrest, and W. Weimer, “Genprog: A
+generic method for automatic software repair,” IEEE Transactions on
+Software Engineering, vol. 38, no. 1, pp. 54–72, 2011.
+[56] M. P. Gissurarson, L. Applis, A. Panichella, A. van Deursen, and
+D. Sands, “Propr: property-based automatic program repair,” in Pro-
+ceedings of the 44th International Conference on Software Engineering,
+2022, pp. 1768–1780.
+[57] A. Arcuri, “On the automation of fixing software bugs,” in Companion
+of the 30th International Conference on Software Engineering, 2008,
+pp. 1003–1006.
+[58] V. Dallmeier, A. Zeller, and B. Meyer, “Generating fixes from object
+behavior anomalies,” in 2009 IEEE/ACM International Conference on
+Automated Software Engineering.
+IEEE, 2009, pp. 550–554.
+[60] T. Ji, L. Chen, X. Mao, and X. Yi, “Automated program repair by using
+similar code containing fix ingredients,” in 2016 IEEE 40th Annual
+Computer Software and Applications Conference (COMPSAC), vol. 1.
+IEEE, 2016, pp. 197–202.
+[61] H. D. T. Nguyen, D. Qi, A. Roychoudhury, and S. Chandra, “Semfix:
+Program repair via semantic analysis,” in 2013 35th International
+Conference on Software Engineering (ICSE).
+IEEE, 2013, pp. 772–
+781.
+[62] Y. Qi, X. Mao, Y. Lei, Z. Dai, and C. Wang, “The strength of
+random search on automated program repair,” in Proceedings of the 36th
+International Conference on Software Engineering, 2014, pp. 254–265.
+[63] Z. Qi, F. Long, S. Achour, and M. Rinard, “An analysis of patch
+plausibility and correctness for generate-and-validate patch generation
+systems,” in Proceedings of the 2015 International Symposium on
+Software Testing and Analysis, 2015, pp. 24–36.
+[64] W. Weimer, Z. P. Fry, and S. Forrest, “Leveraging program equivalence
+for adaptive program repair: Models and first results,” in 2013 28th
+IEEE/ACM International Conference on Automated Software Engineer-
+ing (ASE).
+IEEE, 2013, pp. 356–366.
+[65] Y. Lou, A. Ghanbari, X. Li, L. Zhang, H. Zhang, D. Hao, and
+L. Zhang, “Can automated program repair refine fault localization? a
+unified debugging approach,” in Proceedings of the 29th ACM SIGSOFT
+International Symposium on Software Testing and Analysis, 2020, pp.
+75–87.
+[66] S. Benton, X. Li, Y. Lou, and L. Zhang, “Evaluating and improving
+unified debugging,” IEEE Transactions on Software Engineering, 2021.
+

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+MNRAS 000, 1–6 (2022)
+Preprint 1 February 2023
+Compiled using MNRAS LATEX style file v3.0
+Supernova connection of unidentified ultra high energy gamma-ray source
+LHAASO J2108+5157
+Agnibha De Sarkar,1★
+1Astronomy & Astrophysics group, Raman Research Institute
+C. V. Raman Avenue, 5th Cross Road, Sadashivanagar, Bengaluru 560080, Karnataka, India
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+We present a simple phenomenological model of hadronic interaction between protons accelerated in an old supernova remnant
+(SNR) and cold protons situated within the associated molecular clouds (MCs). The accelerated protons from the old SNR
+escaped the SNR shock front, and got injected into the MCs at an earlier time, producing ultra high energy gamma-rays and
+neutrinos through inelastic proton-proton interaction. We also take into account the acceleration and subsequent escape of
+electrons from the SNR shock front. The escaped electrons produce gamma-rays through various radiative cooling mechanisms,
+after getting injected into the MCs. We use the model discussed in this letter to explain the multiwavelength (MWL) spectral
+energy distribution (SED) of unidentified Galactic ultra high energy gamma-ray source LHAASO J2108+5157. We also discuss
+the feasibility of applying this model in other cases as well. Future observations can test the viability of the model discussed in
+this letter, which will in turn confirm that the SNRs can, in fact, accelerate particles up to PeV energies.
+Key words: radiation mechanisms: non-thermal – ISM: individual objects: LHAASO J2108+5157 – gamma-rays: ISM – ISM:
+supernova remnants
+1 INTRODUCTION
+Observations by the Large High Altitude Air Shower Observatory
+(LHAASO), located in China, have opened a new era of gamma-
+ray astrophysics (Cao 2010). Since it has become operational on
+2020 April, LHAASO has detected more than a dozen ultra high
+energy (UHE; E𝛾 ≥ 100 TeV) gamma-ray sources, most of which are
+unidentified (Cao et al. 2021a). The detection of these UHE gamma-
+ray sources indicates the presence of cosmic ray (CR) accelerators in
+our Milky Way Galaxy, which can accelerate particles up to PeV (=
+1015 eV) energies, more commonly known as “PeVatrons”. Several
+classes of Galactic sources such as supernova remnants (SNRs), pul-
+sar wind nebulae (PWNe), young stellar clusters have been posited to
+be potential PeVatron candidates. Although it is still an open question
+as to what class of source is responsible for accelerating particles up
+to PeV energies, most of the UHE gamma-ray sources detected by
+LHAASO, along with their high energy (HE; E𝛾 < 100 GeV) and
+very high energy (VHE; 100 GeV ≤ E𝛾 < 100 TeV) gamma-ray
+counterparts, have been associated with PWNe in previous studies,
+due to their close proximity with an energetic pulsar, and their typ-
+ically extended spatial morphology (Abdalla et al. 2018). This idea
+gained steam after it was confirmed that Crab pulsar wind nebula
+is indeed a PeVatron source (Cao et al. 2021a). However, in spite
+of the notion that energetic pulsars with high spin-down luminosity
+( �𝐸 > 1036 erg s−1) coinciding or remaining in a very close spatial
+proximity of UHE gamma-ray sources, may be a universal feature
+★ E-mail: [email protected]
+(Albert et al. 2021), further theoretical analyses of these LHAASO
+detected UHE gamma-ray sources seem to tell a different story.
+Recent studies have modeled a few of the significantly detected
+LHAASO sources in detail. For example, De Sarkar & Gupta (2022)
+found that the UHE gamma-ray emission observed from LHAASO
+J1908+0621 is most likely hadronic in origin, emanated from the
+interaction between SNR G40.5-0.5 and the associated MCs. On the
+other hand, in De Sarkar et al. (2022), another significantly detected
+source, LHAASO J2226+6057, was extensively modeled assuming
+that the UHE gamma-ray emission is coming from the PWN associ-
+ated with PSR J2229+6114. As caveats of the model, it was found
+that the PWN interpretation of LHAASO J2226+6057 leads to a very
+high radius of PWN, as well as a very small value of magnetic field.
+Naturally, these results are in contrast with the observational results
+(Ge et al. 2021; Liang et al. 2022), thus indicating that the PWN may
+not be the contributing source to power the UHE gamma-ray source
+detected. This indicates interaction between SNRs and associated
+MCs may be the primary reason behind particle acceleration to PeV
+energies in Galactic sources.
+With this factors in mind, we focus on the emission of the recent
+LHAASO detected unidentified UHE gamma-ray sources: LHAASO
+J2108+5157 (Cao et al. 2021c) and LHAASO J0341+5258 (Cao
+et al. 2021b). Both of these sources were found to be associated with
+MCs, but no apparent association with energetic pulsars or SNRs
+were established. Scenarios including leptonic emission from TeV
+halo (Abe et al. 2022), injection of particles from past explosions
+(Kar & Gupta 2022), hadronic interaction between SNR and MCs
+(Cao et al. 2021c) were discussed in previous literatures. But most of
+these models do not explain the HE-VHE-UHE gamma-ray spectral
+© 2022 The Authors
+arXiv:2301.13451v1  [astro-ph.HE]  31 Jan 2023
+
+2
+A. De Sarkar
+features entirely. Moreover, recent reveal of VHE gamma-ray upper
+limits observed by the Large-Sized Telescope - Cherenkov Telescope
+Array (LST-CTA) (Abe et al. 2022) has overruled some of these
+models for the case of LHAASO J2108+5157. The absence of a
+powerful pulsar or supernova remnant adds to the mystery as well,
+leaving one asking what is the possible emission mechanism at play
+in case of these unidentified UHE gamma-ray sources.
+To that end, in this letter, we discuss and apply a phenomeno-
+logical model, in which accelerated particles, escaped from an old,
+shell-type SNR (now invisible), interact with the associated MCs to
+produce the observed HE-VHE-UHE gamma-ray data for the case
+of LHAASO J2108+5157. We also provide the possible age of the
+old SNR, and account for the disappearance of the SNR in question.
+Our simple model is also consistent with the X-ray 2𝜎 upper limits
+given by Abe et al. (2022). We also discuss the applicability of the
+model in other unidentified Galactic UHE gamma-ray source such
+as LHAASO J0341+5258. Furthermore, we report that the neutrino
+flux produced from the hadronic interaction considered in this model,
+will be non-detectable, even by the next generation observatory such
+as ICECUBE-Gen2 (Aartsen et al. 2021).
+2 THE MODEL
+In this section, we discuss the essentials of the model used to calculate
+the hadronic and leptonic components produced from the interaction
+between an old, now invisible SNR and the associated MCs. A more
+detailed discussion of the model can be found in De Sarkar & Gupta
+(2022), where we developed and applied our model to explain the
+peculiar HE-VHE-UHE gamma-ray SED of LHAASO J1908+0621.
+Our simple model assumes that the supernova had exploded at the
+center of the cavity of a shell-like structure, which is surrounded by
+dense MCs. After this explosion, the SNR shock front expands inside
+the shell cavity, and finally hits the surrounding MCs. During the
+collision between the shock front and associated MCs, the accelerated
+particles get injected into the MCs to produce further emissions.
+After the explosion, the supernova (SN) shock front expands freely
+during its free expansion phase. When the amount of swept-up inter-
+stellar medium (ISM) material becomes equal to that of the ejected
+material at t = t𝑆𝑒𝑑𝑜𝑣, the SN enters the adiabatic Sedov phase. Fi-
+nally after t = t𝑟𝑎𝑑, the SN enters the radiative phase, in which the
+cooling timescales is less than the dynamic timescales. During its
+evolution, the time dependence of the shock velocity and radius can
+be given by the following simple relations (Fujita et al. 2009; Ohira
+et al. 2012; De Sarkar & Gupta 2022),
+𝑣𝑠ℎ(𝑡) =
+�
+𝑣𝑖
+(𝑡 < 𝑡𝑆𝑒𝑑𝑜𝑣)
+𝑣𝑖(𝑡/𝑡𝑆𝑒𝑑𝑜𝑣)−3/5
+(𝑡𝑆𝑒𝑑𝑜𝑣 < 𝑡)
+(1)
+and,
+𝑅𝑠ℎ(𝑡) ∝
+�
+(𝑡/𝑡𝑆𝑒𝑑𝑜𝑣)
+(𝑡 < 𝑡𝑆𝑒𝑑𝑜𝑣)
+(𝑡/𝑡𝑆𝑒𝑑𝑜𝑣)2/5
+(𝑡𝑆𝑒𝑑𝑜𝑣 < 𝑡)
+(2)
+We note that for the entirety of this work, we have assumed the
+following values: initial velocity of the shock v𝑖 = 109 cm s−1 (Fujita
+et al. 2009), radius of the shock and time at the beginning of the Sedov
+phase, R𝑆𝑒𝑑𝑜𝑣 and t𝑆𝑒𝑑𝑜𝑣, to be 2.1 pc and 210 yr, respectively
+(Ohira et al. 2011; Makino et al. 2019).
+The CR protons are accelerated through Diffusive Shock Acceler-
+ation (DSA) mechanism when the SN is in the Sedov phase, where
+the CR protons accelerate by scattering back and forth across the
+shock front, while the shock is expanding towards the surround-
+ing MCs. We assume an escape-limited acceleration scenario (Ohira
+et al. 2010), in which the CR protons need to escape a geometrical
+confinement region around the SN shock front produced by strong
+magnetic turbulence, in order to get injected into the MCs and take
+part in further interactions. The radius of the outermost boundary
+of this confinement region (i.e., the escape boundary) is called the
+escape radius, and it can be denoted by,
+𝑅𝑒𝑠𝑐(𝑡) = (1 + 𝜅)𝑅𝑠ℎ(𝑡),
+(3)
+where 𝜅 ≈ 0.04 (Ohira et al. 2010; Makino et al. 2019), and is
+defined by the geometrical confinement condition D𝑠ℎ/v𝑠ℎ ∼ l𝑒𝑠𝑐 =
+𝜅R𝑠ℎ, where l𝑒𝑠𝑐 is the distance of the escape boundary from the
+shock front and D𝑠ℎ is the diffusion coefficient around the shock
+(Ohira et al. 2010).
+After the explosion, the escape boundary in front of the shock
+front eventually hits the surrounding MCs after traversing a distance
+of R𝑀𝐶, the distance of MC surface from the cavity center. This
+essentially means that at the time of collision t𝑐𝑜𝑙𝑙, the escape ra-
+dius is equal to the MC surface distance, i.e. R𝑒𝑠𝑐 (t𝑐𝑜𝑙𝑙) = R𝑀𝐶
+≈ R𝑠ℎ (t𝑐𝑜𝑙𝑙), and at the time of collision, the velocity of the shock
+is denoted by v𝑠ℎ(t𝑐𝑜𝑙𝑙). We assume that the particle acceleration
+stops at t = t𝑐𝑜𝑙𝑙 (Fujita et al. 2009). Consequently, protons accel-
+erated at t ≤ t𝑐𝑜𝑙𝑙 take part in further interactions inside the MCs.
+Moreover, only the protons with sufficiently high energies will es-
+cape the confinement region around the SNR shock front, whereas
+the low energy protons will remain confined around the SNR. So a
+suppression of fluxes in the lower energies, as well as a dominant con-
+tribution of fluxes in the highest energies should be expected in this
+scenario. The confinement condition invoked in this model changes
+the spectral shape of the injected proton population by constraining
+the minimum energy limit.
+We estimate the minimum energy limit of the injected proton pop-
+ulation by assuming that the escape energy is a decreasing function
+of the shock radius (Makino et al. 2019). This approach is based
+on the assumption that the maximum energy of CR protons, E𝑝
+𝑚𝑎𝑥
+is expected to increase up to knee energy (≈ 1015.5 eV) until the
+beginning of the Sedov phase, and then decrease from that epoch
+(Gabici et al. 2009; Ohira et al. 2012). The minimum energy re-
+quired by protons to escape the confinement region can be given by
+the phenomenological relation,
+𝐸 𝑝
+𝑒𝑠𝑐 = 𝐸 𝑝
+𝑚𝑎𝑥
+�
+𝑅𝑠ℎ
+𝑅𝑆𝑒𝑑𝑜𝑣
+�−𝛼
+,
+(4)
+where 𝛼 signifies the evolution of the maximum energy during the
+Sedov phase (Makino et al. 2019). We treat 𝛼 as a free parameter in
+this work. After putting R𝑠ℎ ≈ R𝑒𝑠𝑐 = R𝑀𝐶 at the time of collision,
+we find the minimum energy required to escape the confinement
+zone, which also gives us the minimum energy threshold for the
+proton population that gets injected inside the surrounding MCs, i.e.,
+E𝑝
+𝑒𝑠𝑐 = E𝑝
+𝑚𝑖𝑛. Since protons are accelerated by DSA mechanism, we
+can expect the CR proton spectrum at the shock front ∝ E−𝑠. Then,
+in an escape-limited particle acceleration scenario, the protons with
+energies greater than 𝐸 𝑝
+𝑒𝑠𝑐 have a spectrum (Ohira et al. 2010),
+𝑁 𝑝
+𝑒𝑠𝑐(𝐸) ∝ 𝐸−[𝑠+(𝛽/𝛼)],
+(5)
+where 𝛽 represents a thermal leakage model of CR injection and
+is given by 𝛽 = 3(3–s)/2 (Makino et al. 2019). For a typical value of
+s = 2, we get the value of 𝛽 = 1.5. Note that the spectral shape as
+MNRAS 000, 1–6 (2022)
+
+Supernova connection of LHAASO J2108+5157
+3
+well as the minimum energy of the proton population are calculated
+at the time when the escape boundary hits the surrounding MCs at t
+= t𝑐𝑜𝑙𝑙.
+At t > t𝑐𝑜𝑙𝑙, the shock enters the momentum conserving “snow-
+plow” phase. The time evolution of the radius of the shocked shell
+R𝑠ℎ𝑒𝑙𝑙 (t) inside the MCs can be found using momentum conserva-
+tion equation (Fujita et al. 2009; De Sarkar & Gupta 2022),
+4𝜋
+3
+�
+𝑛𝑀𝐶 (𝑅𝑠ℎ𝑒𝑙𝑙(𝑡)3 − 𝑅𝑠ℎ(𝑡𝑐𝑜𝑙𝑙)3) + 𝑛𝑐𝑎𝑣 𝑅𝑠ℎ(𝑡𝑐𝑜𝑙𝑙)3�
+�𝑅𝑠ℎ𝑒𝑙𝑙(𝑡)
+= 4𝜋
+3 𝑛𝑐𝑎𝑣 𝑅𝑠ℎ(𝑡𝑐𝑜𝑙𝑙)3𝑣𝑠ℎ(𝑡𝑐𝑜𝑙𝑙),
+(6)
+with R𝑠ℎ𝑒𝑙𝑙 = R𝑀𝐶 at t = t𝑐𝑜𝑙𝑙, n𝑀𝐶 is the number density of
+the MCs, and n𝑐𝑎𝑣 (≈ 1 cm−3) is the number density inside the
+cavity of the shell. We solve equation 6 numerically for t > t𝑐𝑜𝑙𝑙,
+to estimate the current age of the SNR. We estimate the current
+age by considering the fact that the velocity of the shocked shell
+at the current age must be similar or even smaller than the internal
+gas velocity of the MCs. This approach takes into account the non-
+detection of any SNR shell in unidentified UHE gamma-ray sources
+discussed above, as the shell of the SNR becomes invisible owing to
+the higher internal gas velocity of the MCs as compared to that of the
+shocked shell. We consider the above discussed proton population
+and total number density of the cold protons inside the surrounding
+MCs (n𝑀𝐶) to calculate total gamma-ray flux produced through
+hadronic interaction (Kafexhiu et al. 2014).
+Similar to protons, electrons can also get accelerated in the SNR
+shock front and subsequently escape the confinement region to get
+injected in the associated MCs. Moreover, electrons also lose energy
+through radiative cooling very efficiently. Hence, the injected electron
+population was considered to be escape-limited, as well as loss-
+limited (Yamazaki et al. 2006). We consider the spectral index of
+the escaped electron population to be same as that of protons (Ohira
+et al. 2012; De Sarkar & Gupta 2022). To take into accout loss-limited
+nature of injected electron population, we consider a power law with
+exponential cutoff as the spectral shape of the escaped electrons,
+𝑁𝑒
+𝑒𝑠𝑐(���) ∝ 𝐸−[𝑠+(𝛽/𝛼)]𝑒𝑥𝑝(−𝐸/𝐸𝑒
+𝑚𝑎𝑥),
+(7)
+where, maximum energy of the electron population has been de-
+termined by synchrotron cooling (Yamazaki et al. 2006; Fujita et al.
+2009),
+𝐸𝑒
+𝑚𝑎𝑥 = 14ℎ−1/2
+�
+𝑣𝑠ℎ
+108 cm/s
+� �
+𝐵
+10 𝜇G
+�−1/2
+TeV,
+(8)
+where, v𝑠ℎ is the velocity of the shock front and B is the down-
+stream magnetic field. The parameter h (= 0.05𝑟 ( 𝑓 +𝑟𝑔)
+𝑟−1
+, where r is
+the density compression ratio, f and g are functions of shock angle
+and gyro-factors) is used as a factor to calculate the acceleration time
+scale of DSA. We take h ∼ 1, considering the SNR in Sedov phase
+and neglecting non-linear effects, similar to Yamazaki et al. (2006).
+We consider v𝑠ℎ = v𝑠ℎ (t𝑐𝑜𝑙𝑙) since we calculate the maximum en-
+ergy of the lepton population at the collision time and B = B𝑀𝐶,
+the magnetic field inside the MCs. The minimum energy of the elec-
+tron population was considered to be E𝑒
+𝑚𝑖𝑛 ≈ 500 MeV (De Sarkar
+& Gupta 2022). Furthermore, we consider bremsstrahlung, Inverse-
+Compton (IC) and synchrotron cooling (Blumenthal & Gould 1970;
+Ghisellini et al. 1988; Baring et al. 1999) of the injected lepton
+population to calculate the gamma-ray flux produced. For IC inter-
+action, we consider interstellar radiation field (ISRF) from Popescu
+et al. (2017) at the source position, and the Cosmic Microwave Back-
+ground (temperature T𝐶𝑀 𝐵 = 2.7 K, energy density U𝐶𝑀 𝐵 = 0.25
+eV cm−3) contribution as well. The number density was considered
+to be same as that of the MCs.
+Finally we note that in this particular model, we have neglected
+the effect of diffusion of particles inside the MCs and assumed that
+the CR particles, both protons and electrons, lose energy through
+rapid cooling before escaping the cloud. This assumption can be
+realized by considering the idea that inside MCs, the diffusion is
+considerably suppressed (D ≈ 1025−26 cm2 s−1) as compared to that
+observed in the ISM (D ≈ 1028 cm2 s−1) (Gabici et al. 2007, 2009;
+Fujita et al. 2009; De Sarkar et al. 2021). Generation of plasma waves
+by CR streaming can be the reason behind the slow diffusion inside
+the MCs (Wentzel 1974). On the other hand, if the trapping of CR
+particles occurs due to some particular orientation of the magnetic
+field inside the MCs, then also the escape of the particles from the
+MCs will not be effective and can be neglected (Makino et al. 2019).
+Consequently, we have considered a steady-state proton and electron
+population to explain the SED of LHAASO J2108+5157, details of
+which are given in the next section.
+3 APPLICATION OF THE MODEL: LHAASO J2108+5157
+LHAASO J2108+5157 is an UHE gamma-ray source detected by
+LHAASO at R.A. = 317.22◦ ± 0.07◦
+𝑠𝑡𝑎𝑡 and decl. = 51.95◦ ± 0.05◦
+𝑠𝑡𝑎𝑡
+(Cao et al. 2021c) with a significance of 6.4𝜎 above 100 TeV. The
+source is reported to have a 95% confidence level extension upper
+limit of 0.26◦ with a 2D symmetrical Gaussian template, and its
+spectrum above 25 TeV can be well described by a power law with a
+photon index of 2.83 ± 0.18 (Cao et al. 2021c). Although no X-ray
+counterpart within 0.26◦ radius of the source was found, a spatially
+extended, HE counterpart 4FGL J2108.0+5155e (extension ∼ 0.48◦)
+(Abdollahi et al. 2020) was observed to be situated at an angular
+distance of 0.13◦ (Cao et al. 2021c). A new hard spectrum GeV
+source was also found at l = 92.35◦ and b = 2.56◦ by Fermi-LAT
+data analysis (Abe et al. 2022), but its large angular separation (∼
+0.27◦) from the LHAASO source indicates that this new source can
+hardly be a counterpart. Although no VHE component within 0.5◦
+radius was confirmed previously, recent observations by LST-CTA
+has hinted towards an existence of a source with 3.67𝜎 detection
+significance in the energy range of 3 - 100 TeV (Abe et al. 2022).
+Future observations may confirm an existence of a VHE counterpart
+with hard spectral index. The UHE source is located near the center
+of a GMC labeled [MML2017]4607 (Miville-Deschênes et al. 2017),
+which has an average angular radius and mass of 0.236◦ and 8469
+M⊙, respectively, and is situated at a distance of 3.28 kpc from
+Earth. The average number density of the GMC was estimated to
+be n𝑀𝐶 ≈ 30 cm−3 (Cao et al. 2021c). The presence of the GMC,
+spatially coincident with the UHE gamma-ray source points towards
+the hadronic origin, but leptonic origin can not be neglected. The
+absence of any energetic pulsar, its wind nebula or SNR warrants a
+cautious approach in unveiling the true nature of emission regarding
+this UHE source.
+Two young open stellar clusters Kronberger 80 and Kronberger 82
+are in the vicinity of the LHAASO source (with angular distances of
+0.62◦ and 0.45◦, respectively) (Cao et al. 2021c). But large angular
+separation between these clusters and LHAASO source centroid, as
+well as absence of proper distance estimation hint that the contri-
+bution of these clusters are unlikely (Cao et al. 2021c; Abe et al.
+MNRAS 000, 1–6 (2022)
+
+4
+A. De Sarkar
+2022). Cao et al. (2021c) suggested that UHE gamma-ray emission
+is due to an interaction of escaping CRs with MCs, whereas the GeV
+counterpart maybe due to an old SNR. However, Abe et al. (2022)
+pointed out that photon index of GeV counterpart spectrum is too
+soft compared to the observations of old SNRs interacting with MCs
+(Yuan et al. 2012), and to produce UHE gamma-ray spectrum, the
+required spectral index of the proton population has to be very hard
+as compared to the standard DSA theory. Instead, Abe et al. (2022)
+proposed an alternate leptonic scenario, in which UHE gamma-ray
+emission is due to TeV halo emission, and the GeV counterpart is due
+to a tentative, previously undetected pulsar. But a very low associated
+magnetic field (even lower than the average Galactic magnetic field),
+and non-detection of a pulsar make the TeV halo interpretation ques-
+tionable, and open the source up for further exploration. To that end,
+we apply the model discussed in Section 2 to explain the gamma-ray
+data from HE to UHE energy range, while being consistent with the
+X-ray 2𝜎 upper limits. We note that these 2𝜎 X-ray upper limits
+correspond to a uniform, circular source with a radius of 6′ centered
+on the position of the LHAASO source (Abe et al. 2022). We explain
+the VHE-UHE gamma-ray data with hadronic component produced
+from the interaction between protons, accelerated and escaped at an
+early time from a now old SNR shock front, with protons inside
+the surrounding MCs, whereas the HE gamma-ray data is explained
+using bremsstrahlung cooling of accelerated and escaped electrons
+inside the medium of the MCs. Our model also shows that the main
+contribution in X-ray range comes from the synchrotron cooling of
+the same accelerated and escaped electrons.
+In this work, we have considered the free parameter 𝛼 = 1.875, and
+then let the total energy budgets of proton and electron populations
+vary to explain the MWL SED. Considering the value of 𝛼, and the
+values of s and 𝛽 discussed in Section 2, we get the spectral indices
+of the escaped electron and proton populations as p = [s + (𝛽/𝛼)]
+= 2.8. The distance of the source was taken to be d ∼ 3 kpc. The
+model spectrum components, as well as the considered MWL SED
+are shown in Figure 1. Also, we calculate the time evolution of SNR
+shocked shell inside the associated MCs using equation 6, and find
+that the SNR, with a final radius of ∼ 30 pc, has to be ∼ 4.4 × 105
+years old, for the shock velocity to be lower than the internal gas
+velocity of MC [MML2017]4607 (∼ 13 km s−1) (Cao et al. 2021c),
+and the SNR shell to disappear. The time evolution of the shocked
+shell is shown in Figure 2. Finally, the model parameters required to
+explain the gamma-ray data are shown in Table 1. We have used open
+source code GAMERA (Hahn 2016) to calculate the model spectrum
+of different components.
+4 DISCUSSION AND CONCLUSION
+In this letter, we have discussed and applied a simple, analytical and
+phenomenological model to explain the HE-VHE-UHE gamma-ray
+data observed from the direction of LHAASO J2108+5157. By only
+adjusting the index 𝛼, not only we show that the model components
+are consistent with gamma-ray and X-ray observations, the results
+also naturally explain the observed morphology of the source re-
+gion, e.g., the disappearance of the SNR at current age. As expected,
+the SNR was found be old (> 105 years). This also explains why
+no pulsar has been seen in the source region, as the pulsar is ex-
+pected to leave the source region due to its natal kick velocity (∼
+400-500 km s−1) (Gaensler & Slane 2006). Similar nature and emis-
+sion were also found in another UHE gamma-ray source, LHAASO
+J1908+0621, details of which were explained by this model in De
+Sarkar & Gupta (2022). The fact that the emission of multiple UHE
+10
+12
+10
+10
+10
+8
+10
+6
+10
+4
+10
+2
+100
+102
+Energy (TeV)
+10
+16
+10
+15
+10
+14
+10
+13
+10
+12
+10
+11
+E2 J(E)[erg cm
+2 s
+1]
+pp
+synchrotron
+bremsstrahlung
+inverse-compton
+Fermi-LAT (Abe et al. 2022)
+Fermi-LAT (Cao et al. 2021)
+LHAASO
+LST-CTA
+XMM-Newton
+Figure 1. MWL SED of LHAASO J2108+5157. Gamma-ray data points and
+upper limits obtained from different observatories such as Fermi-LAT (red
+(Abe et al. 2022), purple (Cao et al. 2021c)), LHAASO (blue (Cao et al.
+2021c)), and LST-CTA (green (Abe et al. 2022)) are shown in the figure. The
+XMM-Newton X-ray 2𝜎 upper limits (Abe et al. 2022) are given in teal. The
+model p-p interaction (solid line), bremsstrahlung (dashed), IC (dotted), and
+synchrotron (dot-dashed) components are also shown in the figure.
+103
+104
+105
+106
+Time (years)
+16
+18
+20
+22
+24
+26
+28
+30
+32
+Shock radius (pc)
+LHAASO J2108+5157
+Figure 2. Time evolution of the shocked shell associated with the old SNR,
+inside the surrounding MCs.
+gamma-ray sources were explained by the same model hints towards
+its validity in a larger context. Interestingly, another unidentified
+UHE gamma-ray source, LHAASO J0341+5258, also shows similar
+characteristics shown by LHAASO J2108+5157 (Cao et al. 2021b).
+It is very likely that this model is applicable in that case as well.
+However, in that case, the VHE counterpart has not been properly
+constrained, and the High Altitude Water Cherenkov (HAWC) upper
+limit provided in Cao et al. (2021b) corresponds to only a 2𝜎 detec-
+tion significance. Further observations by CTA and detailed analysis
+by Fermi-LAT will be necessary to properly constrain the emission
+of LHAASO J0341+5258.
+From Figure 1, we can see that the hadronic component adequately
+explain the VHE-UHE gamma-ray data, whereas the bremsstrahlung
+component, originated from the cooling of the electron population,
+explains the gamma-ray data in the HE range. The bremsstrahlung
+MNRAS 000, 1–6 (2022)
+
+Supernova connection of LHAASO J2108+5157
+5
+Table 1. Parameters Used in The Model.
+Definition
+Parameter
+Value
+SNR/MC structure and evolution:
+Initial shock velocity
+v𝑖 (cm/s)
+109
+Time at the start of Sedov phase
+t𝑆𝑒𝑑𝑜𝑣 (years)
+210
+Shock radius at the start of Sedov phase R𝑆𝑒𝑑𝑜𝑣 (pc)
+2.1
+Time of collision
+t𝑐𝑜𝑙𝑙 (years)
+3.83 × 103
+Shock radius at time of collision
+R𝑠ℎ (t𝑐𝑜𝑙𝑙) (pc)
+16.77 (= R𝑀𝐶)
+Shock velocity at time of collision
+v𝑠ℎ (t𝑐𝑜𝑙𝑙) (cm/s) 1.75 × 108
+Current age of SNR
+t𝑎𝑔𝑒 (years)
+4.4 × 105
+Final radius of shock
+R𝑠ℎ (t𝑎𝑔𝑒) (pc)
+30
+Final velocity of shock
+v𝑠ℎ (t𝑎𝑔𝑒) (cm/s) 1.2 × 106
+Distance
+d (kpc)
+3
+MC number density
+n𝑀𝐶 (cm−3)
+30
+MC magnetic field
+B𝑀𝐶 (𝜇G)
+25
+Cavity number density
+n𝑐𝑎𝑣 (cm−3)
+1
+Hadronic component:
+Minimum energy
+E𝑝
+𝑚𝑖𝑛 (TeV)
+63
+Maximum energy
+E𝑝
+𝑚𝑎𝑥 (TeV)
+3.1 × 103
+Spectral index
+p
+2.8
+Energy budget
+W𝑝 (erg)
+3.6 × 1047
+Leptonic component:
+Minimum energy
+E𝑒
+𝑚𝑖𝑛 (TeV)
+5 × 10−4
+Maximum energy
+E𝑒𝑚𝑎𝑥 (TeV)
+15.5
+Spectral index
+p
+2.8
+Energy budget
+W𝑒 (erg)
+3.6 × 1047
+component is expected to dominate the IC component, as the in-
+teraction is taking place inside MCs with a high number density
+of cold protons. Moreover, the synchrotron component does not
+violate the X-ray 2𝜎 upper limits. We note that no proper radio
+counterpart has been associated with the LHAASO J2108+5157 yet.
+An extended radio source associated with nearby star-forming re-
+gion (Cao et al. 2021c), as well as point-like radio source NVSS
+210803+515255 or WENSS B2106.4+5140 (Abe et al. 2022) were
+found within 95% extension upper limit of LHAASO J2108+5157
+and 4FGL J2108.0+5155e. Since no proper association was estab-
+lished between these sources and the gamma-ray source, we refrain
+from including their radio data in this study to further constrain the
+model, and we follow the MWL SED discussed in (Abe et al. 2022)
+to ascertain the feasibility of the model discussed in this letter.
+As discussed earlier, we have neglected the effect of particle diffu-
+sion in this model. We note that such an assumption may likely lead
+to an overestimation, and the aspect of suppressed diffusion inside
+the MCs is highly uncertain (Xu et al. 2016; Dogiel et al. 2015). In-
+troducing an energy-independent diffusion coefficient, as discussed
+in Dogiel et al. (2015), will lead to higher energy budgets required
+by the electron and proton populations to explain the data. The sup-
+pressed diffusion coefficient introduced by Gabici et al. (2009) has
+similar energy dependence as to that observed in ISM, but the exact
+energy dependence of diffusion coefficient inside clouds is not well
+constrained. So, to avoid further complications, we have neglected
+the effect of diffusion in this model, similar to Fujita et al. (2009);
+Makino et al. (2019), and assumed that the injected particles quickly
+cool down before escaping the MC medium.
+We further note that we do not consider the contribution of accel-
+erated and escaped particles, when the shock front is within the MC
+medium, in calculating the total gamma-ray SED. Even if the SNR
+is still in the Sedov phase when the shock is within the MCs, the
+corresponding contribution was found to be negligible. Moreover,
+the acceleration and subsequent escape of particles, in that case, will
+depend on the evolution of the confinement region within the high-
+density, turbulent medium of the MCs, details of which is beyond the
+scope of the simple model discussed in this letter. Furthermore, as
+the SNR enters its radiative phase at t𝑟𝑎𝑑 ∼ 4 × 104 years (Blondin
+et al. 1998), the particle acceleration becomes ineffective as the small
+shock velocity at that age, as obtained from equation 6 (< 1.1 × 107
+cm/s), prevents full ionization of the pre-shock gas (Shull & McKee
+1979). So no significant contribution to the total gamma-ray SED is
+expected in the radiative phase of the SNR as well.
+Since hadronic component primarily dominates in the VHE-UHE
+gamma-ray range, neutrinos can be produced from the hadronic in-
+teraction as well. This neutrino flux can be a smoking gun evidence
+for the dominant hadronic interaction. We have calculated the neu-
+trino flux resulting from the hadronic interaction discussed above,
+and found that the corresponding neutrino flux is too low to be de-
+tected by current generation neutrino telescope such as ICECUBE.
+Furthermore, we have found that the model neutrino flux does not
+exceed the 5𝜎 discovery potential after 10 years of observation by
+next generation neutrino observatory ICECUBE-Gen2 for two de-
+clinations, 𝛿 = 0◦ and 30◦ (Aartsen et al. 2021), which indicates
+that it is unlikely to confirm the hadronic nature of UHE gamma-ray
+emission through neutrino observations, even in the near future, for
+this source.
+In conclusion, in this letter, we have shown that by essentially
+tuning the 𝛼 index, the emission of the LHAASO source can be
+explained. We note that we do not intend to “fit” the MWL SED,
+as the SED, in various energy ranges (VHE, X-ray, radio), is poorly
+constrained and in need of further observations. In this work, we
+have only applied a simple phenomenological model, while also
+minimizing the free parameters, which naturally explains the spec-
+tral features and spatial morphology of LHAASO J2108+5157. Fu-
+ture observations can confirm the viability of this model to ex-
+plain LHAASO J2108+5157, or other unidentified UHE gamma-ray
+source LHAASO 0341+5258, and sources detected in future as well,
+which show similar nature and emission signatures. If confirmed,
+then it can be posited that SNRs as a source class, similar to PWNe,
+can likely be a strong candidate for being the Galactic PeVatrons.
+ACKNOWLEDGEMENTS
+I thank the anonymous reviewer for helpful comments and construc-
+tive criticism. I thank Nayantara Gupta for encouragement.
+DATA AVAILABILITY
+The simulated data underlying this paper will be shared on reasonable
+request to the corresponding author.
+REFERENCES
+Aartsen M. G., et al., 2021, Journal of Physics G Nuclear Physics, 48, 060501
+Abdalla H., et al., 2018, A&A, 612, A1
+Abdollahi S., et al., 2020, ApJS, 247, 33
+Abe S., et al., 2022, arXiv e-prints, p. arXiv:2210.00775
+Albert A., et al., 2021, ApJ, 911, L27
+Baring M. G., Ellison D. C., Reynolds S. P., Grenier I. A., Goret P., 1999,
+ApJ, 513, 311
+Blondin J. M., Wright E. B., Borkowski K. J., Reynolds S. P., 1998, ApJ, 500,
+342
+Blumenthal G. R., Gould R. J., 1970, Reviews of Modern Physics, 42, 237
+Cao Z., 2010, Chinese Physics C, 34, 249
+Cao Z., et al., 2021a, Nature, 594, 33
+Cao Z., et al., 2021b, ApJ, 917, L4
+MNRAS 000, 1–6 (2022)
+
+6
+A. De Sarkar
+Cao Z., et al., 2021c, ApJ, 919, L22
+De Sarkar A., Gupta N., 2022, ApJ, 934, 118
+De Sarkar A., Biswas S., Gupta N., 2021, Journal of High Energy Astro-
+physics, 29, 1
+De Sarkar A., Zhang W., Martín J., Torres D. F., Li J., Hou X., 2022, A&A,
+668, A23
+Dogiel V. A., et al., 2015, ApJ, 809, 48
+Fujita Y., Ohira Y., Tanaka S. J., Takahara F., 2009, ApJ, 707, L179
+Gabici S., Aharonian F. A., Blasi P., 2007, Ap&SS, 309, 365
+Gabici S., Aharonian F. A., Casanova S., 2009, MNRAS, 396, 1629
+Gaensler B. M., Slane P. O., 2006, ARA&A, 44, 17
+Ge C., Liu R.-Y., Niu S., Chen Y., Wang X.-Y., 2021, The Innovation, 2,
+100118
+Ghisellini G., Guilbert P. W., Svensson R., 1988, ApJ, 334, L5
+Hahn J., 2016, PoS, ICRC2015, 917
+Kafexhiu E., Aharonian F., Taylor A. M., Vila G. S., 2014, Phys. Rev. D, 90,
+123014
+Kar A., Gupta N., 2022, ApJ, 926, 110
+Liang X.-H., Li C.-M., Wu Q.-Z., Pan J.-S., Liu R.-Y., 2022, Universe, 8, 547
+Makino K., Fujita Y., Nobukawa K. K., Matsumoto H., Ohira Y., 2019, PASJ,
+71, 78
+Miville-Deschênes M.-A., Murray N., Lee E. J., 2017, ApJ, 834, 57
+Ohira Y., Murase K., Yamazaki R., 2010, A&A, 513, A17
+Ohira Y., Murase K., Yamazaki R., 2011, MNRAS, 410, 1577
+Ohira Y., Yamazaki R., Kawanaka N., Ioka K., 2012, MNRAS, 427, 91
+Popescu C. C., Yang R., Tuffs R. J., Natale G., Rushton M., Aharonian F.,
+2017, MNRAS, 470, 2539
+Shull J. M., McKee C. F., 1979, ApJ, 227, 131
+Wentzel D. G., 1974, ARA&A, 12, 71
+Xu S., Yan H., Lazarian A., 2016, ApJ, 826, 166
+Yamazaki R., Kohri K., Bamba A., Yoshida T., Tsuribe T., Takahara F., 2006,
+MNRAS, 371, 1975
+Yuan Q., Liu S., Bi X., 2012, ApJ, 761, 133
+This paper has been typeset from a TEX/LATEX file prepared by the author.
+MNRAS 000, 1–6 (2022)
+

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+page_content='MNRAS 000, 1–6 (2022)  Preprint 1 February 2023  Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='0  Supernova connection of unidentified ultra high energy gamma-ray source  LHAASO J2108+5157  Agnibha De Sarkar,1★  1Astronomy & Astrophysics group, Raman Research Institute  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Raman Avenue, 5th Cross Road, Sadashivanagar, Bengaluru 560080, Karnataka, India  Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' in original form ZZZ  ABSTRACT  We present a simple phenomenological model of hadronic interaction between protons accelerated in an old supernova remnant  (SNR) and cold protons situated within the associated molecular clouds (MCs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' The accelerated protons from the old SNR  escaped the SNR shock front, and got injected into the MCs at an earlier time, producing ultra high energy gamma-rays and  neutrinos through inelastic proton-proton interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' We also take into account the acceleration and subsequent escape of  electrons from the SNR shock front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' The escaped electrons produce gamma-rays through various radiative cooling mechanisms,  after getting injected into the MCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' We use the model discussed in this letter to explain the multiwavelength (MWL) spectral  energy distribution (SED) of unidentified Galactic ultra high energy gamma-ray source LHAASO J2108+5157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' We also discuss  the feasibility of applying this model in other cases as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Future observations can test the viability of the model discussed in  this letter, which will in turn confirm that the SNRs can, in fact, accelerate particles up to PeV energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  Key words: radiation mechanisms: non-thermal – ISM: individual objects: LHAASO J2108+5157 – gamma-rays: ISM – ISM:  supernova remnants  1 INTRODUCTION  Observations by the Large High Altitude Air Shower Observatory  (LHAASO), located in China, have opened a new era of gamma-  ray astrophysics (Cao 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Since it has become operational on  2020 April, LHAASO has detected more than a dozen ultra high  energy (UHE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' E𝛾 ≥ 100 TeV) gamma-ray sources, most of which are  unidentified (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' The detection of these UHE gamma-  ray sources indicates the presence of cosmic ray (CR) accelerators in  our Milky Way Galaxy, which can accelerate particles up to PeV (=  1015 eV) energies, more commonly known as “PeVatrons”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Several  classes of Galactic sources such as supernova remnants (SNRs), pul-  sar wind nebulae (PWNe), young stellar clusters have been posited to  be potential PeVatron candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Although it is still an open question  as to what class of source is responsible for accelerating particles up  to PeV energies, most of the UHE gamma-ray sources detected by  LHAASO, along with their high energy (HE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' E𝛾 < 100 GeV) and  very high energy (VHE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 100 GeV ≤ E𝛾 < 100 TeV) gamma-ray  counterparts, have been associated with PWNe in previous studies,  due to their close proximity with an energetic pulsar, and their typ-  ically extended spatial morphology (Abdalla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' This idea  gained steam after it was confirmed that Crab pulsar wind nebula  is indeed a PeVatron source (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' However, in spite  of the notion that energetic pulsars with high spin-down luminosity  ( �𝐸 > 1036 erg s−1) coinciding or remaining in a very close spatial  proximity of UHE gamma-ray sources, may be a universal feature  ★ E-mail: agnibha@rri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='in  (Albert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2021), further theoretical analyses of these LHAASO  detected UHE gamma-ray sources seem to tell a different story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  Recent studies have modeled a few of the significantly detected  LHAASO sources in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' For example, De Sarkar & Gupta (2022)  found that the UHE gamma-ray emission observed from LHAASO  J1908+0621 is most likely hadronic in origin, emanated from the  interaction between SNR G40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='5-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='5 and the associated MCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' On the  other hand, in De Sarkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' (2022), another significantly detected  source, LHAASO J2226+6057, was extensively modeled assuming  that the UHE gamma-ray emission is coming from the PWN associ-  ated with PSR J2229+6114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' As caveats of the model, it was found  that the PWN interpretation of LHAASO J2226+6057 leads to a very  high radius of PWN, as well as a very small value of magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  Naturally, these results are in contrast with the observational results  (Ge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2022), thus indicating that the PWN may  not be the contributing source to power the UHE gamma-ray source  detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' This indicates interaction between SNRs and associated  MCs may be the primary reason behind particle acceleration to PeV  energies in Galactic sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  With this factors in mind, we focus on the emission of the recent  LHAASO detected unidentified UHE gamma-ray sources: LHAASO  J2108+5157 (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2021c) and LHAASO J0341+5258 (Cao  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Both of these sources were found to be associated with  MCs, but no apparent association with energetic pulsars or SNRs  were established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Scenarios including leptonic emission from TeV  halo (Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2022), injection of particles from past explosions  (Kar & Gupta 2022), hadronic interaction between SNR and MCs  (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2021c) were discussed in previous literatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' But most of  these models do not explain the HE-VHE-UHE gamma-ray spectral  © 2022 The Authors  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='13451v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='HE]  31 Jan 2023  2  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' De Sarkar  features entirely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Moreover, recent reveal of VHE gamma-ray upper  limits observed by the Large-Sized Telescope - Cherenkov Telescope  Array (LST-CTA) (Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2022) has overruled some of these  models for the case of LHAASO J2108+5157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' The absence of a  powerful pulsar or supernova remnant adds to the mystery as well,  leaving one asking what is the possible emission mechanism at play  in case of these unidentified UHE gamma-ray sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  To that end, in this letter, we discuss and apply a phenomeno-  logical model, in which accelerated particles, escaped from an old,  shell-type SNR (now invisible), interact with the associated MCs to  produce the observed HE-VHE-UHE gamma-ray data for the case  of LHAASO J2108+5157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' We also provide the possible age of the  old SNR, and account for the disappearance of the SNR in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  Our simple model is also consistent with the X-ray 2𝜎 upper limits  given by Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' We also discuss the applicability of the  model in other unidentified Galactic UHE gamma-ray source such  as LHAASO J0341+5258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Furthermore, we report that the neutrino  flux produced from the hadronic interaction considered in this model,  will be non-detectable, even by the next generation observatory such  as ICECUBE-Gen2 (Aartsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  2 THE MODEL  In this section, we discuss the essentials of the model used to calculate  the hadronic and leptonic components produced from the interaction  between an old, now invisible SNR and the associated MCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' A more  detailed discussion of the model can be found in De Sarkar & Gupta  (2022), where we developed and applied our model to explain the  peculiar HE-VHE-UHE gamma-ray SED of LHAASO J1908+0621.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  Our simple model assumes that the supernova had exploded at the  center of the cavity of a shell-like structure, which is surrounded by  dense MCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' After this explosion, the SNR shock front expands inside  the shell cavity, and finally hits the surrounding MCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' During the  collision between the shock front and associated MCs, the accelerated  particles get injected into the MCs to produce further emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  After the explosion, the supernova (SN) shock front expands freely  during its free expansion phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' When the amount of swept-up inter-  stellar medium (ISM) material becomes equal to that of the ejected  material at t = t𝑆𝑒𝑑𝑜𝑣, the SN enters the adiabatic Sedov phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Fi-  nally after t = t𝑟𝑎𝑑, the SN enters the radiative phase, in which the  cooling timescales is less than the dynamic timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' During its  evolution, the time dependence of the shock velocity and radius can  be given by the following simple relations (Fujita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Ohira  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' De Sarkar & Gupta 2022),  𝑣𝑠ℎ(𝑡) =  �  𝑣𝑖  (𝑡 < 𝑡𝑆𝑒𝑑𝑜𝑣)  𝑣𝑖(𝑡/𝑡𝑆𝑒𝑑𝑜𝑣)−3/5  (𝑡𝑆𝑒𝑑𝑜𝑣 < 𝑡)  (1)  and,  𝑅𝑠ℎ(𝑡) ∝  �  (𝑡/𝑡𝑆𝑒𝑑𝑜𝑣)  (𝑡 < 𝑡𝑆𝑒𝑑𝑜𝑣)  (𝑡/𝑡𝑆𝑒𝑑𝑜𝑣)2/5  (𝑡𝑆𝑒𝑑𝑜𝑣 < 𝑡)  (2)  We note that for the entirety of this work, we have assumed the  following values: initial velocity of the shock v𝑖 = 109 cm s−1 (Fujita  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2009), radius of the shock and time at the beginning of the Sedov  phase, R𝑆𝑒𝑑𝑜𝑣 and t𝑆𝑒𝑑𝑜𝑣, to be 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='1 pc and 210 yr, respectively  (Ohira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Makino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  The CR protons are accelerated through Diffusive Shock Acceler-  ation (DSA) mechanism when the SN is in the Sedov phase, where  the CR protons accelerate by scattering back and forth across the  shock front, while the shock is expanding towards the surround-  ing MCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' We assume an escape-limited acceleration scenario (Ohira  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2010), in which the CR protons need to escape a geometrical  confinement region around the SN shock front produced by strong  magnetic turbulence, in order to get injected into the MCs and take  part in further interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' The radius of the outermost boundary  of this confinement region (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', the escape boundary) is called the  escape radius, and it can be denoted by,  𝑅𝑒𝑠𝑐(𝑡) = (1 + 𝜅)𝑅𝑠ℎ(𝑡),  (3)  where 𝜅 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='04 (Ohira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Makino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2019), and is  defined by the geometrical confinement condition D𝑠ℎ/v𝑠ℎ ∼ l𝑒𝑠𝑐 =  𝜅R𝑠ℎ, where l𝑒𝑠𝑐 is the distance of the escape boundary from the  shock front and D𝑠ℎ is the diffusion coefficient around the shock  (Ohira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  After the explosion, the escape boundary in front of the shock  front eventually hits the surrounding MCs after traversing a distance  of R𝑀𝐶, the distance of MC surface from the cavity center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' This  essentially means that at the time of collision t𝑐𝑜𝑙𝑙, the escape ra-  dius is equal to the MC surface distance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' R𝑒𝑠𝑐 (t𝑐𝑜𝑙𝑙) = R𝑀𝐶  ≈ R𝑠ℎ (t𝑐𝑜𝑙𝑙), and at the time of collision, the velocity of the shock  is denoted by v𝑠ℎ(t𝑐𝑜𝑙𝑙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' We assume that the particle acceleration  stops at t = t𝑐𝑜𝑙𝑙 (Fujita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Consequently, protons accel-  erated at t ≤ t𝑐𝑜𝑙𝑙 take part in further interactions inside the MCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  Moreover, only the protons with sufficiently high energies will es-  cape the confinement region around the SNR shock front, whereas  the low energy protons will remain confined around the SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' So a  suppression of fluxes in the lower energies, as well as a dominant con-  tribution of fluxes in the highest energies should be expected in this  scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' The confinement condition invoked in this model changes  the spectral shape of the injected proton population by constraining  the minimum energy limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  We estimate the minimum energy limit of the injected proton pop-  ulation by assuming that the escape energy is a decreasing function  of the shock radius (Makino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' This approach is based  on the assumption that the maximum energy of CR protons, E𝑝  𝑚𝑎𝑥  is expected to increase up to knee energy (≈ 1015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='5 eV) until the  beginning of the Sedov phase, and then decrease from that epoch  (Gabici et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Ohira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' The minimum energy re-  quired by protons to escape the confinement region can be given by  the phenomenological relation,  𝐸 𝑝  𝑒𝑠𝑐 = 𝐸 𝑝  𝑚𝑎𝑥  �  𝑅𝑠ℎ  𝑅𝑆𝑒𝑑𝑜𝑣  �−𝛼  ,  (4)  where 𝛼 signifies the evolution of the maximum energy during the  Sedov phase (Makino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' We treat 𝛼 as a free parameter in  this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' After putting R𝑠ℎ ≈ R𝑒𝑠𝑐 = R𝑀𝐶 at the time of collision,  we find the minimum energy required to escape the confinement  zone, which also gives us the minimum energy threshold for the  proton population that gets injected inside the surrounding MCs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=',  E𝑝  𝑒𝑠𝑐 = E𝑝  𝑚𝑖𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Since protons are accelerated by DSA mechanism, we  can expect the CR proton spectrum at the shock front ∝ E−𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Then,  in an escape-limited particle acceleration scenario, the protons with  energies greater than 𝐸 𝑝  𝑒𝑠𝑐 have a spectrum (Ohira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2010),  𝑁 𝑝  𝑒𝑠𝑐(𝐸) ∝ 𝐸−[𝑠+(𝛽/𝛼)],  (5)  where 𝛽 represents a thermal leakage model of CR injection and  is given by 𝛽 = 3(3–s)/2 (Makino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' For a typical value of  s = 2, we get the value of 𝛽 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Note that the spectral shape as  MNRAS 000, 1–6 (2022)  Supernova connection of LHAASO J2108+5157  3  well as the minimum energy of the proton population are calculated  at the time when the escape boundary hits the surrounding MCs at t  = t𝑐𝑜𝑙𝑙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  At t > t𝑐𝑜𝑙𝑙, the shock enters the momentum conserving “snow-  plow” phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' The time evolution of the radius of the shocked shell  R𝑠ℎ𝑒𝑙𝑙 (t) inside the MCs can be found using momentum conserva-  tion equation (Fujita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' De Sarkar & Gupta 2022),  4𝜋  3  �  𝑛𝑀𝐶 (𝑅𝑠ℎ𝑒𝑙𝑙(𝑡)3 − 𝑅𝑠ℎ(𝑡𝑐𝑜𝑙𝑙)3) + 𝑛𝑐𝑎𝑣 𝑅𝑠ℎ(𝑡𝑐𝑜𝑙𝑙)3�  �𝑅𝑠ℎ𝑒𝑙𝑙(𝑡)  = 4𝜋  3 𝑛𝑐𝑎𝑣 𝑅𝑠ℎ(𝑡𝑐𝑜𝑙𝑙)3𝑣𝑠ℎ(𝑡𝑐𝑜𝑙𝑙),  (6)  with R𝑠ℎ𝑒𝑙𝑙 = R𝑀𝐶 at t = t𝑐𝑜𝑙𝑙, n𝑀𝐶 is the number density of  the MCs, and n𝑐𝑎𝑣 (≈ 1 cm−3) is the number density inside the  cavity of the shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' We solve equation 6 numerically for t > t𝑐𝑜𝑙𝑙,  to estimate the current age of the SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' We estimate the current  age by considering the fact that the velocity of the shocked shell  at the current age must be similar or even smaller than the internal  gas velocity of the MCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' This approach takes into account the non-  detection of any SNR shell in unidentified UHE gamma-ray sources  discussed above, as the shell of the SNR becomes invisible owing to  the higher internal gas velocity of the MCs as compared to that of the  shocked shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' We consider the above discussed proton population  and total number density of the cold protons inside the surrounding  MCs (n𝑀𝐶) to calculate total gamma-ray flux produced through  hadronic interaction (Kafexhiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  Similar to protons, electrons can also get accelerated in the SNR  shock front and subsequently escape the confinement region to get  injected in the associated MCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Moreover, electrons also lose energy  through radiative cooling very efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Hence, the injected electron  population was considered to be escape-limited, as well as loss-  limited (Yamazaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' We consider the spectral index of  the escaped electron population to be same as that of protons (Ohira  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' De Sarkar & Gupta 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' To take into accout loss-limited  nature of injected electron population, we consider a power law with  exponential cutoff as the spectral shape of the escaped electrons,  𝑁𝑒  𝑒𝑠𝑐(𝐸) ∝ 𝐸−[𝑠+(𝛽/𝛼)]𝑒𝑥𝑝(−𝐸/𝐸𝑒  𝑚𝑎𝑥),  (7)  where, maximum energy of the electron population has been de-  termined by synchrotron cooling (Yamazaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Fujita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  2009),  𝐸𝑒  𝑚𝑎𝑥 = 14ℎ−1/2  �  𝑣𝑠ℎ  108 cm/s  � �  𝐵  10 𝜇G  �−1/2  TeV,  (8)  where, v𝑠ℎ is the velocity of the shock front and B is the down-  stream magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' The parameter h (= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='05𝑟 ( 𝑓 +𝑟𝑔)  𝑟−1  , where r is  the density compression ratio, f and g are functions of shock angle  and gyro-factors) is used as a factor to calculate the acceleration time  scale of DSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' We take h ∼ 1, considering the SNR in Sedov phase  and neglecting non-linear effects, similar to Yamazaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  We consider v𝑠ℎ = v𝑠ℎ (t𝑐𝑜𝑙𝑙) since we calculate the maximum en-  ergy of the lepton population at the collision time and B = B𝑀𝐶,  the magnetic field inside the MCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' The minimum energy of the elec-  tron population was considered to be E𝑒  𝑚𝑖𝑛 ≈ 500 MeV (De Sarkar  & Gupta 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Furthermore, we consider bremsstrahlung, Inverse-  Compton (IC) and synchrotron cooling (Blumenthal & Gould 1970;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  Ghisellini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Baring et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 1999) of the injected lepton  population to calculate the gamma-ray flux produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' For IC inter-  action, we consider interstellar radiation field (ISRF) from Popescu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' (2017) at the source position, and the Cosmic Microwave Back-  ground (temperature T𝐶𝑀 𝐵 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='7 K, energy density U𝐶𝑀 𝐵 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='25  eV cm−3) contribution as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' The number density was considered  to be same as that of the MCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  Finally we note that in this particular model, we have neglected  the effect of diffusion of particles inside the MCs and assumed that  the CR particles, both protons and electrons, lose energy through  rapid cooling before escaping the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' This assumption can be  realized by considering the idea that inside MCs, the diffusion is  considerably suppressed (D ≈ 1025−26 cm2 s−1) as compared to that  observed in the ISM (D ≈ 1028 cm2 s−1) (Gabici et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2007, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  Fujita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' De Sarkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Generation of plasma waves  by CR streaming can be the reason behind the slow diffusion inside  the MCs (Wentzel 1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' On the other hand, if the trapping of CR  particles occurs due to some particular orientation of the magnetic  field inside the MCs, then also the escape of the particles from the  MCs will not be effective and can be neglected (Makino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  Consequently, we have considered a steady-state proton and electron  population to explain the SED of LHAASO J2108+5157, details of  which are given in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  3 APPLICATION OF THE MODEL: LHAASO J2108+5157  LHAASO J2108+5157 is an UHE gamma-ray source detected by  LHAASO at R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' = 317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='22◦ ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='07◦  𝑠𝑡𝑎𝑡 and decl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' = 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='95◦ ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='05◦  𝑠𝑡𝑎𝑡  (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2021c) with a significance of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='4𝜎 above 100 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' The  source is reported to have a 95% confidence level extension upper  limit of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='26◦ with a 2D symmetrical Gaussian template, and its  spectrum above 25 TeV can be well described by a power law with a  photon index of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='18 (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2021c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Although no X-ray  counterpart within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='26◦ radius of the source was found, a spatially  extended, HE counterpart 4FGL J2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='0+5155e (extension ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='48◦)  (Abdollahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2020) was observed to be situated at an angular  distance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='13◦ (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2021c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' A new hard spectrum GeV  source was also found at l = 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='35◦ and b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='56◦ by Fermi-LAT  data analysis (Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2022), but its large angular separation (∼  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='27◦) from the LHAASO source indicates that this new source can  hardly be a counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Although no VHE component within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='5◦  radius was confirmed previously, recent observations by LST-CTA  has hinted towards an existence of a source with 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='67𝜎 detection  significance in the energy range of 3 - 100 TeV (Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  Future observations may confirm an existence of a VHE counterpart  with hard spectral index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' The UHE source is located near the center  of a GMC labeled [MML2017]4607 (Miville-Deschênes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2017),  which has an average angular radius and mass of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='236◦ and 8469  M⊙, respectively, and is situated at a distance of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='28 kpc from  Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' The average number density of the GMC was estimated to  be n𝑀𝐶 ≈ 30 cm−3 (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2021c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' The presence of the GMC,  spatially coincident with the UHE gamma-ray source points towards  the hadronic origin, but leptonic origin can not be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' The  absence of any energetic pulsar, its wind nebula or SNR warrants a  cautious approach in unveiling the true nature of emission regarding  this UHE source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  Two young open stellar clusters Kronberger 80 and Kronberger 82  are in the vicinity of the LHAASO source (with angular distances of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='62◦ and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='45◦, respectively) (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2021c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' But large angular  separation between these clusters and LHAASO source centroid, as  well as absence of proper distance estimation hint that the contri-  bution of these clusters are unlikely (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2021c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  MNRAS 000, 1–6 (2022)  4  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' De Sarkar  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' (2021c) suggested that UHE gamma-ray emission  is due to an interaction of escaping CRs with MCs, whereas the GeV  counterpart maybe due to an old SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' However, Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' (2022)  pointed out that photon index of GeV counterpart spectrum is too  soft compared to the observations of old SNRs interacting with MCs  (Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2012), and to produce UHE gamma-ray spectrum, the  required spectral index of the proton population has to be very hard  as compared to the standard DSA theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Instead, Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' (2022)  proposed an alternate leptonic scenario, in which UHE gamma-ray  emission is due to TeV halo emission, and the GeV counterpart is due  to a tentative, previously undetected pulsar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' But a very low associated  magnetic field (even lower than the average Galactic magnetic field),  and non-detection of a pulsar make the TeV halo interpretation ques-  tionable, and open the source up for further exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' To that end,  we apply the model discussed in Section 2 to explain the gamma-ray  data from HE to UHE energy range, while being consistent with the  X-ray 2𝜎 upper limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' We note that these 2𝜎 X-ray upper limits  correspond to a uniform, circular source with a radius of 6′ centered  on the position of the LHAASO source (Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' We explain  the VHE-UHE gamma-ray data with hadronic component produced  from the interaction between protons, accelerated and escaped at an  early time from a now old SNR shock front, with protons inside  the surrounding MCs, whereas the HE gamma-ray data is explained  using bremsstrahlung cooling of accelerated and escaped electrons  inside the medium of the MCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Our model also shows that the main  contribution in X-ray range comes from the synchrotron cooling of  the same accelerated and escaped electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  In this work, we have considered the free parameter 𝛼 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='875, and  then let the total energy budgets of proton and electron populations  vary to explain the MWL SED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Considering the value of 𝛼, and the  values of s and 𝛽 discussed in Section 2, we get the spectral indices  of the escaped electron and proton populations as p = [s + (𝛽/𝛼)]  = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' The distance of the source was taken to be d ∼ 3 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' The  model spectrum components, as well as the considered MWL SED  are shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Also, we calculate the time evolution of SNR  shocked shell inside the associated MCs using equation 6, and find  that the SNR, with a final radius of ∼ 30 pc, has to be ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='4 × 105  years old, for the shock velocity to be lower than the internal gas  velocity of MC [MML2017]4607 (∼ 13 km s−1) (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2021c),  and the SNR shell to disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' The time evolution of the shocked  shell is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Finally, the model parameters required to  explain the gamma-ray data are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' We have used open  source code GAMERA (Hahn 2016) to calculate the model spectrum  of different components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  4 DISCUSSION AND CONCLUSION  In this letter, we have discussed and applied a simple, analytical and  phenomenological model to explain the HE-VHE-UHE gamma-ray  data observed from the direction of LHAASO J2108+5157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' By only  adjusting the index 𝛼, not only we show that the model components  are consistent with gamma-ray and X-ray observations, the results  also naturally explain the observed morphology of the source re-  gion, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', the disappearance of the SNR at current age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' As expected,  the SNR was found be old (> 105 years).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' This also explains why  no pulsar has been seen in the source region, as the pulsar is ex-  pected to leave the source region due to its natal kick velocity (∼  400-500 km s−1) (Gaensler & Slane 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Similar nature and emis-  sion were also found in another UHE gamma-ray source, LHAASO  J1908+0621, details of which were explained by this model in De  Sarkar & Gupta (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' The fact that the emission of multiple UHE  10  12  10  10  10  8  10  6  10  4  10  2  100  102  Energy (TeV)  10  16  10  15  10  14  10  13  10  12  10  11  E2 J(E)[erg cm  2 s  1]  pp  synchrotron  bremsstrahlung  inverse-compton  Fermi-LAT (Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2022)  Fermi-LAT (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2021)  LHAASO  LST-CTA  XMM-Newton  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' MWL SED of LHAASO J2108+5157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Gamma-ray data points and  upper limits obtained from different observatories such as Fermi-LAT (red  (Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2022), purple (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2021c)), LHAASO (blue (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  2021c)), and LST-CTA (green (Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2022)) are shown in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' The  XMM-Newton X-ray 2𝜎 upper limits (Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2022) are given in teal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' The  model p-p interaction (solid line), bremsstrahlung (dashed), IC (dotted), and  synchrotron (dot-dashed) components are also shown in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  103  104  105  106  Time (years)  16  18  20  22  24  26  28  30  32  Shock radius (pc)  LHAASO J2108+5157  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Time evolution of the shocked shell associated with the old SNR,  inside the surrounding MCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  gamma-ray sources were explained by the same model hints towards  its validity in a larger context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Interestingly, another unidentified  UHE gamma-ray source, LHAASO J0341+5258, also shows similar  characteristics shown by LHAASO J2108+5157 (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  It is very likely that this model is applicable in that case as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  However, in that case, the VHE counterpart has not been properly  constrained, and the High Altitude Water Cherenkov (HAWC) upper  limit provided in Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' (2021b) corresponds to only a 2𝜎 detec-  tion significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Further observations by CTA and detailed analysis  by Fermi-LAT will be necessary to properly constrain the emission  of LHAASO J0341+5258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  From Figure 1, we can see that the hadronic component adequately  explain the VHE-UHE gamma-ray data, whereas the bremsstrahlung  component, originated from the cooling of the electron population,  explains the gamma-ray data in the HE range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' The bremsstrahlung  MNRAS 000, 1–6 (2022)  Supernova connection of LHAASO J2108+5157  5  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Parameters Used in The Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  Definition  Parameter  Value  SNR/MC structure and evolution:  Initial shock velocity  v𝑖 (cm/s)  109  Time at the start of Sedov phase  t𝑆𝑒𝑑𝑜𝑣 (years)  210  Shock radius at the start of Sedov phase R𝑆𝑒𝑑𝑜𝑣 (pc)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='1  Time of collision  t𝑐𝑜𝑙𝑙 (years)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='83 × 103  Shock radius at time of collision  R𝑠ℎ (t𝑐𝑜𝑙𝑙) (pc)  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='77 (= R𝑀𝐶)  Shock velocity at time of collision  v𝑠ℎ (t𝑐𝑜𝑙𝑙) (cm/s) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='75 × 108  Current age of SNR  t𝑎𝑔𝑒 (years)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='4 × 105  Final radius of shock  R𝑠ℎ (t𝑎𝑔𝑒) (pc)  30  Final velocity of shock  v𝑠ℎ (t𝑎𝑔𝑒) (cm/s) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='2 × 106  Distance  d (kpc)  3  MC number density  n𝑀𝐶 (cm−3)  30  MC magnetic field  B𝑀𝐶 (𝜇G)  25  Cavity number density  n𝑐𝑎𝑣 (cm−3)  1  Hadronic component:  Minimum energy  E𝑝  𝑚𝑖𝑛 (TeV)  63  Maximum energy  E𝑝  𝑚𝑎𝑥 (TeV)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='1 × 103  Spectral index  p  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='8  Energy budget  W𝑝 (erg)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='6 × 1047  Leptonic component:  Minimum energy  E𝑒  𝑚𝑖𝑛 (TeV)  5 × 10−4  Maximum energy  E𝑒𝑚𝑎𝑥 (TeV)  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='5  Spectral index  p  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='8  Energy budget  W𝑒 (erg)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='6 × 1047  component is expected to dominate the IC component, as the in-  teraction is taking place inside MCs with a high number density  of cold protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Moreover, the synchrotron component does not  violate the X-ray 2𝜎 upper limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' We note that no proper radio  counterpart has been associated with the LHAASO J2108+5157 yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  An extended radio source associated with nearby star-forming re-  gion (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2021c), as well as point-like radio source NVSS  210803+515255 or WENSS B2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='4+5140 (Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2022) were  found within 95% extension upper limit of LHAASO J2108+5157  and 4FGL J2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='0+5155e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Since no proper association was estab-  lished between these sources and the gamma-ray source, we refrain  from including their radio data in this study to further constrain the  model, and we follow the MWL SED discussed in (Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2022)  to ascertain the feasibility of the model discussed in this letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  As discussed earlier, we have neglected the effect of particle diffu-  sion in this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' We note that such an assumption may likely lead  to an overestimation, and the aspect of suppressed diffusion inside  the MCs is highly uncertain (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Dogiel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' In-  troducing an energy-independent diffusion coefficient, as discussed  in Dogiel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' (2015), will lead to higher energy budgets required  by the electron and proton populations to explain the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' The sup-  pressed diffusion coefficient introduced by Gabici et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' (2009) has  similar energy dependence as to that observed in ISM, but the exact  energy dependence of diffusion coefficient inside clouds is not well  constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' So, to avoid further complications, we have neglected  the effect of diffusion in this model, similar to Fujita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  Makino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' (2019), and assumed that the injected particles quickly  cool down before escaping the MC medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  We further note that we do not consider the contribution of accel-  erated and escaped particles, when the shock front is within the MC  medium, in calculating the total gamma-ray SED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Even if the SNR  is still in the Sedov phase when the shock is within the MCs, the  corresponding contribution was found to be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Moreover,  the acceleration and subsequent escape of particles, in that case, will  depend on the evolution of the confinement region within the high-  density, turbulent medium of the MCs, details of which is beyond the  scope of the simple model discussed in this letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Furthermore, as  the SNR enters its radiative phase at t𝑟𝑎𝑑 ∼ 4 × 104 years (Blondin  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 1998), the particle acceleration becomes ineffective as the small  shock velocity at that age, as obtained from equation 6 (< 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='1 × 107  cm/s), prevents full ionization of the pre-shock gas (Shull & McKee  1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' So no significant contribution to the total gamma-ray SED is  expected in the radiative phase of the SNR as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  Since hadronic component primarily dominates in the VHE-UHE  gamma-ray range, neutrinos can be produced from the hadronic in-  teraction as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' This neutrino flux can be a smoking gun evidence  for the dominant hadronic interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' We have calculated the neu-  trino flux resulting from the hadronic interaction discussed above,  and found that the corresponding neutrino flux is too low to be de-  tected by current generation neutrino telescope such as ICECUBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  Furthermore, we have found that the model neutrino flux does not  exceed the 5𝜎 discovery potential after 10 years of observation by  next generation neutrino observatory ICECUBE-Gen2 for two de-  clinations, 𝛿 = 0◦ and 30◦ (Aartsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' 2021), which indicates  that it is unlikely to confirm the hadronic nature of UHE gamma-ray  emission through neutrino observations, even in the near future, for  this source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  In conclusion, in this letter, we have shown that by essentially  tuning the 𝛼 index, the emission of the LHAASO source can be  explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' We note that we do not intend to “fit” the MWL SED,  as the SED, in various energy ranges (VHE, X-ray, radio), is poorly  constrained and in need of further observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' In this work, we  have only applied a simple phenomenological model, while also  minimizing the free parameters, which naturally explains the spec-  tral features and spatial morphology of LHAASO J2108+5157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Fu-  ture observations can confirm the viability of this model to ex-  plain LHAASO J2108+5157, or other unidentified UHE gamma-ray  source LHAASO 0341+5258, and sources detected in future as well,  which show similar nature and emission signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' If confirmed,  then it can be posited that SNRs as a source class, similar to PWNe,  can likely be a strong candidate for being the Galactic PeVatrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  I thank the anonymous reviewer for helpful comments and construc-  tive criticism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' I thank Nayantara Gupta for encouragement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  DATA AVAILABILITY  The simulated data underlying this paper will be shared on reasonable  request to the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  REFERENCES  Aartsen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 2021, Journal of Physics G Nuclear Physics, 48, 060501  Abdalla H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 2018, A&A, 612, A1  Abdollahi S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 2020, ApJS, 247, 33  Abe S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='00775  Albert A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 2021, ApJ, 911, L27  Baring M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Ellison D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Reynolds S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Grenier I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Goret P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 1999,  ApJ, 513, 311  Blondin J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Wright E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Borkowski K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Reynolds S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 1998, ApJ, 500,  342  Blumenthal G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Gould R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 1970, Reviews of Modern Physics, 42, 237  Cao Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 2010, Chinese Physics C, 34, 249  Cao Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 2021a, Nature, 594, 33  Cao Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 2021b, ApJ, 917, L4  MNRAS 000, 1–6 (2022)  6  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' De Sarkar  Cao Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 2021c, ApJ, 919, L22  De Sarkar A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Gupta N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 2022, ApJ, 934, 118  De Sarkar A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Biswas S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Gupta N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 2021, Journal of High Energy Astro-  physics, 29, 1  De Sarkar A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Zhang W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Martín J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Torres D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Li J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Hou X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 2022, A&A,  668, A23  Dogiel V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 2015, ApJ, 809, 48  Fujita Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Ohira Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Tanaka S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Takahara F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 2009, ApJ, 707, L179  Gabici S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Aharonian F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Blasi P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 2007, Ap&SS, 309, 365  Gabici S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Aharonian F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Casanova S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 2009, MNRAS, 396, 1629  Gaensler B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Slane P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 2006, ARA&A, 44, 17  Ge C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Liu R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Niu S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Chen Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Wang X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 2021, The Innovation, 2,  100118  Ghisellini G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Guilbert P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Svensson R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 1988, ApJ, 334, L5  Hahn J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 2016, PoS, ICRC2015, 917  Kafexhiu E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Aharonian F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Taylor A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Vila G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 2014, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' D, 90,  123014  Kar A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Gupta N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 2022, ApJ, 926, 110  Liang X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Li C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Wu Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Pan J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Liu R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 2022, Universe, 8, 547  Makino K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Fujita Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Nobukawa K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Matsumoto H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Ohira Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 2019, PASJ,  71, 78  Miville-Deschênes M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Murray N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Lee E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 2017, ApJ, 834, 57  Ohira Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Murase K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Yamazaki R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 2010, A&A, 513, A17  Ohira Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Murase K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Yamazaki R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 2011, MNRAS, 410, 1577  Ohira Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Yamazaki R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Kawanaka N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Ioka K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 2012, MNRAS, 427, 91  Popescu C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Yang R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Tuffs R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Natale G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Rushton M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Aharonian F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=',  2017, MNRAS, 470, 2539  Shull J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', McKee C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 1979, ApJ, 227, 131  Wentzel D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 1974, ARA&A, 12, 71  Xu S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Yan H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 2016, ApJ, 826, 166  Yamazaki R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Kohri K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Bamba A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Yoshida T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Tsuribe T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Takahara F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 2006,  MNRAS, 371, 1975  Yuan Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Liu S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', Bi X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content=', 2012, ApJ, 761, 133  This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}
+page_content='  MNRAS 000, 1–6 (2022)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'}

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